On best proximity point of ψ-Geraghty contractions

Fixed Point Theory and Applications20132013:200

DOI: 10.1186/1687-1812-2013-200

Received: 12 March 2013

Accepted: 17 May 2013

Published: 24 July 2013

Abstract

Very recently, Caballero, Harjani and Sadarangani (Fixed Point Theory Appl. 2012:231, 2012) observed some best proximity point results for Geraghty contractions by using the P-property. In this paper, we introduce the notion of ψ-Geraghty contractions and show the existence and uniqueness of the best proximity point of such contractions in the setting of a metric space. We state examples to illustrate our result.

MSC: 41A65, 90C30, 47H10.

Keywords

best proximity point non-self mapping partial order metric space fixed point

1 Introduction and preliminaries

In nonlinear functional analysis, fixed point theory and best proximity point theory play a crucial role in the establishment of the existence of certain differential and integral equations. As a consequence, fixed point theory is very useful for various quantitative sciences that involve such equations. To list a few, certain branches of computer sciences, engineering and economics are well-known examples in which fixed point theory is used.

The most remarkable paper in this field was reported by Banach [1] in 1922. In this paper, Banach proved that every contraction in a complete metric space has a unique fixed point. Following this outstanding paper, many authors have extended, generalized and improved this remarkable fixed point theorem of Banach by changing either the conditions of the mappings or the construction of the space. In particular, one of the notable generalizations of Banach fixed point theorem was reported by Geraghty [2].

Theorem 1.1 (Geraghty [2])

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq1_HTML.gif be a complete metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq2_HTML.gif be an operator. Suppose that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq3_HTML.gif satisfying the condition
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equa_HTML.gif
If T satisfies the following inequality:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ1_HTML.gif
(1)

then T has a unique fixed point.

It is very natural that some mappings, especially non-self-mappings defined on a complete metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq4_HTML.gif , do not necessarily possess a fixed point, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq5_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq6_HTML.gif . In such situations, it is reasonable to search for the existence (and uniqueness) of a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq8_HTML.gif is an approximation of an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq6_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq9_HTML.gif . In other words, one speculates to determine an approximate solution http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq10_HTML.gif that is optimal in the sense that the distance between http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq11_HTML.gif is minimum. Here, the point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq12_HTML.gif is called a best proximity point.

This research subject has attracted attention of a number of authors; for example, see [223]. In this paper we generalize and improve certain results of Caballero et al. in [6]. Notice also that in the best proximity point theory, we usually consider a non-self-mapping. In fixed point theory, almost all maps are self-mappings. For the sake of completeness, we recall some basic definitions and fundamental results on the best proximity theory.

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq1_HTML.gif be a metric space and A and B be nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq1_HTML.gif . A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq13_HTML.gif is called a k-contraction if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq14_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq15_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq16_HTML.gif . It is clear that a k-contraction coincides with the celebrated Banach fixed point theorem if one takes http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq17_HTML.gif , where A is a complete subset of X.

Let A and B be two nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq4_HTML.gif . We denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq19_HTML.gif the following sets:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ2_HTML.gif
(2)

where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq20_HTML.gif .

In [13], the authors presented sufficient conditions which determine when the sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq19_HTML.gif are nonempty. In [19], the author introduced the following definition.

Definition 1.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif be a pair of nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq4_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq22_HTML.gif . Then the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif is said to have the P-property if and only if for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq24_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ3_HTML.gif
(3)
It can be easily seen that for any nonempty subset A of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq4_HTML.gif , the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq25_HTML.gif has the P-property. In [19], the author proved that any pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property. Now, we introduce the class F of those functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq3_HTML.gif satisfying the following condition:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ4_HTML.gif
(4)

Definition 1.2 (See [6])

Let A, B be two nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq4_HTML.gif . A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq26_HTML.gif is said to be a Geraghty-contraction if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq27_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ5_HTML.gif
(5)

Theorem 1.2 (See [6])

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif be a pair of nonempty closed subsets of a complete metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq4_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq18_HTML.gif is nonempty. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq26_HTML.gif be a continuous Geraghty-contraction satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq28_HTML.gif . Suppose that the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif has the P-property. Then there exists a unique http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq29_HTML.gif in A such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq30_HTML.gif .

