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On mappings with φ-contractive iterate at a point on generalized metric spaces

Ljiljana Gajić1 and Mila Stojaković2*

Author Affiliations

1 Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Novi Sad, Serbia

2 Department of Mathematics, Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

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Fixed Point Theory and Applications 2014, 2014:46  doi:10.1186/1687-1812-2014-46

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2014/1/46


Received:18 July 2013
Accepted:6 February 2014
Published:21 February 2014

© 2014 Gaji¿ and Stojakovi¿; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove a set of fixed point theorems for mappings with φ-contractive iterate at a point in a class of generalized metric spaces. This is a generalization of some well-known results for Guseman and Matkowski types of fixed point results in metric and generalized metric spaces.

MSC: 47H10, 54H25.

Keywords:
φ-contraction; fixed point; mapping with contractive iterate at a point; generalized metric space

1 Introduction and preliminaries

In 1975 Matkowski introduced the following class of mappings.

Definition 1.1[1]

Let T be a mapping on a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M1">View MathML</a>. Then T is called a weak contraction if there exists a function φ from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M2">View MathML</a> to itself satisfying the following:

(i) φ is nondecreasing,

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M3">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4">View MathML</a>,

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M5">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M6">View MathML</a>.

In the same paper he proved the existence and uniqueness of a fixed point for such type of mappings. This result is significant because the concept of weak contraction of Matkowski type is independent of the Meir-Keeler contraction [2], and it was generalized in different directions [3-10]. Matkowski generalized his own result proving a theorem of Segal-Guseman type [11].

Theorem 1.1[4]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M1">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M8">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9">View MathML</a>. Ifφis nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M10">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M11">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4">View MathML</a>, and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13">View MathML</a>there is a positive integer<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M14">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M15">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M16">View MathML</a>

thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M17">View MathML</a>. Moreover, for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M19">View MathML</a>.

The aim of this paper is to show that this result is valid in a more general class of spaces and wide class of functions φ.

In 1963, Gähler introduced 2-metric spaces, but other authors proved that there is no relation between the two distance functions and there is no easy relationship between results obtained in the two settings. Dhage introduced a new concept of the measure of nearness between three or more objects. But the topological structure of so-called D-metric spaces was incorrect. Finally, Mustafa and Sims [12] introduced the correct definition of a generalized metric space as follows.

Definition 1.2[12]

Let X be a nonempty set, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M20">View MathML</a> be a function satisfying the following properties:

(G1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M21">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M22">View MathML</a>;

(G2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M23">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M24">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M25">View MathML</a>;

(G3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M26">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M27">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M28">View MathML</a>;

(G4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M29">View MathML</a> (symmetry in all three variables);

(G5) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M30">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M31">View MathML</a>.

Then the function G is called a generalized metric, abbreviated G-metric, on X, and the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a> is called a G-metric space.

Clearly these properties are satisfied when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M33">View MathML</a> is the perimeter of the triangle with vertices at x, y, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M34">View MathML</a>. Moreover, taking a in the interior of the triangle shows that (G5) is the best possible.

Example 1.1[12]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M1">View MathML</a> be an ordinary metric space, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M1">View MathML</a> defines G-metrics on X by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M37">View MathML</a>

Example 1.2[12]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M38">View MathML</a>. Define G on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M39">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M40">View MathML</a>

and extend G to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M39">View MathML</a> by using the symmetry in the variables. Then it is clear that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a> is a G-metric space.

The following useful properties of a G-metric are readily derived from the axioms.

Proposition 1.1[12]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>be aG-metric space, then for anyx, y, zandafromXit follows that:

1. if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M44">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M22">View MathML</a>,

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M46">View MathML</a>,

3. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M47">View MathML</a>,

4. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M48">View MathML</a>,

5. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M49">View MathML</a>,

6. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M50">View MathML</a>.

Definition 1.3[12]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a> be a G-metric space, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52">View MathML</a> be a sequence of points of X. A point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53">View MathML</a> is said to be the limit of the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M55">View MathML</a>, and one says that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52">View MathML</a> is G-convergent to x.

