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Best proximity point theorems for rational proximal contractions
Fixed Point Theory and Applications volume 2013, Article number: 95 (2013)
Abstract
We provide sufficient conditions which warrant the existence and uniqueness of the best proximity point for two new types of contractions in the setting of metric spaces. The presented results extend, generalize and improve some known results from best proximity point theory and fixedpoint theory. We also give some examples to illustrate and validate our definitions and results.
MSC:41A65, 46B20, 47H10.
1 Introduction
Let (\mathcal{X},d) be a metric space and \mathcal{T} be a selfmapping defined on a subset of \mathcal{X}. In this setting, the fixedpoint theory is an important tool for solving equations of the kind \mathcal{T}x=x, whose solutions are the fixed points of the mapping \mathcal{T}. On the other hand, if \mathcal{T} is not a selfmapping, say \mathcal{T}:\mathcal{A}\to \mathcal{B} where \mathcal{A} and ℬ are nonempty subsets of \mathcal{X}, then \mathcal{T} does not necessarily have a fixed point. Consequently, the equation \mathcal{T}x=x could have no solutions, and in this case, it is of a certain interest to determine an element x that is in some sense closest to \mathcal{T}x. Thus, we can say that the aim of the best proximity point theorems is to provide sufficient conditions to solve a minimization problem. In view of the fact that d(x,\mathcal{T}x) is at least d(\mathcal{A},\mathcal{B}):=inf\{d(x,y):x\in \mathcal{A}\text{and}y\in \mathcal{B}\}, a best proximity point theorem concerns the global minimum of the real valued function x\to d(x,\mathcal{T}x), that is, an indicator of the error involved for an approximate solution of the equation \mathcal{T}x=x, by complying the condition d(x,\mathcal{T}x)=d(\mathcal{A},\mathcal{B}). The notation of best proximity point is introduced in [1] but one of the most interesting results in this direction is due to Fan [2] and can be stated as follows.
Theorem 1.1 Let \mathcal{K} be a nonempty, compact and convex subset of a normed space ℰ. Then for any continuous mapping \mathcal{T}:\mathcal{K}\to \mathcal{E}, there exists x\in \mathcal{K} with \parallel x\mathcal{T}x\parallel ={inf}_{y\in \mathcal{K}}\parallel \mathcal{T}xy\parallel.
Some generalizations and extensions of this theorem appeared in the literature by Prolla [3], Reich [4], Sehgal and Singh [5, 6], Vetrivel et al. [7] and others. It turns out that many of the contractive conditions which are investigated for fixed points ensure the existence of best proximity points. Some results of this kind are obtained in [1, 5–40]. Note that the authors often, in proving these results, assume restrictive compactness hypotheses on the domain and codomain of the involved nonselfmapping. Inspired by [29], we consider these hypotheses too restrictive in dealing with proximal contractions and so we prove that the compactness hypotheses can be successfully replaced by standard completeness hypotheses. Following this idea, we propose a new type of condition to study the existence and uniqueness of the best proximity point of a nonselfmapping by assuming both compactness hypotheses and standard completeness hypotheses. Precisely, we introduce the notions of rational proximal contractions of the first and second kinds, then we establish some corresponding best proximity point theorems for such contractions. Our definitions include some earlier definitions as special cases. In particular, the presented theorems contain the results given in [29].
2 Preliminaries
In this section, we give some basic notations and definitions that will be used in the sequel.
Let (\mathcal{X},d) be a metric space, \mathcal{A} and ℬ be two nonempty subsets of \mathcal{X} and \mathcal{T}:\mathcal{A}\to \mathcal{B} be a nonselfmapping. We denote by {B}_{\mathrm{est}}(\mathcal{T}) the set of all best proximity points of \mathcal{T}, that is,
Also, let
and
Sufficient conditions to ensure that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty are given in [41]. Also, observe that if \mathcal{A} and ℬ are closed subsets of a normed linear space such that d(\mathcal{A},\mathcal{B})>0, then {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are contained in the boundaries of \mathcal{A} and ℬ, respectively; see [27].
Now, we give sequentially two definitions that are essential to state and prove our main results.
Definition 2.1 Let (\mathcal{X},d) be a metric space and \mathcal{A} and ℬ be two nonempty subsets of \mathcal{X}. Then \mathcal{T}:\mathcal{A}\to \mathcal{B} is said to be a rational proximal contraction of the first kind if there exist nonnegative real numbers α, β, γ, δ with \alpha +\beta +2\gamma +2\delta <1, such that the conditions
imply that
for all {u}_{1},{u}_{2},{x}_{1},{x}_{2}\in \mathcal{A}.
