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Best proximity points of generalized almost ψGeraghty contractive nonselfmappings
Fixed Point Theory and Applications volume 2014, Article number: 32 (2014)
Abstract
In this paper, we introduce the new notion of almost ψGeraghty contractive mappings and investigate the existence of a best proximity point for such mappings in complete metric spaces via the weak Pproperty. 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.
1 Introduction and preliminaries
Nonselfmappings 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 nonselfcontraction 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 kcontraction 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 kcontraction 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 selfmapping 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 selfmapping defined on an abstract space. It is quite natural to investigate the existence and uniqueness of a nonselfmapping $T:A\to B$ which does not possess a fixed point. If a nonselfmapping $T:A\to B$ 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 nonselfmapping $T:A\to B$ 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 selfmapping.
Let $(X,d)$ be a metric space. Suppose that 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.
In 1973 Geraghty [33] introduced the class 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])
Let A, B be two nonempty subsets of a metric space $(X,d)$. A mapping $T:A\to B$ is said to be a Geraghtycontraction 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, Geraghtycontraction satisfying $T({A}_{0})\subseteq {B}_{0}$. Suppose that the pair $(A,B)$ has the Pproperty, then there exists a unique ${x}^{\ast}$ in A such that $d({x}^{\ast},T{x}^{\ast})=d(A,B)$.
The Pproperty 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 Pproperty 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 Pproperty. In [29], the author proved that any pair $(A,B)$ of nonempty closed convex subsets of a real Hilbert space H satisfies the Pproperty.
Recently, Zhang et al. [34] defined the following notion, which is weaker than the Pproperty.
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 Pproperty if and only if for any ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$,
Let Ψ denote the class of functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfying the following conditions:

(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 ψGeraghty 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 ψGeraghty 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 ψGeraghty contraction satisfying $T({A}_{0})\subseteq {B}_{0}$. Suppose that the pair $(A,B)$ has the Pproperty. 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 ψGeraghty contraction if there exist $\beta \in S$ and $\psi \in \mathrm{\Psi}$ such that
for all $x,y\in A$ where $L\ge 0$,
Now, we state and prove our main theorem about existence and uniqueness of a best proximity point for a nonselfmapping satisfying a generalized almost ψGeraghty 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 ψGeraghty contraction satisfying $T({A}_{0})\subseteq {B}_{0}$. Assume that the pair $(A,B)$ has the weak Pproperty. 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
Since the pair $(A,B)$ has the weak Pproperty, we deduce
Due to the triangle inequality together with the equality (11) we have
Analogously, combining the equalities (11) and (12) with the triangle inequality we obtain
Consequently, we have
Also note that
If there exists ${n}_{0}\in \mathbb{N}$ such that $d({x}_{{n}_{0}},{x}_{{n}_{0}+1})=0$, then the proof is completed. Indeed,
and consequently, $T{x}_{{n}_{0}1}=T{x}_{{n}_{0}}$. Therefore, we conclude that
For the rest of the proof, we suppose that $d({x}_{n},{x}_{n+1})>0$ for all $n\in \mathbb{N}$. In view of the fact that T is a generalized almost ψGeraghty contraction, we have
Taking into account the inequalities (14) and (18), we deduce that
If for some n, $max\{d({x}_{n1},{x}_{n}),d({x}_{n},{x}_{n+1})\}=d({x}_{n},{x}_{n+1})$, then we get
which is a contradiction. Therefore, we must have
for all $n\in \mathbb{N}$. Regarding the inequality (18), we see that
holds for all $n\in \mathbb{N}$. Since ψ is nondecreasing, then $d({x}_{n},{x}_{n+1})<d({x}_{n1},{x}_{n})$ 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
for each $n\ge 1$. Taking the limit of both sides as $n\to \mathrm{\infty}$, we find
On the other hand, since $\beta \in S$, we conclude ${lim}_{n\to \mathrm{\infty}}\psi (M({x}_{n},{x}_{n+1}))=0$, that is,
Since $d({x}_{n},T{x}_{n1})=d(A,B)$ holds for all $n\in \mathbb{N}$ and $(A,B)$ satisfies the weak Pproperty, then for all $m,n\in \mathbb{N}$, we can write
From (13), we deduce
By using ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=0$, we get
On the other hand,
Due to the fact that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=0$, we obtain
We shall show next that $\{{x}_{n}\}$ is a Cauchy sequence. Assume on the contrary that
Employing the triangular inequality and (22), we get
Combining (10) and (27), and regarding the properties of ψ, we obtain
From (23), (25), (28), and by using ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=0$, we have
So by (26), we get
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.
Since $\{{x}_{n}\}\subset A$ and 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
Taking the limit as $n\to \mathrm{\infty}$ in the above inequality, we obtain
Since ψ is nondecreasing and continuous, then
Also, letting $n\to \mathrm{\infty}$ in (13) results in
that is, ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=d(A,B)$. Then we get
and therefore
Further,
which implies
Therefore, combining (10), (29), (30), and (31) we deduce
Now, since $\psi (d({x}^{\ast},T{x}^{\ast})d(A,B))>0$, and making use of (32), we get
that is,
which implies
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.
We shall show next the uniqueness of the best proximity point of 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
where $d({x}^{\ast},{y}^{\ast})>0$. Due to the weak Pproperty of the pair $(A,B)$, we have
Observe that in this case
Also, note that
Using the fact that T is a generalized almost ψGeraghty contraction, we derive
If $M({x}^{\ast},{y}^{\ast})=d(A,B)$, due to the fact that $d({x}^{\ast},{y}^{\ast})>0$, the inequality above becomes
which implies $d({x}^{\ast},{y}^{\ast})=0$ and contradicts the assumption $d({x}^{\ast},{y}^{\ast})>0$. Else, if $M({x}^{\ast},{y}^{\ast})=d({x}^{\ast},{y}^{\ast})$, we deduce
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 Pproperty.
Consider
and $\psi (t)=\alpha t$ (with $\alpha \ge \frac{1}{2}$) for all $t\ge 0$. Note that $\beta \in S$ and $\psi \in \mathrm{\Psi}$. For all $x,y\in A$, we have
We shall show that T is a generalized almost ψGeraghty contraction. Without loss of generality, consider the case where $x\ge y$. Then we have $M(x,y)=2y$ and $d(Tx,Ty)=xy$.
In this case, we see that
Therefore
On the other hand, we know that $\psi (M(x,y))=2\alpha y\ge 1$ for all $x,y\in A$ with $x\ge y$. Hence,
and from (37) we deduce
Thus, all hypotheses of Theorem 2.1 are satisfied, and ${x}^{\ast}=1$ is the unique best proximity point of the map T.
On the other hand, 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 ψGeraghty 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 nonselfmapping 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 Pproperty. 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 nonselfmapping 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 Pproperty. 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 selfmapping 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.
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Keywords
 fixed point
 metric space
 best proximity point
 generalized almost ψGeraghty contractions