Proposition 1.2[12]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>be aG-metric space, then for a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M58">View MathML</a>and a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13">View MathML</a>the following are equivalent:

1. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52">View MathML</a>isG-convergent tox,

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M61">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M62">View MathML</a>,

3. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M63">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M62">View MathML</a>.

Definition 1.4[12]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a> be a G-metric space, a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52">View MathML</a> is called G-Cauchy if for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M67">View MathML</a>, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M68">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M69">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M70">View MathML</a>, that is, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M71">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M72">View MathML</a>.

Proposition 1.3[12]

In aG-metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>, the following are equivalent:

1. the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52">View MathML</a>isG-Cauchy,

2. for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M75">View MathML</a>, there exists an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M76">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M77">View MathML</a>, for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M78">View MathML</a>.

A G-metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a> is G-complete (or complete G-metric), if every G-Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a> is G-convergent in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>.

Proposition 1.4[12]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>be aG-metric space, then the function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M33">View MathML</a>is jointly continuous in all three of its variables.

Definition 1.5<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a> is symmetric G-metric space if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M85">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M6">View MathML</a>.

Fixed point theorems in symmetric G-metric space are mostly consequences of the related fixed point results in metric spaces. In this paper we discuss the non-symmetric case.

In [13] it was shown that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a> is a G-metric space, putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M88">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M89">View MathML</a> is a quasi-metric space (δ is not symmetric). It is well known that any quasi-metric induces different metrics and mostly used are

(μ) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M90">View MathML</a>,

(ρ) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M91">View MathML</a>.

The following result is an immediate consequence of above definitions and relations.

Theorem 1.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>be aG-metric space and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M93">View MathML</a>. Then

1. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M94">View MathML</a>isG-convergent to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13">View MathML</a>if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52">View MathML</a>is convergent toxin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M97">View MathML</a>;

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M94">View MathML</a>isG-Cauchy if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M52">View MathML</a>is Cauchy in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M97">View MathML</a>;

3. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>isG-complete if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M97">View MathML</a>is complete.

Recently, Samet et al.[14] and Jleli-Samet [13] observed that some fixed point theorems in the context of a G-metric space can be proved (by simple transformation) using related existing results in the setting of a (quasi-) metric space. Namely, if the contraction condition of the fixed point theorem on G-metric space can be reduced to two variables, then one can construct an equivalent fixed point theorem in setting of usual metric space. This idea is not completely new, but it was not successfully used before (see [15]). Very recently, Karapinar and Agarwal suggest new contraction conditions in G-metric space in a way that the techniques in [13,14] are not applicable. In this approach [16], contraction conditions cannot be expressed in two variables. So, in some cases, as is noticed even in Jleli-Samet’s paper [13], when the contraction condition is of nonlinear type, this strategy cannot be always successfully used. This is exactly the case in our paper.

For more fixed point results for mappings defined in G-metric spaces, we refer the reader to [3,6,9,10,15-36].

2 Main result

A generalization of the contraction principle can be obtained using a different type of a nondecreasing function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9">View MathML</a>. The most usual additional properties imposed on φ are given using a combination of the next seven conditions:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M104">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M105">View MathML</a>,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M107">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4">View MathML</a>,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M110">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4">View MathML</a>,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M113">View MathML</a> is a sequence such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M114">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M115">View MathML</a>,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116">View MathML</a>) for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M117">View MathML</a> there exists a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M118">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M119">View MathML</a>,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M121">View MathML</a>,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M123">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4">View MathML</a>.

Some of the noted properties of φ are equivalent, some of them imply others, some of them are incompatible. The next lemma discusses some of the relations between properties (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M104">View MathML</a>)-(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>), especially those which are used in this paper to define a generalized contraction.

Lemma 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9">View MathML</a>be a nondecreasing function. Then

(i) (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) ⇔ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>),

(ii) ifφis right continuous, then (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>) ⇔ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) ⇔ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>),

(iii) (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M135">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M136">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M104">View MathML</a>), where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M138">View MathML</a>,

(iv) (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116">View MathML</a>) ⇔ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>),

(v) (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>) ⇏ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>), (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>) ⇏ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>),

(vi) (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116">View MathML</a>) + (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) ⇏ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>) and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>) + (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) ⇏ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>),

(vii) (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>) ⇏ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116">View MathML</a>) and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>) ⇏ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>).