Note that, if \beta =0, then from (1) we get the definition of the generalized proximal contraction of the first kind with \alpha +2\gamma +2\delta <1; see [29].
Moreover, if \mathcal{T} is a selfmapping on \mathcal{A}, then the requirement in Definition 2.1 reduces to the following generalized contractive condition of rational type useful in establishing a fixedpoint theorem:
Definition 2.2 Let (\mathcal{X},d) be a metric space and \mathcal{A} and ℬ be two nonempty subsets of \mathcal{X}. Then \mathcal{T}:\mathcal{A}\to \mathcal{B} is said to be a rational proximal contraction of the second kind if there exist nonnegative real numbers α, β, γ, δ with \alpha +\beta +2\gamma +2\delta <1 such that the conditions
imply that
for all {u}_{1},{u}_{2},{x}_{1},{x}_{2}\in \mathcal{A}.
Note that, if \beta =0, then from (2) we get the definition of the generalized proximal contraction of the second kind with \alpha +2\gamma +2\delta <1, see [29].
The following example illustrates that a rational proximal contraction of the second kind is not necessarily a rational proximal contraction of the first kind. Therefore, both Definitions 2.1 and 2.2 are consistent.
Example 2.1 Let \mathcal{X}=\mathbb{R}\times \mathbb{R} endowed with the usual metric
for all ({x}_{1},{x}_{2}),({y}_{1},{y}_{2})\in \mathbb{R}\times \mathbb{R}. Define \mathcal{A}:=\{(x,1):x\in \mathbb{R}\} and \mathcal{B}:=\{(x,1):x\in \mathbb{R}\}. Also define \mathcal{T}:\mathcal{A}\to \mathcal{B} by
Then d(\mathcal{A},\mathcal{B})=2 and \mathcal{T} is a rational proximal contraction of the second kind but not a rational proximal contraction of the first kind. Indeed, using Definition 2.2 and after routine calculations, one can show that the lefthand side of inequality (2) is equal to 0. On the other hand, using Definition 2.1 and after routine calculations, one can show that the lefthand side of inequality (1) is equal to 2 and so inequality (1) is not satisfied for all nonnegative real numbers α, β, γ, δ with \alpha +\beta +2\gamma +2\delta <1.
It is well known that the notion of approximative compactness plays an important role in the theory of approximation [12]. In particular, the notion of an approximatively compact set was introduced by Efimov and Stechkin [16] and the properties of approximatively compact sets have been largely studied. The boundendly compact sets that are the sets whose intersection with any closed ball is compact are useful examples of approximatively compact sets. It is shown in [14] that in every infinitedimensional separable Banach space there exists a bounded approximatively compact set, which is not compact.
Remark 2.1 Since (\mathcal{X},d) is a metric space, the bounded compactness of a set is equivalent to its closure and the possibility of selecting from any bounded sequence contained in it a converging subsequence.
Here, for our further use, we give the following definition.
Definition 2.3 Let (\mathcal{X},d) be a metric space and \mathcal{A} and ℬ be two nonempty subsets of \mathcal{X}. Then ℬ is said to be approximatively compact with respect to \mathcal{A} if every sequence \{{y}_{n}\} of ℬ, satisfying the condition d(x,{y}_{n})\to d(x,\mathcal{B}) for some x in \mathcal{A}, has a convergent subsequence.
Obviously, any set is approximatively compact with respect to itself.
3 Rational proximal contractions
Our first main result is the following best proximity point theorem for a rational proximal contraction of the first kind.
Theorem 3.1 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that ℬ is approximatively compact with respect to \mathcal{A}. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a rational proximal contraction of the first kind;

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists x\in \mathcal{A} such that {B}_{\mathrm{est}}(\mathcal{T})=\{x\}. Further, for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x.
