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A note on common fixed points for (ψ,α,β)-weakly contractive mappings in generalized metric spaces

Nurcan Bilgili12*, Erdal Karapınar3 and Duran Turkoglu12

Author Affiliations

1 Department of Mathematics, Faculty of Science and Arts, Amasya University, Ipekkoy, Amasya, 05000, Turkey

2 Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, Ankara, 06500, Turkey

3 Department of Mathematics, Atilim University, İncek, Ankara, 06836, Turkey

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Fixed Point Theory and Applications 2013, 2013:287  doi:10.1186/1687-1812-2013-287

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/287


Received:20 May 2013
Accepted:18 September 2013
Published:8 November 2013

© 2013 Bilgili 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

Very recently, Isik and Turkoglu (Fixed Point Theory Appl. 2013:131, 2013) proved a common fixed point theorem in a rectangular metric space by using three auxiliary distance functions. In this paper, we note that this result can be derived from the recent paper of Lakzian and Samet (Appl. Math. Lett. 25:902-906, 2012).

MSC: 47H10, 54H25.

Keywords:
fixed point; partial metric space; GP-metric space

1 Introduction and preliminaries

In 2012, Lakzian and Samet [1] proved a fixed point theorem of a self-mapping with certain conditions in the context of a rectangular metric space via two auxiliary functions. Very recently, as a generalization of the main result of [1], Isik and Turkoglu [2] reported a common fixed point result of two self-mappings in the setting of a rectangular metric space by using three auxiliary functions. In this paper, unexpectedly, we conclude that the main result of Isik and Turkoglu [2] is a consequence of the main results of [1]. The obtained results are inspired by the techniques and ideas of, e.g., [3-11].

Throughout the paper, we follow the notations used in [2]. For the sake of completeness, we recall some basic definitions, notations and results.

Definition 1.1 Let X be a nonempty set, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M2">View MathML</a> satisfy the following conditions for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3">View MathML</a> and all distinct <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M4">View MathML</a>, each of which is different from x and y:

(RM1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M5">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M6">View MathML</a>,

(RM2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M7">View MathML</a>,

(RM3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M8">View MathML</a>.

Then the map d is called a rectangular metric and the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9">View MathML</a> is called a rectangular metric space (or, for short, RMS).

We note that a rectangular metric space is also known as a generalized metric space (g.m.s.) in some sources.

We first recall the definitions of the following auxiliary functions: Let ℱ be the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M10">View MathML</a> satisfying the condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M11">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M12">View MathML</a>. We denote by Ψ the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M13">View MathML</a> such that ψ is continuous and nondecreasing. We reserve Φ for the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M14">View MathML</a> such that α is continuous. Finally, by Γ we denote the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M15">View MathML</a> satisfying the following condition: β is lower semi-continuous.

Lakzian and Samet [1] proved the following fixed point theorem.

Theorem 1.1[1]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9">View MathML</a>be a Hausdorff and completeRMS, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17">View MathML</a>be a self-map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M18">View MathML</a>

(1)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M20">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M21">View MathML</a>. ThenThas a unique fixed point inX.

Lemma 1.1[3]

LetXbe a nonempty set and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17">View MathML</a>be a function. Then there exists a subset<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M23">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M24">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M25">View MathML</a>is one-to-one.

Definition 1.2 Let X be a nonempty set, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M26">View MathML</a> be self-mappings. The mappings are said to be weakly compatible if they commute at their coincidence points, that is, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M27">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M28">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M29">View MathML</a>.

Theorem 1.2[2]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9">View MathML</a>be a Hausdorff and completeRMS, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M26">View MathML</a>be self-mappings such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M32">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M33">View MathML</a>is a closed subspace ofX, and that the following condition holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M34">View MathML</a>

(2)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M36">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M38">View MathML</a>, and these mappings satisfy the condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M39">View MathML</a>

(3)

ThenTandFhave a unique coincidence point inX. Moreover, ifTandFare weakly compatible, thenTandFhave a unique common fixed point.

Remark 1.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9">View MathML</a> be RMS. Then d is continuous (see, e.g., Proposition 2 in [5]).

2 Main results

We start this section with the following theorem which is a slightly improved version of Theorem 1.1, obtained by replacing the continuity condition of ϕ with a lower semi-continuity.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9">View MathML</a>be a Hausdorff and completeRMS, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17">View MathML</a>be a self-map satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M43">View MathML</a>

(4)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M20">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M46">View MathML</a>. ThenThas a unique fixed point inX.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M47">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M48">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M49">View MathML</a> . Following the lines of the proof of Theorem 1.1 in [1], we conclude that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M50">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M51">View MathML</a>

(5)

We can easily derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M52">View MathML</a>

(6)

by replacing <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M53">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M54">View MathML</a> in inequality (4).

