Open Access Research

Suzuki-type fixed point results in metric type spaces

Nawab Hussain1*, Dragan Ðorić2, Zoran Kadelburg3 and Stojan Radenović4

Author Affiliations

1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

2 Faculty of Organizational Sciences, University of Belgrade, Jove Ilića 154, Beograd, 11000, Serbia

3 Faculty of Mathematics, University of Belgrade, Studentski trg 16, Beograd, 11000, Serbia

4 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Beograd, 11120, Serbia

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Fixed Point Theory and Applications 2012, 2012:126  doi:10.1186/1687-1812-2012-126


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/126


Received:5 February 2012
Accepted:18 July 2012
Published:31 July 2012

© 2012 Hussain et al.; 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

Suzuki’s fixed point results from (Suzuki, Proc. Am. Math. Soc. 136:1861-1869, 2008) and (Suzuki, Nonlinear Anal. 71:5313-5317, 2009) are extended to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones.

MSC: 47H10, 54H25.

Keywords:
metric type space; cone metric space; normal cone; fixed point

1 Introduction and preliminaries

In 2008 Suzuki proved the following refinement of Banach’s fixed point principle.

Theorem 1 ([1], Theorem 2])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1">View MathML</a>be a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M3">View MathML</a>be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M4">View MathML</a>

(1.1)

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a>such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>,

thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10">View MathML</a>converges to z.

There were various extensions of Suzuki’s result, such as Kikkawa-Suzuki’s version of Kannan’s theorem [2] and Popescu’s version of Ćirić’s theorem [3].

Suzuki proved also the following version of Edelstein’s fixed point theorem.

Theorem 2 ([4], Theorem 3])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1">View MathML</a>be a compact metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap, satisfying for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M14">View MathML</a>the condition

ThenThas a unique fixed point inX.

This theorem was generalized in [5].

Let E be a real Banach space with the zero vector θ. A subset P of E is called a cone if: (a) P is closed, non-empty and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M16">View MathML</a>; (b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M17">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M18">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M19">View MathML</a> imply that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M20">View MathML</a>; (c) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M21">View MathML</a>. Given a cone P, we define the partial ordering ⪯ with respect to P by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M22">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M23">View MathML</a>. We shall write <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M24">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M25">View MathML</a>, where intP stands for the interior of P and use <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M26">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M22">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M28">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M29">View MathML</a>, then P is called a solid cone. It is said to be normal if there is a number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M30">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M31">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M32">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M33">View MathML</a>. Such a minimal constant K is called the normal constant of P.

Huang and Zhang re-introduced cone metric spaces in [6] (this notion was known under various names since the mid of the 20th century, see a survey in [7]), replacing the set of real numbers by an ordered Banach space as the codomain for a metric. Cone metric spaces over normal cones inspired another generalization of metric spaces that were called metric type spaces by Khamsi [8] (see also [9-12]; note that, in fact, spaces of this kind were used earlier under the name of b-spaces by Czerwik [13]). Cvetković et al.[14] and Shah et al.[15] extended Khamsi’s definition and defined cone metric type spaces as follows:

Definition 1 ([14,15])

Let X be a nonempty set, E a Banach space with the solid cone P and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M34">View MathML</a> be a real number. If the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M35">View MathML</a> satisfies the following properties:

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M36">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M37">View MathML</a>;

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M38">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>;

(c) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M40">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M41">View MathML</a>,

then D is called a cone metric type function and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a> is called a cone metric type space (CMTS).

In particular, when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M43">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M44">View MathML</a>, CMTS <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a> reduces to a metric type space (MTS) of [8,9,12].

Of course, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M46">View MathML</a> we get the cone metric space (CMS) of [6], resp. the usual metric space.

Example 1 ([14])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M47">View MathML</a> be an orthonormal basis of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M48">View MathML</a> with inner product <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M49">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M50">View MathML</a>. Define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M51">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M52">View MathML</a> is the class of functions being equal to the function f a.e. Further, let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M53">View MathML</a>

and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M54">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M55">View MathML</a>

It was shown in [14] that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M56">View MathML</a> is a solid cone in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M48">View MathML</a> and that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M58">View MathML</a> is a CMTS. In particular, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M59">View MathML</a> we get an MTS and for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M60">View MathML</a> a CMS.

