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A short note on the equivalence between ‘best proximity’ points and ‘fixed point’ results
Journal of Inequalities and Applications volume 2014, Article number: 246 (2014)
Abstract
In this short note, we notice that, unexpectedly, some existing fixed point results and recently announced best proximity point results are equivalent.
MSC:41A65, 90C30, 47H10.
1 Introduction and preliminaries
In 1973 Geraghty [1] 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 [1])
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.
Theorem 1.1 was generalized in several ways, see e.g. [2–6]. Recently, Caballero et al. [2] introduced the following contraction.
Definition 1.1 ([7])
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
For the sake of completeness, we recall some basic definitions and fundamental results.
Let (X,d) be a metric space and (A,B) a pair of nonempty subsets of X. We consider the following notations:
Through this paper, ℕ denotes the set of natural numbers.
In [8], Sadiq Basha introduced the following concept.
Definition 1.2 We say that B is approximatively compact with respect to A if and only if every sequence \{{y}_{n}\}\subset B satisfying the condition that {lim}_{n\to \mathrm{\infty}}d(x,{y}_{n})=d(x,B) for some x in A, has a convergent subsequence.
Definition 1.3 A mapping g:A\to A is called an isometry if
Definition 1.4 (see e.g. [9])
Given a mapping T:A\to B and an isometry g:A\to A, the mapping T is said to preserve the isometric distance with respect to g if and only if
Denote by Ξ the set of functions \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying the following conditions:

(I)
φ is continuous and nondecreasing;

(II)
\phi (0)=0;

(III)
{lim}_{t\to \mathrm{\infty}}\phi (t)=\mathrm{\infty}.
The following notions were introduced by Sadiq Basha [9].
Definition 1.5 A mapping T:A\to B is said to be a generalized proximal contraction of the first kind if and only if
where x,y,u,v\in A and \phi \in \mathrm{\Xi}.
Definition 1.6 A mapping T:A\to B is said to be a generalized proximal contraction of the second kind if and only if
where x,y,u,v\in A and \phi \in \mathrm{\Xi}.
Inspired by these definitions, AminiHarandi [3] introduced the following definition.
Denote by Ψ the set of functions \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying the following conditions:

(I)
ψ is continuous and nondecreasing;

(II)
\psi (0)=0;

(III)
t\le \psi (t) for each t\ge 0.
Definition 1.7 A mapping T:A\to B is said to be a generalized Geraghty proximal contraction of the first kind if and only if
where x,y,u,v\in A and \psi \in \mathrm{\Psi}, \beta \in S.
Definition 1.8 A mapping T:A\to B is said to be a generalized Geraghty proximal contraction of the second kind if and only if
where x,y,u,v\in A and \psi \in \mathrm{\Psi}, \beta \in S.
The main result in [3] is the following.
Theorem 1.2 Let A and B be two nonempty closed subsets of a complete metric space (X,d) such that {A}_{0}\ne \mathrm{\varnothing}, {B}_{0}\ne \mathrm{\varnothing} and B is approximatively compact with respect to A. Suppose that the mappings g:A\to A and T:A\to B satisfy the following conditions:

(i)
T is a generalized Geraghty proximal contraction of the first kind;

(ii)
T({A}_{0})\subseteq {B}_{0};

(iii)
g is an isometry;

(iv)
{A}_{0}\subseteq g({A}_{0}).
Then there exists a unique element {x}^{\ast}\in A such that
Further, for any fixed element {x}_{0}\in {A}_{0}, the iterative sequence \{{x}_{n}\}\subset {A}_{0}, defined by
converges to {x}^{\ast}.
In this manuscript, we shall show that Theorem 1.2 is a particular case of existing fixed point theorems in the literature. Hence, the main result of [3] is not a real generalization.
2 Some useful results
Denote by Φ the set of functions \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying the following conditions:

(I)
ϕ is continuous and nondecreasing;

(II)
\varphi (t)=0 if and only if t=0.
First we show that we get the more general form of the main result in [3] by replacing the class of distance functions Ψ by Φ in Definition 1.7.
Theorem 2.1 Let A and B be two nonempty closed subsets of a complete metric space (X,d) such that {A}_{0}\ne \mathrm{\varnothing}, {B}_{0}\ne \mathrm{\varnothing}, and B is approximatively compact with respect to A. Suppose that the mappings g:A\to A and T:A\to B satisfy the following conditions:

(i)
\begin{array}{r}d(u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B)\end{array}\}\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\psi (d(u,v))\le \beta (d(x,y))\psi (d(x,y)),
where x,y,u,v\in A and \psi \in \mathrm{\Phi}, \beta \in S.