In the following section, we improve the theorem above by using a distance function ψ in Definition 1.2. In particular, we introduce Definition 2.1 and broaden the scope of Theorem 1.2 to ψ-Geraghty-contractions.

2 Main results

We start this section with the definition of the following auxiliary function. Let Ψ denote the class of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq31_HTML.gif which satisfy the following conditions:
  1. (a)

    ψ is nondecreasing;

     
  2. (b)

    ψ is subadditive, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq32_HTML.gif ;

     
  3. (c)

    ψ is continuous;

     
  4. (d)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq33_HTML.gif .

     

We introduce the following contraction.

Definition 2.1 Let A, B be two nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq4_HTML.gif . A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq26_HTML.gif is said to be a ψ-Geraghty contraction if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq27_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ6_HTML.gif
(6)
Remark 2.1 Notice that since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq3_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ7_HTML.gif
(7)

We are now ready to state and prove our main theorem.

Theorem 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif be a pair of nonempty closed subsets of a complete metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq4_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq18_HTML.gif is nonempty. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq26_HTML.gif be a ψ-Geraghty contraction satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq28_HTML.gif . Suppose that the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif has the P-property. Then there exists a unique http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq29_HTML.gif in A such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq30_HTML.gif .

Proof Regarding that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq18_HTML.gif is nonempty, we take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq34_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq35_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq36_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq37_HTML.gif . Analogously, regarding the assumption http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq38_HTML.gif , we determine http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq39_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq40_HTML.gif . Recursively, we obtain a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq41_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq18_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ8_HTML.gif
(8)
Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif has the P-property, we derive that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ9_HTML.gif
(9)
If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq42_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq43_HTML.gif , then the proof is completed. Indeed,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ10_HTML.gif
(10)
and consequently, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq44_HTML.gif . On the other hand, due to (8) we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equb_HTML.gif
Therefore, we conclude that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ11_HTML.gif
(11)
For the rest of the proof, we suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq45_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq46_HTML.gif . Since T is a ψ-Geraghty contraction, for any ℕ, we have that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ12_HTML.gif
(12)
Consequently, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq47_HTML.gif is a nonincreasing sequence and bounded below, and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq48_HTML.gif exists. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq49_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq50_HTML.gif . Then, from (6), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equc_HTML.gif
for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq51_HTML.gif , which implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equd_HTML.gif
On the other hand, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq27_HTML.gif , we conclude http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq52_HTML.gif , that is,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ13_HTML.gif
(13)
Notice that since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq53_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq54_HTML.gif , for fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq55_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq56_HTML.gif , and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif satisfies the P-property, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq57_HTML.gif . In what follows, we prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq41_HTML.gif is a Cauchy sequence. On the contrary, assume that we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ14_HTML.gif
(14)
By using the triangular inequality,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ15_HTML.gif
(15)
By (12) and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq58_HTML.gif , by the comment mentioned above, regarding the discussion on the P-property above together with (12), (15) and the property of the function ψ, we derive that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ16_HTML.gif
(16)
By a simple manipulation, (16) yields that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ17_HTML.gif
(17)
By taking the properties of the function ψ into account, together with (13) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq60_HTML.gif , the last inequality yields
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ18_HTML.gif
(18)
Therefore http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq61_HTML.gif . By taking the fact http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq27_HTML.gif into account, we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Eque_HTML.gif

Regarding the properties of the function ψ, the limit above contradicts the assumption (14). Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq41_HTML.gif is a Cauchy sequence.

Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq62_HTML.gif and A is a closed subset of the complete metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq4_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq63_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq64_HTML.gif .