Taking lim sup in inequality (6) as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M55">View MathML</a>, we find that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M56">View MathML</a>

(7)

and using the continuity of ψ and lower semi-continuity of ϕ, thus, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M57">View MathML</a>

(8)

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M58">View MathML</a> and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M59">View MathML</a>. Consequently, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M60">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M61">View MathML</a>.

Next, we shall prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M62">View MathML</a>

(9)

By using inequality (4), we derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M63">View MathML</a>

(10)

From the monotone property of the function ψ, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M64">View MathML</a> is monotone decreasing. Thus, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M65">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M66">View MathML</a>

(11)

Taking lim sup of inequality (10) as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M55">View MathML</a>, we derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M68">View MathML</a>

(12)

Then, by using the continuity of ψ and lower semi-continuity of ϕ, we find

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M69">View MathML</a>

(13)

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M70">View MathML</a>. So, we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M71">View MathML</a> and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M72">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M61">View MathML</a>.

As in Theorem 1.1 in [1], we notice that T has no periodic point.

We assert that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M74">View MathML</a> is a Cauchy sequence. Suppose, on the contrary, that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M75">View MathML</a> for which we can obtain subsequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M76">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M77">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M74">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M79">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M80">View MathML</a>

(14)

Again, repeating the steps of Theorem 1.1 in [1], we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M81">View MathML</a>

(15)

Now, letting lim sup in inequality (15) as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M55">View MathML</a>, we observe that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M83">View MathML</a>

(16)

Using the continuity of ψ and lower semi-continuity of ϕ, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M84">View MathML</a>

(17)

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M85">View MathML</a> and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M86">View MathML</a>, a contradiction with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M87">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M74">View MathML</a> is a Cauchy sequence. The rest of the proof is the mimic of the proof of Theorem 1.1 in [1] and hence we omit the details. □

Inspired by Theorem 1.2, one can state the following theorem.

Theorem 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M9">View MathML</a>be a Hausdorff and completeRMS, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17">View MathML</a>be self-mappings such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M91">View MathML</a>

(18)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M36">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M38">View MathML</a>and these mappings satisfy the condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M96">View MathML</a>

(19)

ThenThas a unique fixed point inX.

Since the proof is the mimic of the proof of Theorem 1.2, we omit it.

We first prove that the above theorem is equivalent to Theorem 2.1.

Theorem 2.3Theorem 2.2 is a consequence of Theorem 2.1.

Proof Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M97">View MathML</a> in Theorem 2.2, we obtain immediately Theorem 2.1. Now, we shall prove that Theorem 2.2 can be deduced from Theorem 2.1. Indeed, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M17">View MathML</a> be a mapping satisfying (18) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M36">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M38">View MathML</a>, and let these mappings satisfy condition (19). From (18), for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M103">View MathML</a>

(20)

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M104">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M105">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M106">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M107">View MathML</a>

(21)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M3">View MathML</a>. Due to the definition of θ, we observe that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M109">View MathML</a>. Now, Theorem 2.2 follows immediately from Theorem 2.1. □

By regarding the techniques in [3], we conclude the following result.

Theorem 2.4Theorem 1.2 is a consequence of Theorem 2.2.

Proof By Lemma 1.1, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M23">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M111">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M112">View MathML</a> is one-to-one. Now, define a map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M113">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M114">View MathML</a>. Since F is one-to-one on E, h is well defined. Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M115">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M116">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M111">View MathML</a> is complete, by using Theorem 2.2, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M47">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/287/mathml/M119">View MathML</a>. Hence, T and F have a point of coincidence, which is also unique. It is clear that T and F have a unique common fixed point whenever T and F are weakly compatible. □

Theorem 2.5Theorem 1.2 is a consequence of Theorem 2.1.

Proof It is evident from Theorem 2.3 and Theorem 2.4. □

3 Conclusion

In this paper, we first slightly improve the main result of Lakzian and Samet, Theorem 1.1. Then, we conclude that the main result (Theorem 1.2) of Isik-Turkoglu [2] is a consequence of our improved result, Theorem 2.1.

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Authors’ contributions

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

Acknowledgements

The authors thank anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.

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