(ii)
T({A}_{0})\subseteq {B}_{0};

(iii)
g is an isometry;

(iv)
{A}_{0}\subseteq g({A}_{0}).
Then there exists a unique element {x}^{\ast}\in A such that
Further, for any fixed element {x}_{0}\in {A}_{0}, the iterative sequence \{{x}_{n}\}\subset {A}_{0}, defined by
converges to {x}^{\ast}.
Proof By following the lines in the proof of Theorem 3.1 in the paper of AminiHarandi [3], we conclude that
It is sufficient to prove that \{{x}_{n}\} is a Cauchy sequence.
Suppose, on the contrary, that \{{x}_{n}\} is not a Cauchy sequence. Then there exists \epsilon >0 for which we can find subsequences \{{x}_{m(k)}\} and \{{x}_{n(k)}\} of \{{x}_{n}\} such that n(k)>m(k)>k and
Furthermore, we can choose n(k), associated with m(k), is the smallest integer which satisfies n(k)>m(k)>k and (5). Consequently, we have
Due to Lemma 2.4, we conclude that
So, we obtain
Letting k\to \mathrm{\infty} in the inequality above, we get
since ψ is continuous and (7) holds. Due to the property of ψ, \psi (\epsilon )>0, we derive
from the last inequality above. Since \beta \in S, we conclude that
Due to (7) we get \epsilon =0, a contradiction. Hence, \{{x}_{n}\} is Cauchy.
The rest follows from the corresponding lines in the proof of Theorem 3.1 in the paper of AminiHarandi [3]. □
The following theorem is due to [10].
Theorem 2.2 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 and \psi \in \mathrm{\Phi}, then T has a unique fixed point.
The following concept was introduced by Ćirić in [11].
Definition 2.1 Let (X,d) be a metric space and f:X\to X be a selfmapping. We say that X is forbitally complete if and only if for any x\in X, if \{{f}^{n}x\} is a Cauchy sequence, then it converges to some element in X.
It is evident that in Theorem 2.2, the notion of completeness of the metric space (X,d) can be replaced by the notion of forbitally completeness. Consequently, we derive the following fixed point result.
Lemma 2.1 (cf. [7])
Let (X,d) be a forbitally complete metric space, where f:X\to X is a selfmapping satisfying the following condition:
where \psi \in \mathrm{\Psi}, \beta \in S. Then f has a unique fixed point {x}^{\ast}\in X. Moreover, for any x\in X, the sequence \{{f}^{n}x\} converges to {x}^{\ast}.
Regarding the analogy with the proof of Lemma 2.2 in [7].
Lemma 2.2 (cf. [7])
Let (A,B) be a pair of closed subsets of a metric space (X,d). Suppose that the following conditions hold:

(i)
{A}_{0}\ne \mathrm{\varnothing};

(ii)
B is approximatively compact with respect to A.
Then the set {A}_{0} is closed.
Lemma 2.3 (cf. [7])
Let A and B be nonempty subsets of a metric space (X,d) such that {A}_{0}\ne \mathrm{\varnothing}. Suppose that the mappings g:A\to A and T:A\to B satisfy the following conditions:

(i)
T is a generalized Geraghtyproximal contraction of the first kind;

(ii)
T({A}_{0})\subseteq {B}_{0};

(iii)
g is an isometry;

(iv)
{A}_{0}\subseteq g({A}_{0}).
Then there exists a selfmapping f:{A}_{0}\to {A}_{0} satisfying the condition:
Regarding the analogy with the proof of Lemma 2.4 in [7].
Lemma 2.4 ([12])
Let (X,d) be a metric space and let ({x}_{n}) be a sequence in X such that (d({x}_{n+1},{x}_{n})) is nonincreasing and
If ({x}_{n}) is not a Cauchy sequence, then there exist \epsilon >0 and two sequences ({m}_{k}) and ({n}_{k}) of positive integers such that {m}_{k}>{n}_{k}\ge k and the following four sequences tend to ε when k\to +\mathrm{\infty}:
3 Main results
Theorem 3.1 Theorem 2.1 is a consequence of Theorem 2.2.
Proof Suppose that all the assumptions of Theorem 1.2 are satisfied. From Lemma 2.3, there exists a selfmapping f:{A}_{0}\to {A}_{0} satisfying (13). Then, for all (x,y)\in {A}_{0}\times {A}_{0}, we have
Since T is a generalized Geraghtyproximal contraction of the first kind and g is an isometry, we obtain
for every pair (x,y)\in {A}_{0}\times {A}_{0}. Thus f satisfies inequality (11).
Since (X,d) is complete, ({A}_{0},d) is also complete. From Theorem 2.2, the selfmapping h:{A}_{0}\to {A}_{0} has a unique fixed point {x}^{\ast}\in {A}_{0}.
Note that from (13), since T is a generalized Geraghtyproximal contraction of the first kind and g is an isometry, we have {x}^{\ast}\in {A}_{0} is a fixed point of f if and only if {x}^{\ast}\in A and d(g{x}^{\ast},T{x}^{\ast})=d(A,B). Then there exists a unique {x}^{\ast}\in {A}_{0} such that d(g{x}^{\ast},T{x}^{\ast})=d(A,B). Now, let a\in {A}_{0} be an arbitrary point. Consider a sequence \{{a}_{n}\}\subset {A}_{0} satisfying
Since T is a generalized Geraghtyproximal contraction of the first kind and g is an isometry, it follows from (13) that
From Lemma 2.1, we have
This ends the proof. □
4 Consequences
Theorem 4.1 Let A and B be two nonempty closed subsets of a complete metric space (X,d) such that {A}_{0}\ne \mathrm{\varnothing} and {B}_{0}\ne \mathrm{\varnothing}. Suppose that the mappings g:A\to A and T:A\to B satisfy the following conditions:

