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# Best proximity points of generalized almost *ψ*-Geraghty contractive non-self-mappings

- Hassen Aydi
^{1}, - Erdal Karapınar
^{2, 3}, - İnci M Erhan
^{2}Email author and - Peyman Salimi
^{4}

**2014**:32

https://doi.org/10.1186/1687-1812-2014-32

© Aydi et al.; licensee Springer. 2014

**Received:**11 November 2013**Accepted:**24 January 2014**Published:**11 February 2014

## Abstract

In this paper, we introduce the new notion of almost *ψ*-Geraghty contractive mappings and investigate the existence of a best proximity point for such mappings in complete metric spaces via the weak *P*-property. We provide an example to validate our best proximity point theorem. The obtained results extend, generalize, and complement some known fixed and best proximity point results from the literature.

**MSC:**47H10, 54H25, 46J10, 46J15.

## Keywords

- fixed point
- metric space
- best proximity point
- generalized almost
*ψ*-Geraghty contractions

## 1 Introduction and preliminaries

Non-self-mappings are among the intriguing research directions in fixed point theory. This is evident from the increase of the number of publications related with such maps. A great deal of articles on the subject investigate the non-self-contraction mappings on metric spaces. Let $(X,d)$ be a metric space and *A* and *B* be nonempty subsets of *X*. A mapping $T:A\to B$ is said to be a *k*-contraction if there exists $k\in [0,1)$ such that $d(Tx,Ty)\le kd(x,y)$ for any $x,y\in A$. It is clear that a *k*-contraction coincides with the celebrated Banach fixed point theorem (Banach contraction principle) [1] if one takes $A=B$ where the induced metric space $(A,d{|}_{A})$ is complete.

In nonlinear analysis, the theory of fixed points is an essential instrument to solve the equation $Tx=x$ for a self-mapping *T* defined on a subset of an abstract space such as a metric space, a normed linear space or a topological vector space. Following the Banach contraction principle, most of the fixed point results have been proved for a self-mapping defined on an abstract space. It is quite natural to investigate the existence and uniqueness of a non-self-mapping $T:A\to B$ which does not possess a fixed point. If a non-self-mapping $T:A\to B$ has no fixed point, then the answer of the following question makes sense: Is there a point $x\in X$ such that the distance between *x* and *Tx* is closest in some sense? Roughly speaking, best proximity theory investigates the existence and uniqueness of such a closest point *x*. We refer the reader to [2–9] and [10–32] for further discussion of best proximity.

**Definition 1.1**Let $(X,d)$ be a metric space and $A,B\subset X$. We say that ${x}^{\ast}\in A$ is a best proximity point of the non-self-mapping $T:A\to B$ if the following equality holds:

where $d(A,B)=inf\{d(x,y):x\in A,y\in B\}$.

It is clear that the notion of a fixed point coincided with the notion of a best proximity point when the underlying mapping is a self-mapping.

*A*and

*B*are nonempty subsets of a metric space $(X,d)$. We define the following sets:

In [17], the authors presented sufficient conditions for the sets ${A}_{0}$ and ${B}_{0}$ to be nonempty.

*S*of functions $\beta :[0,\mathrm{\infty})\to [0,1)$ satisfying the following condition:

The author defined contraction mappings via functions from this class and proved the following result.

**Theorem 1.1** (Geraghty [33])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be an operator*.

*If*

*T*

*satisfies the following inequality*:

*where* $\beta \in S$, *then* *T* *has a unique fixed point*.

Recently, Caballero *et al.* [6] introduced the following contraction.

**Definition 1.2** ([6])

*A*,

*B*be two nonempty subsets of a metric space $(X,d)$. A mapping $T:A\to B$ is said to be a Geraghty-contraction if there exists $\beta \in S$ such that

Based on Definition 1.2, the authors [6] obtained the following result.