Proof (i) (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>): If for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M158">View MathML</a>, then, knowing that φ is nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M159">View MathML</a>. It means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M160">View MathML</a>, which contradicts (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>).

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>): Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M113">View MathML</a> be any sequence such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M114">View MathML</a>. Using the implication (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M168">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M169">View MathML</a>.

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>): We assume that for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M173">View MathML</a>. Since (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>), the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M176">View MathML</a> satisfies condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M177">View MathML</a>, but it converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M178">View MathML</a>. That contradicts (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>).

(ii) It is enough to prove that (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>): We assume that for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M173">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M184">View MathML</a> is a nonincreasing sequence, by the right continuity of φ, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M185">View MathML</a>, i.e.<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M186">View MathML</a>, which contradicts (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>).

(iii), (iv) are obvious, so the proof is omitted.

(v) The function

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M188">View MathML</a>

satisfies (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>), but not (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>), nor (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>), since for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M192">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M193">View MathML</a>.

(vi) The function

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M194">View MathML</a>

satisfies (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116">View MathML</a>), (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>), and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>), but not (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>).

(vii) The function

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M199">View MathML</a>

satisfies (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>), but not (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116">View MathML</a>), nor (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>). □

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M203">View MathML</a>be a completeG-metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M204">View MathML</a>, where the nondecreasing functionφsatisfies (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M205">View MathML</a>) or (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>) together with (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) or (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>) and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53">View MathML</a>there exists a positive integer<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M14">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M211">View MathML</a>

(1)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M212">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213">View MathML</a>. Moreover, for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M215">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216">View MathML</a>is continuous ata.

Proof Let the nondecreasing function φ satisfy (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>) together with (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) (weak contraction in the sense of Matkowski). Then by Proposition 1.1(3), in a non-symmetric G-metric space we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M219">View MathML</a>

The last inequality together with (1) implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M220">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M6">View MathML</a>. So, one can apply the Matkowski fixed point theorem if the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M222">View MathML</a> satisfies the conditions (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>). Since there exist functions φ which satisfy (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>), but 2φ does not (for example <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M227">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M228">View MathML</a>), the Jleli-Samet technique [13] is not applicable. We are going to prove our theorem using the G-metric G.

We first prove by mathematical induction that, for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53">View MathML</a>, the orbit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M230">View MathML</a> is bounded.

Fix <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13">View MathML</a>, fix the integer s, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M232">View MathML</a> and put

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M233">View MathML</a>

By (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>) there exist c, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M235">View MathML</a>, such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M236">View MathML</a>

The last inequalities imply that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M237">View MathML</a>. Suppose that there exists a positive integer j such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M238">View MathML</a>, but <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M239">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M240">View MathML</a>.

Using (1), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M241">View MathML</a>

i.e.<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M242">View MathML</a>, which contradicts the choice of c. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M243">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M244">View MathML</a> , and consequently the orbit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M230">View MathML</a> is bounded, so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M246">View MathML</a>.

For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M247">View MathML</a>, we define sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M248">View MathML</a> as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M249">View MathML</a>

(2)

We shall prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M248">View MathML</a> is a Cauchy sequence. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M251">View MathML</a>. From (2) we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M252">View MathML</a>

With the notation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M253">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M254">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M255">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M248">View MathML</a> is a Cauchy sequence in a complete G-metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M257">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213">View MathML</a>.

In order to prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M259">View MathML</a>, we assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M260">View MathML</a>. Using the same arguments as in the previous part of the proof, we see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M261">View MathML</a>

meaning that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M262">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M263">View MathML</a>

Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M264">View MathML</a>

From the last contradiction we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M259">View MathML</a>.