(i)
T is a generalized Geraghty proximal contraction of the first and second kind,
\begin{array}{c}\begin{array}{r}d(u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B)\end{array}\}\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\psi (d(u,v))\le \beta (d(x,y))\psi (d(x,y)),\hfill \\ \begin{array}{r}d(u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B)\end{array}\}\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d(Tu,Tv)\le \beta (d(Tx,Ty))\psi (d(Tx,Ty)),\hfill \end{array}
where x,y,u,v\in A and \psi \in \mathrm{\Phi}, \beta \in S;

(ii)
T({A}_{0})\subseteq {B}_{0};

(iii)
g is an isometry;

(iv)
{A}_{0}\subseteq g({A}_{0}).
Then there exists a unique element {x}^{\ast}\in A such that
Further, for any fixed element {x}_{0}\in {A}_{0}, the iterative sequence \{{x}_{n}\}\subset {A}_{0}, defined by
converges to {x}^{\ast}.
Theorem 4.2 Theorem 4.1 is a consequence of Lemma 2.1.
Proof Suppose that all the assumptions of Theorem 4.1 are satisfied. From Lemma 2.3, there exists a selfmapping f:{A}_{0}\to {A}_{0} satisfying (13). Then, for all (x,y)\in {A}_{0}\times {A}_{0}, we have
Since T is a generalized Geraghtyproximal contraction of the first kind and g is an isometry, we obtain
for every pair (x,y)\in {A}_{0}\times {A}_{0}. Thus f satisfies inequality (11).
Now, we shall prove that ({A}_{0},d) is forbitally complete. Indeed, let {x}_{0}\in {A}_{0} and consider the sequence \{{x}_{n}\}\subset {A}_{0} defined by {x}_{n}={f}^{n}{x}_{0} for all n\in \mathbb{N}. Suppose that \{{x}_{n}\} is a Cauchy sequence, we have to prove that \{{x}_{n}\} converges to some element in {A}_{0}. Since (X,d) is complete and A is closed, there exists some z\in A such that
By the definition of f, for all n\in \mathbb{N}, we have
which implies (since T is a generalized Geraghtyproximal contraction of the second kind) that
for all n\in \mathbb{N}. Since T preserves the isometric distance with respect to g, we obtain
Following the same lines as the proof of Theorem 3.1 in [3], one can show that \{T{x}_{n}\} is a Cauchy sequence in the complete metric space (X,d). Since B is closed, there exists some b\in B such that
Now, using (14), (15), and the definition of f, we get
Note that since g is an isometry, it is continuous. Now, we have
This implies that gz\in {A}_{0}. On the other hand, since {A}_{0}\subseteq g({A}_{0}) and g is an isometry, we obtain z\in {A}_{0}. Thus, we proved that {A}_{0} is forbitally complete. Now, applying Lemma 2.1, we find that f has a unique fixed point {x}^{\ast}\in {A}_{0}.
The rest follows from the lines of the proof of Theorem 3.1. □
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Acknowledgements
The authors thank the Visiting Professor Programming at King Saud University for funding this work. The authors thank to anonymous referees for their remarkable comments, suggestion and ideas that helps to improve this paper.
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Jleli, M., Karapınar, E. & Samet, B. A short note on the equivalence between ‘best proximity’ points and ‘fixed point’ results. J Inequal Appl 2014, 246 (2014). https://doi.org/10.1186/1029242X2014246
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DOI: https://doi.org/10.1186/1029242X2014246