**Theorem 1.2** (See [6])

*Let* $(A,B)$ *be a pair of nonempty closet subsets of a complete metric space* $(X,d)$ *such that* ${A}_{0}$ *is nonempty*. *Let* $T:A\to B$ *be a continuous*, *Geraghty*-*contraction satisfying* $T({A}_{0})\subseteq {B}_{0}$. *Suppose that the pair* $(A,B)$ *has the* *P*-*property*, *then there exists a unique* ${x}^{\ast}$ *in* *A* *such that* $d({x}^{\ast},T{x}^{\ast})=d(A,B)$.

The *P*-property mentioned in the theorem above has been introduced in [29].

**Definition 1.3**Let $(A,B)$ be a pair of nonempty subsets of a metric space $(X,d)$ with ${A}_{0}\ne \mathrm{\varnothing}$. Then the pair $(A,B)$ is said to have the

*P*-property if and only if for any ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$,

It is easily seen that for any nonempty subset *A* of $(X,d)$, the pair $(A,A)$ has the *P*-property. In [29], the author proved that any pair $(A,B)$ of nonempty closed convex subsets of a real Hilbert space *H* satisfies the *P*-property.

Recently, Zhang *et al.* [34] defined the following notion, which is weaker than the *P*-property.

**Definition 1.4**Let $(A,B)$ be a pair of nonempty subsets of a metric space $(X,d)$ with ${A}_{0}\ne \mathrm{\varnothing}$. Then the pair $(A,B)$ is said to have the weak

*P*-property if and only if for any ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$,

- (a)
*ψ*is nondecreasing; - (b)
*ψ*is subadditive, that is, $\psi (s+t)\le \psi (s)+\psi (t)$; - (c)
*ψ*is continuous; - (d)
$\psi (t)=0\iff t=0$.

The notion of *ψ*-Geraghty contraction has been introduced very recently in [11], as an extension of Definition 1.2.

**Definition 1.5**Let

*A*,

*B*be two nonempty subsets of a metric space $(X,d)$. A mapping $T:A\to B$ is said to be a

*ψ*-Geraghty contraction if there exist $\beta \in S$ and $\psi \in \mathrm{\Psi}$ such that

**Remark 1.1**Notice that since $\beta :[0,\mathrm{\infty})\to [0,1)$, we have

In [11], the author also proved the following best proximity point theorem.

**Theorem 1.3** (See [11])

*Let* $(A,B)$ *be a pair of nonempty closed subsets of a complete metric space* $(X,d)$ *such that* ${A}_{0}$ *is nonempty*. *Let* $T:A\to B$ *be a* *ψ*-*Geraghty contraction satisfying* $T({A}_{0})\subseteq {B}_{0}$. *Suppose that the pair* $(A,B)$ *has the* *P*-*property*. *Then there exists a unique* ${x}^{\ast}$ *in* *A* *such that* $d({x}^{\ast},T{x}^{\ast})=d(A,B)$.

## 2 Main results

Our main results are based on the following definition which is a generalization of Definition 1.5.

**Definition 2.1**Let

*A*,

*B*be two nonempty subsets of a metric space $(X,d)$. A mapping $T:A\to B$ is said to be a generalized almost

*ψ*-Geraghty contraction if there exist $\beta \in S$ and $\psi \in \mathrm{\Psi}$ such that

Now, we state and prove our main theorem about existence and uniqueness of a best proximity point for a non-self-mapping satisfying a generalized almost *ψ*-Geraghty contraction.

**Theorem 2.1** *Let* $(A,B)$ *be a pair of nonempty closed subsets of a complete metric space* $(X,d)$ *such that* ${A}_{0}$ *is nonempty*. *Let* $T:A\to B$ *be a generalized almost* *ψ*-*Geraghty contraction satisfying* $T({A}_{0})\subseteq {B}_{0}$. *Assume that the pair* $(A,B)$ *has the weak* *P*-*property*. *Then* *T* *has a unique best proximity point in* *A*.