Proof Let {x}_{0}\in {\mathcal{A}}_{0} (such a point there exists since {\mathcal{A}}_{0}\ne \mathrm{\varnothing}). Since \mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}, then by the definition of {\mathcal{B}}_{0}, there exists {x}_{1}\in {\mathcal{A}}_{0} such that d({x}_{1},\mathcal{T}{x}_{0})=d(\mathcal{A},\mathcal{B}). Again, since \mathcal{T}{x}_{1}\in {\mathcal{B}}_{0}, it follows that there is {x}_{2}\in {\mathcal{A}}_{0} such that d({x}_{2},\mathcal{T}{x}_{1})=d(\mathcal{A},\mathcal{B}). Continuing this process, we can construct a sequence \{{x}_{n}\} in {\mathcal{A}}_{0}, such that
for every nonnegative integer n. Using the fact that \mathcal{T} is a rational proximal contraction of the first kind, we have
It follows that
where k=\frac{\alpha +\gamma +\delta}{1\beta \gamma \delta}<1. Therefore, \{{x}_{n}\} is a Cauchy sequence and, since (\mathcal{X},d) is complete and \mathcal{A} is closed, the sequence \{{x}_{n}\} converges to some x\in \mathcal{A}.
Moreover, we have
Taking the limit as n\to +\mathrm{\infty}, we get d(x,\mathcal{T}{x}_{n})\to d(x,\mathcal{B}). Since ℬ is approximatively compact with respect to \mathcal{A}, then the sequence \{\mathcal{T}{x}_{n}\} has a subsequence \{\mathcal{T}{x}_{{n}_{k}}\} that converges to some y\in \mathcal{B}. Therefore,
and hence x must be in {\mathcal{A}}_{0}. Since \mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}, then d(u,\mathcal{T}x)=d(\mathcal{A},\mathcal{B}) for some u\in \mathcal{A}. Again, using the fact that \mathcal{T} is a rational proximal contraction of the first kind, we get
Taking the limit as n\to +\mathrm{\infty}, we have
which implies x=u, since \gamma +\delta <1. Thus, it follows that
that is, x\in {B}_{\mathrm{est}}(\mathcal{T}). Now, to prove the uniqueness of the best proximity point (i.e., {B}_{\mathrm{est}}(\mathcal{T}) is singleton), assume that z is another best proximity point of \mathcal{T} so that
Since \mathcal{T} is a rational proximal contraction of the first kind, we have
which implies
It follows immediately that x=z, since \alpha +2\delta <1. Hence, \mathcal{T} has a unique best proximity point. □
As consequences of the Theorem 3.1, we state the following corollaries.
Corollary 3.1 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that ℬ is approximatively compact with respect to \mathcal{A}. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a generalized proximal contraction of the first kind, with \alpha +2\gamma +2\delta <1;

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then, there exists x\in \mathcal{A} such that {B}_{\mathrm{est}}(\mathcal{T})=\{x\}. Further, for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to the best proximity point x.
Corollary 3.2 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that ℬ is approximatively compact with respect to \mathcal{A}. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
There exists a nonnegative real number \alpha <1 such that, for all {u}_{1}, {u}_{2}, {x}_{1}, {x}_{2} in \mathcal{A}, the conditions d({u}_{1},\mathcal{T}{x}_{1})=d(\mathcal{A},\mathcal{B}) and d({u}_{2},\mathcal{T}{x}_{2})=d(\mathcal{A},\mathcal{B}) imply that d({u}_{1},{u}_{2})\le \alpha d({x}_{1},{x}_{2});

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists x\in \mathcal{A} such that {B}_{\mathrm{est}}(\mathcal{T})=\{x\}. Further, for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to the best proximity point x.
The following fixedpoint result can be considered as a special case of the Theorem 3.1, when \mathcal{T} is a selfmapping.
Corollary 3.3 Let (\mathcal{X},d) be a complete metric space and \mathcal{T} be a selfmapping on \mathcal{X}. Assume that there exist nonnegative real numbers α, β, γ, δ with \alpha +\beta +2\gamma +2\delta <1 such that
for all {x}_{1},{x}_{2}\in \mathcal{X}. Then the mapping \mathcal{T} has a unique fixed point.
Remark 3.1 Note that the Corollary 3.3 is a proper extension of the contraction mapping principle [13] because the continuity of the mapping \mathcal{T} is not required. It is well known that a contraction mapping must be continuous.
Now, we state and prove a best proximity point theorem for a rational proximal contraction of the second kind.
Theorem 3.2 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that \mathcal{A} is approximatively compact with respect to ℬ. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a continuous rational proximal contraction of the second kind;

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists x\in {B}_{\mathrm{est}}(\mathcal{T}) and for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x, and \mathcal{T}x=\mathcal{T}z for all x,z\in {B}_{\mathrm{est}}(\mathcal{T}).
