Abstract
Very recently, the notion of a ψ-Geraghty type contraction was defined by Gordji et al. (Fixed Point Theory and Applications 2012:74, 2012). In this short note, we realize that the main result via ψ-Geraghty type contraction is equivalent to an existing related result in the literature. Consequently, all results inspired by the paper of Gordji et al. in (Fixed Point Theory and Applications 2012:74, 2012) can be derived in the same way.
MSC:47H10, 54H25, 46J10, 46J15.
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1 Introduction and preliminaries
One of the celebrated generalizations of the Banach contraction (mapping) principle was given by Geraghty [1].
Theorem 1.1 (Geraghty [1])
Let be a complete metric space and be an operator. Suppose that there exists satisfying the condition
If T satisfies the following inequality:
then T has a unique fixed point.
Let denote the set of all functions satisfying (1). This nice result of Geraghty [1] has been studied by a number of authors, see e.g. [2–10] and references therein.
In the following Harandi and Emami [2] reconsidered Theorem 1.1 in the framework of partially ordered metric spaces (see also [11]).
Theorem 1.2 Let be a partially ordered complete metric space. Let be an increasing mapping such that there exists an element with . If there exists such that
for each with , then f has a fixed point provided that either f is continuous or X is such that if an increasing sequence in X; then , for all n. Besides, if for each there exists which is comparable to x and y, then f has a unique fixed point.
Very recently, Gordji et al. [12] supposedly improved and extended Theorem 1.2 in the following way via the auxiliary function defined below. Let Ψ denote the class of the functions which satisfy the following conditions:
() ψ is nondecreasing;
() ψ is subadditive, that is, ;
() ψ is continuous;
() .
The following is the main theorem of Gordji et al. [12].
Theorem 1.3 Let be a partially ordered complete metric space. Let be a nondecreasing mapping such that there exists with . Suppose that there exist and such that
for all with . Assume that either f is continuous or X is such that if an increasing sequence converges to x, then for each . Then f has a fixed point.
2 Main results
We start this section with the following lemma, which is the skeleton of this note.
Lemma 2.1 Let be a metric space and . Then, a function defined by forms a metric on X. Moreover, is complete if and only if is complete.
Proof
-
(1)
If , then . Due to (), we have . The converse is obtained analogously.
-
(2)
.
-
(3)
Since ψ is nondecreasing, we have . Regarding the subadditivity of ψ, we derived
Notice that the completeness of follows from () and (). □
The following is the main result of this note.
Theorem 2.2 Theorem 1.3 is a consequence of Theorem 1.2.
Proof Due to Lemma 2.1, we derived the result that is a complete metric space. Furthermore, the condition (4) turns into
Hence all conditions of Theorem 1.2 are satisfied. □
3 The best proximity case
Let A and B be two nonempty subsets of a metric space . We denote by and the following sets:
where .
In [13, 14], the author introduces the following definition.
Definition 3.1 Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the P-property if and only if, for any and ,
Caballero et al. proved the following result.
Theorem 3.2 (See [8])
Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a Geraghty contraction, i.e. there exists such that
Suppose that T is continuous and satisfies . Suppose also that the pair has the P-property. Then there exists a unique in A such that .
Inspired by Gordji et al. [12] and Caballero et al. [8], Karapinar [7] reported the following result.
Theorem 3.3 Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be ψ-Geraghty contraction, i.e. there exists such that
Suppose that T is continuous and satisfies . Suppose also that the pair has the P-property. Then there exists a unique in A such that .
The following lemmas belong to Akbar and Gabeleh [15].
Lemma 3.4 [15]
Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty and has the P-property. Then is a closed pair of subsets of X.
Lemma 3.5 [15]
Let be a pair of nonempty closed subsets of a metric space such that is nonempty. Assume that the pair has the P-property. Then there exists a bijective isometry such that .
Very recently, by using Lemma 3.4 and Lemma 3.5, Akbar and Gabeleh [15] proved that the best proximity point results via P-property can be obtained from the associate results in fixed point theory. In particular they proved the following theorem.
Theorem 3.6 Theorem 3.2 is a consequence of Theorem 1.1.
As a consequence of Theorem 2.2 we can observe the following result.
Corollary 3.7 Theorem 3.3 is a consequence of Theorem 3.2.
Regarding the analogy, we omit the proof.
Theorem 3.8 Theorem 3.3 is a consequence of Theorem 1.1.
References
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Acknowledgements
The authors thank the Visiting Professor Programming at King Saud University for funding this work. The authors thank the anonymous referees for their remarkable comments, suggestion, and ideas, which helped to improve this paper.
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Karapınar, E., Samet, B. A note on ‘ψ-Geraghty type contractions’. Fixed Point Theory Appl 2014, 26 (2014). https://doi.org/10.1186/1687-1812-2014-26
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DOI: https://doi.org/10.1186/1687-1812-2014-26