*Proof*Since the subset ${A}_{0}$ is not empty, we can take ${x}_{0}$ in ${A}_{0}$. Taking into account that $T{x}_{0}\in T({A}_{0})\subseteq {B}_{0}$, we can find ${x}_{1}\in {A}_{0}$ such that $d({x}_{1},T{x}_{0})=d(A,B)$. Further, since $T{x}_{1}\in T({A}_{0})\subseteq {B}_{0}$, it follows that there is an element ${x}_{2}$ in ${A}_{0}$ such that $d({x}_{2},T{x}_{1})=d(A,B)$. Recursively, we obtain a sequence $\{{x}_{n}\}$ in ${A}_{0}$ satisfying

*P*-property, we deduce

*T*is a generalized almost

*ψ*-Geraghty contraction, we have

*n*, $max\{d({x}_{n-1},{x}_{n}),d({x}_{n},{x}_{n+1})\}=d({x}_{n},{x}_{n+1})$, then we get

*ψ*is nondecreasing, then $d({x}_{n},{x}_{n+1})<d({x}_{n-1},{x}_{n})$ for all

*n*. Consequently, the sequence $\{d({x}_{n},{x}_{n+1})\}$ is decreasing and is bounded below and hence ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=s\ge 0$ exists. Assume that $s>0$. Rewrite (20) as

*P*-property, then for all $m,n\in \mathbb{N}$, we can write

*ψ*, we obtain

that is, ${lim}_{m,n\to \mathrm{\infty}}\beta (\psi (M({x}_{n},{x}_{m})))=1$. Therefore, ${lim}_{m,n\to \mathrm{\infty}}M({x}_{n},{x}_{m})=0$. This implies that ${lim}_{m,n\to \mathrm{\infty}}d({x}_{n},{x}_{m})=0$, which is a contradiction. Therefore, $\{{x}_{n}\}$ is a Cauchy sequence.

*A*is a closed subset of the complete metric space $(X,d)$, we can find ${x}^{\ast}\in A$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. We shall show that $d({x}^{\ast},T{x}^{\ast})=d(A,B)$. If ${x}^{\ast}=T{x}^{\ast}$, then $A\cap B\ne \mathrm{\varnothing}$, and $d({x}^{\ast},T{x}^{\ast})=d(A,B)=0$,

*i.e.*, ${x}^{\ast}$ is a best proximity point of

*T*. Hence, we assume that $d({x}^{\ast},T{x}^{\ast})>0$. Suppose on the contrary that ${x}^{\ast}$ is not a best proximity point of

*T*, that is, $d({x}^{\ast},T{x}^{\ast})>d(A,B)$. First note that

*ψ*is nondecreasing and continuous, then

and so $d({x}^{\ast},T{x}^{\ast})=0>d(A,B)$, which is a contradiction. Therefore, $d({x}^{\ast},T{x}^{\ast})\le d(A,B)$, that is, $d({x}^{\ast},T{x}^{\ast})=d(A,B)$. In other words, ${x}^{\ast}$ is a best proximity point of *T*. This completes the proof of the existence of a best proximity point.

*T*. Suppose that ${x}^{\ast}$ and ${y}^{\ast}$ are two best proximity points of

*T*, such that ${x}^{\ast}\ne {y}^{\ast}$. This implies that

*P*-property of the pair $(A,B)$, we have

*T*is a generalized almost

*ψ*-Geraghty contraction, we derive

which is not possible, since *ψ* is nondecreasing. Therefore, we must have $d({x}^{\ast},{y}^{\ast})=0$. This completes the proof. □

To illustrate our result given in Theorem 2.1, we present the following example, which shows that Theorem 2.1 is a proper generalization of Theorem 1.2.

**Example 2.1**Consider the space $X=\mathbb{R}$ with Euclidean metric. Take the sets

Obviously, $d(A,B)=2$. Let $T:A\to B$ be defined by $Tx=-x$. Notice that ${A}_{0}=\{-1\}$, ${B}_{0}=\{1\}$ and $T({A}_{0})\subseteq {B}_{0}$. Also, it is clear that the pair $(A,B)$ has the weak *P*-property.

We shall show that *T* is a generalized almost *ψ*-Geraghty contraction. Without loss of generality, consider the case where $x\ge y$. Then we have $M(x,y)=-2y$ and $d(Tx,Ty)=x-y$.

Thus, all hypotheses of Theorem 2.1 are satisfied, and ${x}^{\ast}=-1$ is the unique best proximity point of the map *T*.