We claim that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq65_HTML.gif . Suppose, on the contrary, that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq66_HTML.gif . This means that we can find http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq67_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq68_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq69_HTML.gif with
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ19_HTML.gif
(19)
Due to the properties of ψ, we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ20_HTML.gif
(20)
Using the fact that T is a ψ-Geraghty contraction, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ21_HTML.gif
(21)
for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq68_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq71_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq72_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq73_HTML.gif
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ22_HTML.gif
(22)
Consequently, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq73_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ23_HTML.gif
(23)

a contradiction. Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq65_HTML.gif .

Regarding the fact that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq74_HTML.gif is a constant sequence with value http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq75_HTML.gif , we derive
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ24_HTML.gif
(24)

which is equivalent to saying that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq29_HTML.gif is the best proximity point of T. This completes the proof of the existence of a best proximity point.

We shall show the uniqueness of the best proximity point of T. Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq76_HTML.gif are two distinct best proximity points of T, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq77_HTML.gif . This implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ25_HTML.gif
(25)
Using the P-property, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ26_HTML.gif
(26)
Using the fact that T is a ψ-Geraghty contraction, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equ27_HTML.gif
(27)

a contradiction. This completes the proof. □

Notice that the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq25_HTML.gif satisfies the P-property for any nonempty subset A of X. Consequently, we have the following corollary.

Corollary 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq1_HTML.gif be a complete metric space and A be a nonempty closed subset of X. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq78_HTML.gif be a ψ-Geraghty-contraction. Then T has a unique fixed point.

Proof Apply Theorem 2.1 with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq79_HTML.gif . □

If we take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq80_HTML.gif we obtain Theorem 1.2 as a corollary of Theorem 2.1.

Corollary 2.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq1_HTML.gif be a complete metric space and A be a nonempty closed subset of X. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq78_HTML.gif be a Geraghty-contraction. Then T has a unique fixed point.

Proof Apply Theorem 2.1 with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq80_HTML.gif . □

In order to illustrate our results, we present the following example.

Example 2.1 Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq81_HTML.gif with the metric
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equf_HTML.gif
and consider the closed subsets
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equg_HTML.gif

and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq83_HTML.gif .

Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq84_HTML.gif to be the mapping defined by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equh_HTML.gif

Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq85_HTML.gif , the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif has the P-property.

Notice that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq28_HTML.gif .

Without loss of generality, we assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq88_HTML.gif . Moreover,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equi_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equj_HTML.gif
We have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equk_HTML.gif

where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq89_HTML.gif is defined as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq90_HTML.gif .

Therefore, T is a ψ-Geraghty-contraction. Notice that the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq91_HTML.gif satisfies the P-property. Indeed, if
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equl_HTML.gif
then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq93_HTML.gif and hence
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equm_HTML.gif
Therefore, since the assumptions of Theorem 2.1 are satisfied, by Theorem 2.1 there exists a unique http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq94_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equn_HTML.gif

More precisely, the point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq95_HTML.gif is the best proximity point of T.

Example 2.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq97_HTML.gif be a metric on X. Suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq98_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq99_HTML.gif are two closed subsets of ℝ. Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq84_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq100_HTML.gif . Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq101_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq102_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq103_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq104_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq105_HTML.gif . Now we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equo_HTML.gif
Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq106_HTML.gif . Further, clearly, the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq21_HTML.gif has the P-property. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq107_HTML.gif . Note that, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq108_HTML.gif , then condition (6) holds. Hence, we assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq109_HTML.gif . We shall show that (6) holds. Suppose, on the contrary, there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq110_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equp_HTML.gif
and so
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_Equq_HTML.gif

which yields that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq111_HTML.gif , a contradiction. Therefore condition (6) holds for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq107_HTML.gif . Hence, the conditions of Theorem 2.1 hold and T has a unique best proximity point. Here, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_533_IEq112_HTML.gif is the best proximity point of T.

Declarations

Acknowledgements

The author expresses his gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, Atilim University

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© Karapınar; licensee Springer 2013

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