Proof Following the same lines of the proof of the Theorem 3.1, it is possible to construct a sequence \{{x}_{n}\} in {\mathcal{A}}_{0} such that
for every nonnegative integer n. Using the fact that \mathcal{T} is a rational proximal contraction of the second kind, we have
It follows that
where k=\frac{\alpha +\gamma +\delta}{1\beta \gamma \delta}<1. Therefore, \{\mathcal{T}{x}_{n}\} is a Cauchy sequence and, since (\mathcal{X},d) is complete, then the sequence \{\mathcal{T}{x}_{n}\} converges to some y\in \mathcal{B}.
Moreover, we have
Taking the limit as n\to +\mathrm{\infty}, we get d(y,{x}_{n})\to d(y,\mathcal{A}). Since \mathcal{A} is approximatively compact with respect to ℬ, then the sequence \{{x}_{n}\} has a subsequence \{{x}_{{n}_{k}}\} converging to some x\in \mathcal{A}. Now, using the continuity of \mathcal{T}, we obtain that
that is, x\in {B}_{\mathrm{est}}(\mathcal{T}). Finally, to prove the last assertion of the present theorem, assume that z is another best proximity point of \mathcal{T} so that
Since \mathcal{T} is a rational proximal contraction of the second kind, we have
which implies
It follows immediately that \mathcal{T}x=\mathcal{T}z, since \alpha +2\delta <1. □
As consequences of the Theorem 3.2, we state the following corollaries.
Corollary 3.4 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that \mathcal{A} is approximatively compact with respect to ℬ. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a continuous generalized proximal contraction of the second kind, with \alpha +2\gamma +2\delta <1;

(b)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then, there exists x\in {B}_{\mathrm{est}}(\mathcal{T}) and for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x. Further, \mathcal{T}x=\mathcal{T}z for all x,z\in {B}_{\mathrm{est}}(\mathcal{T}).
Corollary 3.5 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X} such that \mathcal{A} is approximatively compact with respect to ℬ. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
There exists a nonnegative real number \alpha <1 such that, for all {u}_{1}, {u}_{2}, {x}_{1}, {x}_{2} in \mathcal{A}, the conditions d({u}_{1},\mathcal{T}{x}_{1})=d(\mathcal{A},\mathcal{B}) and d({u}_{2},\mathcal{T}{x}_{2})=d(\mathcal{A},\mathcal{B}) imply that d(\mathcal{T}{u}_{1},\mathcal{T}{u}_{2})\le \alpha d(\mathcal{T}{x}_{1},\mathcal{T}{x}_{2});

(b)
\mathcal{T} is continuous;

(c)
\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists x\in {B}_{\mathrm{est}}(\mathcal{T}) and for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x. Further, \mathcal{T}x=\mathcal{T}z for all x,z\in {B}_{\mathrm{est}}(\mathcal{T}).
Remark 3.2 Note that in the Theorem 3.1 is not required the continuity of the mapping \mathcal{T}. On the contrary, the continuity of \mathcal{T} is an hypothesis of the Theorem 3.2.
Our next theorem concerns a nonselfmapping that is a rational proximal contraction of the first kind as well as a rational proximal contraction of the second kind. In this theorem, we consider only a completeness hypothesis without assuming the continuity of the nonselfmapping.
Theorem 3.3 Let (\mathcal{X},d) be a complete metric space and \mathcal{A} and ℬ be two nonempty, closed subsets of \mathcal{X}. Assume that {\mathcal{A}}_{0} and {\mathcal{B}}_{0} are nonempty and \mathcal{T}:\mathcal{A}\to \mathcal{B} is a nonselfmapping such that:

(a)
\mathcal{T} is a rational proximal contraction of the first and second kinds;

(b)
{\mathcal{T}(\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}.
Then there exists a unique x\in {B}_{\mathrm{est}}(\mathcal{T}). Further, for any fixed {x}_{0}\in {\mathcal{A}}_{0}, the sequence \{{x}_{n}\}, defined by d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B}), converges to x.