*T*is not a Geraghty contraction. Indeed, taking $x=-1$ and $y=-2$, we get

Then Theorem 1.2 (the main result of Caballero *et al.* [6]) is not applicable.

Similarly, we cannot apply Theorem 1.3 because *T* is not a *ψ*-Geraghty contraction. Let $x=-1$, $y=-2$ and $\psi (t)=\alpha t$ with $\alpha <2$. Then *T* does not satisfy (8).

If in Theorem 2.1 we take $\psi (t)=t$ for all $t\ge 0$, then we deduce the following corollary.

**Corollary 2.1**

*Let*$(A,B)$

*be a pair of nonempty closed subsets of a complete metric space*$(X,d)$

*such that*${A}_{0}$

*is nonempty*.

*Let*$T:A\to B$

*be a non*-

*self*-

*mapping satisfying*$T({A}_{0})\subseteq {B}_{0}$

*and*

*for all*$x,y\in A$

*where*$\beta \in S$, $L\ge 0$,

*Assume that the pair* $(A,B)$ *has the weak* *P*-*property*. *Then* *T* *has a unique best proximity point in* *A*.

If further in the above corollary we take $\beta (t)=r$ where $0\le r<1$, then we deduce another particular result.

**Corollary 2.2**

*Let*$(A,B)$

*be a pair of nonempty closed subsets of a complete metric space*$(X,d)$

*such that*${A}_{0}$

*is nonempty*.

*Let*$T:A\to B$

*be a non*-

*self*-

*mapping satisfying*$T({A}_{0})\subseteq {B}_{0}$

*and*

*for all*$x,y\in A$

*where*$0\le r<1$, $L\ge 0$,

*Assume that the pair* $(A,B)$ *has the weak* *P*-*property*. *Then* *T* *has a unique best proximity point in* *A*.

## 3 Application to fixed point theory

The case $A=B$ in Theorem 2.1 corresponds to a self-mapping and results in an existence and uniqueness theorem for a fixed point of the map *T*. We state this case in the next theorem.

**Theorem 3.1**

*Let*$(X,d)$

*be a complete metric space*.

*Suppose that*

*A*

*is a nonempty closed subset of*

*X*.

*Let*$T:A\to A$

*be a mapping such that*

*where*$\psi \in \mathrm{\Psi}$, $\beta \in S$, $L\ge 0$,

*Then* *T* *has a unique fixed point*.

Finally, taking $\psi (t)=t$ in Theorem 3.1, we get another fixed point result.

**Corollary 3.1**

*Let*$(X,d)$

*be a complete metric space*.

*Suppose that*

*A*

*is a nonempty closed subset of*

*X*.

*Let*$T:A\to A$

*be a mapping such that*

*where*$\beta \in S$, $L\ge 0$,

*Then* *T* *has a unique fixed point*.

**Remark 3.1** The best proximity theorem given in this work, more precisely Theorem 2.1, is a quite general result. It is a generalization of Theorem 2.1 in [14], Theorem 8 in [5], and also Theorem 1.2 given in Section 1. In addition, Corollary 3.1 improves Theorem 1.1.

**Remark 3.2** Very recently, Karapınar and Samet [15] proved that the function ${d}_{\phi}=\phi \circ d$ on the set *X*, where $\phi \in \mathrm{\Psi}$ is also a metric on *X*. Therefore, some of the fixed theorems regarding contraction mappings defined via auxiliary functions from the set Ψ can be in fact deduced from the existing ones in the literature. However, our main result given in Theorem 2.1 is not a consequence of any existing theorems due to the fact that the contraction condition contains the term $d(A,B)$.

On the other hand, the definition of ${d}_{\phi}=\phi \circ d$ can be used to show that Theorem 3.1 follows from Corollary 3.1. Nevertheless, Corollary 3.1 and hence Theorem 3.1 are still new results.

## Declarations

### Acknowledgements

The authors thank to the referees for their careful reading and valuable comments and remarks which contributed to the improvement of the article.

## Authors’ Affiliations

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