Suppose that there is a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M266">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M267">View MathML</a>, such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M268">View MathML</a>. Then, by (1),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M269">View MathML</a>

This contradiction proves that a is a unique fixed point of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216">View MathML</a>. According to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M271">View MathML</a> and from the uniqueness which has been proved already, we deduce that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M272">View MathML</a>.

Next, we claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M215">View MathML</a>, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13">View MathML</a>. To prove this, fix <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M276">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M277">View MathML</a> and put

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M278">View MathML</a>

Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M279">View MathML</a>

By (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M281">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M282">View MathML</a>.

To prove continuity of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216">View MathML</a> at a, we consider any sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M284">View MathML</a> converging to a. For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M285">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M286">View MathML</a>

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M287">View MathML</a>. Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M288">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M289">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M290">View MathML</a> converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M291">View MathML</a>, meaning that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216">View MathML</a> is continuous at a.

In other cases (when we use (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116">View MathML</a>) instead of (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>) or (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>) instead of (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>)), by Lemma 2.1, the same conclusion can be drawn. □

Corollary 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M203">View MathML</a>be a completeG-metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M204">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13">View MathML</a>there exists a positive integer<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M14">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M301">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M15">View MathML</a>and some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M303">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213">View MathML</a>. Moreover, for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M306">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216">View MathML</a>is continuous ata.

Proof The function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M308">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M309">View MathML</a>, satisfies (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>) and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>), so the corollary is a consequence of Theorem 2.1. □

From the proof of Theorem 2.1 we can see that it would be enough to impose certain assumptions not for all elements from X but only over some subset B of X, just as was done by Guseman [11]. The next theorem is a Guseman type of fixed point theorem in a G-metric space.

Theorem 2.2LetTbe a selfmapping of a completeG-metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>. If there exists a subsetBofXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M313">View MathML</a>, Tsatisfies (1) overBand for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M314">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M315">View MathML</a>, then there exists a unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M316">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M317">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M318">View MathML</a>for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M319">View MathML</a>. Furthermore, ifTsatisfies (1) overX, thenuis unique fixed point inXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M318">View MathML</a>for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M15">View MathML</a>.

Remark 2.1 Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M308">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M323">View MathML</a>, by Theorem 2.2 we obtain the fixed point result from [23] or [24], so Theorem 2.2 is also a generalization of the Guseman fixed point result from [11].

Corollary 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M203">View MathML</a>be a completeG-metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M204">View MathML</a>, and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53">View MathML</a>there exists a positive integer<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M14">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M328">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M212">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213">View MathML</a>. Moreover, for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M215">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216">View MathML</a>is continuous ata.

Proof Since the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M334">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M335">View MathML</a>, satisfies (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>), (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M112">View MathML</a>), (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116">View MathML</a>), and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>), we can apply Theorem 2.1. Also for that φ, the appropriate version of Theorem 2.2 can be formulated in a similar way as was done in this corollary. □

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M340">View MathML</a>, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53">View MathML</a>, it is easy to see that condition (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M116">View MathML</a>) or (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M120">View MathML</a>) in Theorem 2.1 can be omitted. This version of Theorem 1.2 is an improvement and another proof of Theorem 3.1 (Corollary 3.2) from [9]. But in that case it would be more appropriate to use the metric ρ, which reduces (1) to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M344">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M6">View MathML</a> and enables the use of well-known results in metric spaces.

Proposition 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M203">View MathML</a>be a completeG-metric space, whereφis a nondecreasing function satisfying (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>). If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M204">View MathML</a>satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M349">View MathML</a>

(3)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M24">View MathML</a>and some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M351">View MathML</a>, thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M213">View MathML</a>. Moreover, for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M53">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M215">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M216">View MathML</a>is continuous ata.

The next theorem is also a Guseman type of fixed point theorem in a G-metric space. The assumptions about the contractor φ is different with respect to Theorem 2.2. Similarly as in previous analysis, the next theorem can be applied in a metric space and in cases where some special form of function φ is used.