Proof Following the same lines of the proof of the Theorem 3.1, it is possible to construct a sequence \{{x}_{n}\} in {\mathcal{A}}_{0} such that
for every nonnegative integer n. Also, using the same arguments in the proof of the Theorem 3.1, we deduce that the sequence \{{x}_{n}\} is a Cauchy sequence, and hence converges to some x\in \mathcal{A}. Moreover, on the lines of the proof of the Theorem 3.2, we obtain that the sequence \{\mathcal{T}{x}_{n}\} is a Cauchy sequence and hence converges to some y\in \mathcal{B}. Therefore, we have
and hence x must be in {\mathcal{A}}_{0}. Since \mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}, then d(u,\mathcal{T}x)=d(\mathcal{A},\mathcal{B}) for some u\in \mathcal{A}. Using the fact that \mathcal{T} is a rational proximal contraction of the first kind, we get
Taking the limit as n\to +\mathrm{\infty}, we have
which implies that x=u, since \gamma +\delta <1. Thus, it follows that
that is, x\in {B}_{\mathrm{est}}(\mathcal{T}). Again, following the same lines of the proof of the Theorem 3.1, we prove the uniqueness of the best proximity point of the mapping \mathcal{T}. To avoid repetitions, we omit the details. □
Example 3.1 Let \mathcal{X}=\mathbb{R} endowed with the usual metric d(x,y)=xy, for all x,y\in \mathcal{X}. Define \mathcal{A}=[1,1] and \mathcal{B}=[3,2]\cup [2,3]. Then, d(\mathcal{A},\mathcal{B})=1, {\mathcal{A}}_{0}=\{1,1\} and {\mathcal{B}}_{0}=\{2,2\}. Also define \mathcal{T}:\mathcal{A}\to \mathcal{B} by
It is easy to show that \mathcal{T} is a rational proximal contraction of the first and second kinds and \mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}. Then all the hypotheses of the Theorem 3.3 are satisfied and d(1,\mathcal{T}(1))=d(\mathcal{A},\mathcal{B}). Clearly, the Theorem 3.2 is not applicable in this case.
References
Eldred A, Veeramani PL: Existence and convergence of best proximity points. J. Math. Anal. Appl. 2006, 323: 1001–1006. doi:10.1016/j.jmaa.2005.10.081
Fan K: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 1969, 112: 234–240. doi:10.1007/BF01110225
Prolla JB: Fixed point theorems for set valued mappings and existence of best approximations. Numer. Funct. Anal. Optim. 1982, 5: 449–455.
Reich S: Approximate selections, best approximations, fixed points and invariant sets. J. Math. Anal. Appl. 1978, 62: 104–113. doi:10.1016/0022–247X(78)90222–6
Sehgal VM, Singh SP: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Am. Math. Soc. 1988, 102: 534–537.
Sehgal VM, Singh SP: A theorem on best approximations. Numer. Funct. Anal. Optim. 1989, 10: 181–184. doi:10.1080/01630568908816298
Vetrivel V, Veeramani P, Bhattacharyya P: Some extensions of Fan’s best approximation theorem. Numer. Funct. Anal. Optim. 1992, 13: 397–402. doi:10.1080/01630569208816486 10.1080/01630569208816486
AlThagafi MA, Shahzad N: Best proximity sets and equilibrium pairs for a finite family of multimaps. Fixed Point Theory Appl. 2008., 2008: Article ID 457069
AlThagafi MA, Shahzad N: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 2009, 70(3):1209–1216. doi:10.1016/j.na.2008.02.004 10.1016/j.na.2008.02.004
AlThagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Anal. 2009, 70(10):3665–3671. doi:10.1016/j.na.2008.07.022 10.1016/j.na.2008.07.022
Anuradha J, Veeramani P: Proximal pointwise contraction. Topol. Appl. 2009, 156(18):2942–2948. doi:10.1016/j.topol.2009.01.017 10.1016/j.topol.2009.01.017
Balaganskii VS, Vlasov LP: The problem of the convexity of Chebyshev sets. Usp. Mat. Nauk 1996, 51: 125–188.
Banach S: Sur les opérations dans les ensembles absraites et leurs applications. Fundam. Math. 1922, 3: 133–181.
Borodin PA: An example of a bounded approximately compact set that is not compact. Russ. Math. Surv. 1994, 49: 153–154.