Example 2 ([8,10])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1">View MathML</a> be any CMS over a normal cone with normal constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M34">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a> is an MTS, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M64">View MathML</a>. In this case the spaces <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a> have the same topologies (see [10], Theorem 2.7]).

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67">View MathML</a> is a CMTS over a normal cone with a normal constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M68">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M69">View MathML</a> is an MTS, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M70">View MathML</a>. Similarly as above, the spaces <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M69">View MathML</a> have the same topologies.

Notions such as convergent and Cauchy sequences, as well as completeness, are introduced in (cone) metric type spaces in the standard way. The following obviously holds in an arbitrary (cone) metric type space:

We will sometimes need the continuity of metric-type function D in one variable:

or in two variables:

The last property always holds in the case of an MTS <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a> generated by a CMS <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M1">View MathML</a> over a normal cone, see Example 2, but not in general, as the following example shows.

Example 3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M78">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M79">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M80">View MathML</a>

Then it is easy to see that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M81">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M82">View MathML</a>

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M83">View MathML</a> is a metric-type space. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M84">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M86">View MathML</a>

that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M87">View MathML</a>, but <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M88">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89">View MathML</a>.

Recall that a selfmap <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a> is said to have the property (P) [16] if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M91">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M93">View MathML</a> is the set of fixed points of T.

In this paper, we extend Suzuki’s Theorems 1 and 2, as well as Popescu’s results from [3] to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones.

2 Results

2.1 Results in metric type spaces

Theorem 3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a>be a complete MTS whereDis continuous in each variable. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M96">View MathML</a>be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M97">View MathML</a>

(2.1)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M98">View MathML</a>is the positive solution of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M99">View MathML</a>. If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a>such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>,

(2.2)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M103">View MathML</a>

thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10">View MathML</a>converges to z. Moreover, Thas the property (P).

Note that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M46">View MathML</a>, Theorem 3 reduces to a special case of Theorem 2.1 by Popescu [3].

Proof First note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M108">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M109">View MathML</a> and it follows by (2.2) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M110">View MathML</a>

wherefrom

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M111">View MathML</a>

(2.3)

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M113">View MathML</a> be arbitrary and form the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M114">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M115">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M116">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. It follows from (2.3) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M118">View MathML</a>

(2.4)

and, by induction,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M119">View MathML</a>

(2.5)

Using [12], Lemma 3.1] we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M114">View MathML</a> is a Cauchy sequence, tending to some z in the complete space X. Obviously, also <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M121">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89">View MathML</a>.

Let us prove now that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M123">View MathML</a>

(2.6)

holds for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M124">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M125">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M126">View MathML</a> (and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M127">View MathML</a>) and, by continuity of D, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M128">View MathML</a>, it follows that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M129">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M130">View MathML</a>

holds for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M131">View MathML</a>. Assumption (2.2) implies that for such n

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M132">View MathML</a>

Passing to the limit when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89">View MathML</a> (and using continuity of D), we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M134">View MathML</a>

It is easy to see that (2.6) follows from the previous relation.

Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M135">View MathML</a> in (2.3), we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M136">View MathML</a>

(2.7)

holds for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a> (where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M138">View MathML</a>). It follows by induction that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M139">View MathML</a>

(2.8)

We will prove now that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M140">View MathML</a>

(2.9)

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M59">View MathML</a> this relation is obvious. Suppose that it holds for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M144">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M145">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M146">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M147">View MathML</a>, then we can apply (2.6) to obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M148">View MathML</a>

Using (2.8) and the induction hypothesis, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M149">View MathML</a>

and (2.9) is proved by induction.

In order to prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M150">View MathML</a>, we suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M151">View MathML</a> and consider the two possible cases.

Case I. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M152">View MathML</a> (and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M153">View MathML</a>). We will prove first that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M154">View MathML</a>

(2.10)

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M59">View MathML</a> this is obvious and for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M157">View MathML</a> it follows from (2.8). Suppose that (2.10) holds for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M158">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M159">View MathML</a>

wherefrom <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M160">View MathML</a>. It follows (using (2.8)) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M161">View MathML</a>

Assumption (2.2) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M162">View MathML</a>

It is easy to see (using (2.8), (2.9) and the inductive hypothesis) that the last maximum is equal to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M163">View MathML</a>, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M164">View MathML</a> and relation (2.10) is proved by induction.