Theorem 2.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M356">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>is aG-metric space and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9">View MathML</a>be a subadditive mapping satisfying (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>). If for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M314">View MathML</a>the closure of orbit<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M361">View MathML</a>is complete and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M362">View MathML</a>there exists an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M363">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M364">View MathML</a>

(4)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M365">View MathML</a>, then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M366">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M367">View MathML</a>, converges to some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M368">View MathML</a>.

If inequality (4) holds for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M369">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M370">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M371">View MathML</a>for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M369">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M373">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374">View MathML</a>is a fixed point off.

Proof First, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M375">View MathML</a> is a Cauchy sequence. For sufficiently large <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M351">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M377">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M378">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M379">View MathML</a>. Using (4), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M380">View MathML</a>

Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M381">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M351">View MathML</a>, the next inequality holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M383">View MathML</a>

(5)

and consequently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M384">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M351">View MathML</a>. Using the last inequality, for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M386">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M240">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M388">View MathML</a>

implying that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M389">View MathML</a> is a Cauchy sequence. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M361">View MathML</a> is complete, and there exists an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M391">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M392">View MathML</a>.

In the second part of the theorem, inequality (4) holds for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M369">View MathML</a>. Then the elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M394">View MathML</a> of the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M389">View MathML</a> from the previous part of the proof satisfy the next two relations:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M396">View MathML</a>

(6)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M397">View MathML</a>

(7)

By (6)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M398">View MathML</a>

and by (7)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M399">View MathML</a>

Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M370">View MathML</a>.

Next, we claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M401">View MathML</a>, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M369">View MathML</a>. Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M403">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M276">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M405">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M406">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M407">View MathML</a>. Since (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M122">View MathML</a>) ⇒ (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M109">View MathML</a>), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M401">View MathML</a>.

To show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374">View MathML</a> is a unique fixed point of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M412">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M361">View MathML</a>, we assume that there exists another point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M414">View MathML</a> with the same property. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M415">View MathML</a>

that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M416">View MathML</a>. Further, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M373">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M418">View MathML</a>, implying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M419">View MathML</a>. □

In the last theorem in this paper we consider a common fixed point for a family of selfmappings with the property of a contractive iterate at a point. The generalized contractive condition is imposed over a subset of a G-metric space.

Theorem 2.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M32">View MathML</a>beG-metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M421">View MathML</a>. Further, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M422">View MathML</a>be the sequence of selfmappings ofXsuch that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M424">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M13">View MathML</a>there exists an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M363">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M427">View MathML</a>

(8)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M386">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M429">View MathML</a>, and all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M319">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M9">View MathML</a>is a nondecreasing right continuous function satisfying (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M106">View MathML</a>). If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M433">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M434">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374">View MathML</a>is a unique common fixed point for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M422">View MathML</a>inBand for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M438">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M439">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M440">View MathML</a>, converges to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374">View MathML</a>.

Proof First we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374">View MathML</a> is a unique point in B with the property that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M443">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M440">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M445">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M446">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M447">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M449">View MathML</a>

By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M107">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M4">View MathML</a>, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M452">View MathML</a>, we have a contradiction, that is, the assumption <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M453">View MathML</a> is not correct.

Further, since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M454">View MathML</a>

it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M455">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423">View MathML</a>.

Now, for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M438">View MathML</a>, we form the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M458">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M459">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M460">View MathML</a> and the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M461">View MathML</a> converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M463">View MathML</a>, in order to prove that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M461">View MathML</a> converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374">View MathML</a>, we consider the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M466">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M423">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M468">View MathML</a>

If we choose the option that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M469">View MathML</a>

it implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M470">View MathML</a>

(9)

On the other hand, in that case

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M471">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M472">View MathML</a>

(10)

It is obvious that (9) contradicts (10). So,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M473">View MathML</a>

Now, applying that procedure i times and letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M474">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M475">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M476">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M477">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M478">View MathML</a>. The last relation proves that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M461">View MathML</a> converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2014/1/46/mathml/M374">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors have equal contribution in the paper and they read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the anonymous referees for their careful reading of the paper and suggestions, which have contributed to the improvement of the paper. This work is supported by Ministry of Science and Technological Development, Republic of Serbia.

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