Di Bari C, Suzuki T, Vetro C: Best proximity points for cyclic MeirKeeler contractions. Nonlinear Anal. 2008, 69(11):3790–3794. doi:10.1016/j.na.2007.10.014 10.1016/j.na.2007.10.014
Efimov NV, Stechkin SB: Approximative compactness and Chebyshev sets. Dokl. Akad. Nauk SSSR 1961, 140: 522–524. (in Russian)
Eldred A, Kirk WA, Veeramani P: Proximinal normal structure and relatively nonexpanisve mappings. Stud. Math. 2005, 171(3):283–293. doi:10.4064/sm171–35
Karpagam S, Agrawal S: Best proximity point theorems for p cyclic MeirKeeler contractions. Fixed Point Theory Appl. 2009., 2009(9): Article ID 197308
Kim WK, Kum S, Lee KH: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 2008, 68(8):2216–2227. doi:10.1016/j.na.2007.01.057 10.1016/j.na.2007.01.057
Mongkolkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 2012, 155: 215–226. doi:10.1007/s10957–012–9991y 10.1007/s109570129991y
Mongkolkeha C, Kumam P: Some common best proximity points for proximity commuting mappings. Optim. Lett. 2012. doi:10.1007/s11590–012–0525–1
Mongkolkeha C, Cho YJ, Kumam P: Best proximity points for generalized proximal C contraction mappings in metric spaces with partial orders. J. Inequal. Appl. 2013., 2013: Article ID 94. doi:10.1186/1029–242X2013–94
Sadiq Basha S: Extensions of Banach’s contraction principle. Numer. Funct. Anal. Optim. 2010, 31: 569–576. doi:10.1080/01630563.2010.485713
Sadiq Basha S: Best proximity points: global optimal approximate solution. J. Glob. Optim. 2010. doi:10.1007/s10898–009–9521–0
Sadiq Basha S, Shahzad N, Jeyaraj R: Common best proximity points: global optimization of multiobjective functions. Appl. Math. Lett. 2011, 24: 883–886. doi:10.1016/j.aml.2010.12.043
Sadiq Basha S, Veeramani P: Best approximations and best proximity pairs. Acta Sci. Math. 1997, 63: 289–300.
Sadiq Basha S, Veeramani P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103: 119–129. doi:10.1006/jath.1999.3415 10.1006/jath.1999.3415
Sadiq Basha S, Veeramani P, Pai DV: Best proximity pair theorems. Indian J. Pure Appl. Math. 2001, 32: 1237–1246.
Sadiq Basha S, Shahzad N: Best proximity point theorems for generalized proximal contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 42
Sankar Raj V, Veeramani P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 2009, 10(1):21–28.
Shahzad N, Sadiq Basha S, Jeyaraj R: Common best proximity points: global optimal solutions. J. Optim. Theory Appl. 2011, 148: 69–78. doi:10.1007/s10957–010–9745–7 10.1007/s1095701097457
Sanhan W, Mongkolkeha C, Kumam P: Generalized proximal ψ contraction mappings and best proximity points. Abstr. Appl. Anal. 2012., 2012: Article ID 896912
Sintunavarat W, Kumam K: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 93
Srinivasan PS: Best proximity pair theorems. Acta Sci. Math. 2001, 67: 421–429.
Suzuki T, Kikkawa M, Vetro C: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 2009, 71: 2918–2926. doi:10.1016/j.na.2009.01.173 10.1016/j.na.2009.01.173
Suzuki T, Vetro C: Three existence theorems for weak contractions of Matkowski type. Int. J. Math. Stat. 2010, 6: 110–120. 10.3844/jmssp.2010.110.115
Vetro C: Best proximity points: convergence and existence theorems for p cyclic mappings. Nonlinear Anal. 2010, 73: 2283–2291. doi:10.1016/j.na.2010.06.008 10.1016/j.na.2010.06.008
Wlodarczyk K, Plebaniak R, Banach A: Best proximity points for cyclic and noncyclic setvalued relatively quasiasymptotic contractions in uniform spaces. Nonlinear Anal. 2009, 70(9):3332–3341. doi:10.1016/j.na.2008.04.037 10.1016/j.na.2008.04.037
Wlodarczyk K, Plebaniak R, Banach A: Erratum to: best proximity points for cyclic and noncyclic setvalued relatively quasiasymptotic contractions in uniform spaces. Nonlinear Anal. 2009, 71: 3585–3586. doi:10.1016/j.na.2008.11.020
Wlodarczyk K, Plebaniak R, Obczynski C: Convergence theorems, best approximation and best proximity for setvalued dynamic systems of relatively quasiasymptotic contractions in cone uniform spaces. Nonlinear Anal. 2010, 72: 794–805. doi:10.1016/j.na.2009.07.024
Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24: 851–862. doi:10.1081/NFA120026380
Acknowledgements
The third author is supported by Università degli Studi di Palermo, Local University Project R. S. ex 60%. The second author was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkuts University of Technology Thonburi (Grant No. MRG5580213).
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Nashine, H.K., Kumam, P. & Vetro, C. Best proximity point theorems for rational proximal contractions. Fixed Point Theory Appl 2013, 95 (2013). https://doi.org/10.1186/16871812201395
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DOI: https://doi.org/10.1186/16871812201395
Keywords
 best proximity point
 contraction
 fixed point
 generalized proximal contraction
 optimal approximate solution