Now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M151">View MathML</a> and (2.10) implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M147">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. Hence, (2.6) and (2.8) imply that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M168">View MathML</a>

(2.11)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M169">View MathML</a>, it follows from (2.10) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M170">View MathML</a>

There exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M171">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M172">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M173">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M152">View MathML</a>. For such n, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M175">View MathML</a>

It follows from (2.11) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M176">View MathML</a>

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M177">View MathML</a> and, again from (2.10), we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M178">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M179">View MathML</a>, a contradiction.

Case II. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M180">View MathML</a> (and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M181">View MathML</a>). We will prove that there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M182">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M114">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M184">View MathML</a>

(2.12)

holds for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M185">View MathML</a>. From (2.4) we know that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M186">View MathML</a> holds for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. Suppose that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M188">View MathML</a>

both hold for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M190">View MathML</a>

which is impossible. Hence one of the following holds for each n:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M191">View MathML</a>

In particular,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M192">View MathML</a>

holds for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. In other words, there is a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M182">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M114">View MathML</a> such that (2.12) holds for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M185">View MathML</a>. But then assumption (2.2) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M197">View MathML</a>

Passing to the limit when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M198">View MathML</a> we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M199">View MathML</a>, which is possible only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M150">View MathML</a>, a contradiction.

Thus, we have proved that z is a fixed point of T. The uniqueness of the fixed point follows easily from (2.6). Indeed, if yz are two fixed points of T, then (2.6) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M201">View MathML</a>

wherefrom <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M202">View MathML</a>. The property (P) follows from (2.3) (see [16]). □

Suzuki-Banach-type and Suzuki-Kannan-type fixed point results in metric type spaces (versions of [1], Theorem 2] and [2], Theorem 2.2]) are special cases of Theorem 3.

Corollary 1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a>be a complete MTS whereDis continuous in each variable. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205">View MathML</a>be defined by (2.1). If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a>such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>,

thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10">View MathML</a>converges to z. Moreover, Thas the property (P).

Corollary 2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a>be a complete MTS whereDis continuous in each variable. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205">View MathML</a>be defined by (2.1). If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a>such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>,

thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10">View MathML</a>converges to z. Moreover, Thas the property (P).

Corollary 3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a>be a complete MTS whereDis continuous in each variable. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205">View MathML</a>be defined by (2.1). If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a>such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>,

thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10">View MathML</a>converges to z. Moreover, Thas the property (P).

Adapting [1], Example 1] we give now an example of a mapping satisfying the conditions of Theorem 3 (and having a unique fixed point) but not satisfying the respective classical (non-Suzuki-type) condition in metric type spaces (see, e.g., [14], Theorem 3.4]).

Example 4 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M230">View MathML</a>, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M231">View MathML</a> be given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M232">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M233">View MathML</a> is a metric type space (see Example 1). Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a> be given as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M235">View MathML</a>

We will check that condition (2.2) holds true for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M236">View MathML</a> and all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M37">View MathML</a> or if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M239">View MathML</a>, it is trivially satisfied. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M240">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M241">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M242">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M243">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M244">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M245">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M246">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M247">View MathML</a>. Hence, in any case,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M248">View MathML</a>

Let now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M249">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M14">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M251">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M252">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M253">View MathML</a>, and (2.2) is trivially satisfied. Note that in the classical variant, in this case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M254">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M245">View MathML</a>, so the inequality <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M256">View MathML</a> does not hold for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M257">View MathML</a>.

The following is a metric-type version of Theorem 2.

Theorem 4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M42">View MathML</a>be a compact MTS, where the functionDis continuous. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap, satisfying for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M14">View MathML</a>the condition

(2.13)

ThenThas a unique fixed point inX.

Proof Denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M263">View MathML</a> and choose a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M264">View MathML</a> in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M265">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89">View MathML</a>). Since the space X is (sequentially) compact, we can suppose that there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M267">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M268">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M269">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89">View MathML</a>). We will prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M271">View MathML</a>.

Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M272">View MathML</a> and note that continuity of D implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M273">View MathML</a>. Choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M129">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M275">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M276">View MathML</a>

holds true. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M277">View MathML</a> and assumption (2.13) implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M278">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M275">View MathML</a>. Passing to the limit, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M280">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M281">View MathML</a>, the last inequality is impossible by the definition of β. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M46">View MathML</a>, it is possible only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M283">View MathML</a> (recall that we have supposed that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M272">View MathML</a>). But in this case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M285">View MathML</a> and (2.13) implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M286">View MathML</a>, which is again impossible by the definition of β. Hence, in all cases we obtain a contradiction and it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M271">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M288">View MathML</a>.

In order to prove that T has a fixed point, suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M151">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>. Then, in particular, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M291">View MathML</a> and (2.13) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M292">View MathML</a>

It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M293">View MathML</a>

when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M295">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M89">View MathML</a>).

Suppose now that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M297">View MathML</a>

both hold for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M299">View MathML</a>

which is impossible. Thus, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a>, either

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M301">View MathML</a>

holds true. Assumption (2.13) implies that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M85">View MathML</a> either

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M303">View MathML</a>

holds. In other words, there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M304">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M305">View MathML</a> holds for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M185">View MathML</a>, or there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M307">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M308">View MathML</a> holds for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M309">View MathML</a>. In both cases, passing to the limit, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M310">View MathML</a>, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M311">View MathML</a>, a contradiction with the assumption that T has no fixed points.

It follows that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M150">View MathML</a>. Uniqueness follows easily. □

2.2 Results in cone metric type spaces

In this subsection, we formulate cone-metric-type versions of the results from the previous subsection.

Theorem 5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67">View MathML</a>be a complete CMTS with the normal underlying coneP, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315">View MathML</a>is continuous in each variable. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M96">View MathML</a>be defined by (2.1). If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a>such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>,

(2.14)

for some

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M321">View MathML</a>

thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10">View MathML</a>converges to z.

Proof Since the cone P is normal, without loss of generality, we can assume that the normal constant of P is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M325">View MathML</a> and that the given norm in E is monotone, i.e. (see [17], Lemma 2.1]). Denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M70">View MathML</a>. Then D is a (real-valued) metric-type function and the space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M328">View MathML</a> is compact (together with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M329">View MathML</a>, see [10], Theorem 2.7]). Let us prove that the mapping T satisfies for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a> the condition

(2.15)

of Theorem 3. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M332">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M333">View MathML</a> (indeed, if, to the contrary, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M334">View MathML</a>i.e.<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M335">View MathML</a>, it would follow that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M336">View MathML</a>, a contradiction with the assumption). Assumption (2.14) implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M337">View MathML</a> for some

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M338">View MathML</a>

Again by the monotonicity of the norm, this means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M339">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M340">View MathML</a>

Hence, condition (2.15) is satisfied, and the conclusion follows. □

In a similar way, the following corollaries and the theorem can be proved.

Corollary 4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67">View MathML</a>be a complete CMTS where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315">View MathML</a>is continuous in each variable. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205">View MathML</a>be defined by (2.1). If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a>such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>,

thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10">View MathML</a>converges to z.

Corollary 5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67">View MathML</a>be a complete CMTS where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315">View MathML</a>is continuous in each variable. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205">View MathML</a>be defined by (2.1). If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a>such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>,

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M358">View MathML</a>, thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10">View MathML</a>converges toz.

Corollary 6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67">View MathML</a>be a complete CMTS where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315">View MathML</a>is continuous in each variable. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M205">View MathML</a>be defined by (2.1). If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M5">View MathML</a>such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>,

thenThas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M8">View MathML</a>and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M9">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M10">View MathML</a>converges to z.

Example 4 can be easily adapted to a CMTS.

Theorem 6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M67">View MathML</a>be a compact CMTS, where the function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M315">View MathML</a>is continuous. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M2">View MathML</a>be a selfmap satisfying, for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M6">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M14">View MathML</a>the condition

(2.16)

ThenThas a unique fixed point inX.

Note that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/126/mathml/M46">View MathML</a> the above theorem reduces to [5], Theorem 3.8].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

The first author gratefully acknowledges the support provided by the Deanship of Scientific Research (DSR), King Abdulaziz University during this research. The second, third and fourth authors are thankful to the Ministry of Science and Technological Development of Serbia.

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