Best proximity point theorems for rational proximal contractions
© Nashine et al.; licensee Springer 2013
Received: 12 January 2013
Accepted: 26 March 2013
Published: 12 April 2013
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 fixed-point theory. We also give some examples to illustrate and validate our definitions and results.
MSC:41A65, 46B20, 47H10.
Let be a metric space and be a self-mapping defined on a subset of . In this setting, the fixed-point theory is an important tool for solving equations of the kind , whose solutions are the fixed points of the mapping . On the other hand, if is not a self-mapping, say where and ℬ are nonempty subsets of , then does not necessarily have a fixed point. Consequently, the equation 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 . 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 is at least , a best proximity point theorem concerns the global minimum of the real valued function , that is, an indicator of the error involved for an approximate solution of the equation , by complying the condition . The notation of best proximity point is introduced in  but one of the most interesting results in this direction is due to Fan  and can be stated as follows.
Theorem 1.1 Let be a nonempty, compact and convex subset of a normed space ℰ. Then for any continuous mapping , there exists with .
Some generalizations and extensions of this theorem appeared in the literature by Prolla , Reich , Sehgal and Singh [5, 6], Vetrivel et al.  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 nonself-mapping. Inspired by , 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 nonself-mapping 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 .
In this section, we give some basic notations and definitions that will be used in the sequel.
Sufficient conditions to ensure that and are nonempty are given in . Also, observe that if and ℬ are closed subsets of a normed linear space such that , then and are contained in the boundaries of and ℬ, respectively; see .
Now, we give sequentially two definitions that are essential to state and prove our main results.
for all .
Note that, if , then from (1) we get the definition of the generalized proximal contraction of the first kind with ; see .
for all .
Note that, if , then from (2) we get the definition of the generalized proximal contraction of the second kind with , see .
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.
Then and 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 left-hand 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 left-hand 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 .
It is well known that the notion of approximative compactness plays an important role in the theory of approximation . In particular, the notion of an approximatively compact set was introduced by Efimov and Stechkin  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  that in every infinite-dimensional separable Banach space there exists a bounded approximatively compact set, which is not compact.
Remark 2.1 Since 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 be a metric space and and ℬ be two nonempty subsets of . Then ℬ is said to be approximatively compact with respect to if every sequence of ℬ, satisfying the condition for some x in , 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.
is a rational proximal contraction of the first kind;
Then there exists such that . Further, for any fixed , the sequence , defined by , converges to x.
where . Therefore, is a Cauchy sequence and, since is complete and is closed, the sequence converges to some .
It follows immediately that , since . Hence, has a unique best proximity point. □
As consequences of the Theorem 3.1, we state the following corollaries.
is a generalized proximal contraction of the first kind, with ;
Then, there exists such that . Further, for any fixed , the sequence , defined by , converges to the best proximity point x.
There exists a nonnegative real number such that, for all , , , in , the conditions and imply that ;
Then there exists such that . Further, for any fixed , the sequence , defined by , converges to the best proximity point x.
The following fixed-point result can be considered as a special case of the Theorem 3.1, when is a self-mapping.
for all . Then the mapping has a unique fixed point.
Remark 3.1 Note that the Corollary 3.3 is a proper extension of the contraction mapping principle  because the continuity of the mapping 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.
is a continuous rational proximal contraction of the second kind;
Then there exists and for any fixed , the sequence , defined by , converges to x, and for all .
where . Therefore, is a Cauchy sequence and, since is complete, then the sequence converges to some .
It follows immediately that , since . □
As consequences of the Theorem 3.2, we state the following corollaries.
is a continuous generalized proximal contraction of the second kind, with ;
Then, there exists and for any fixed , the sequence , defined by , converges to x. Further, for all .
There exists a nonnegative real number such that, for all , , , in , the conditions and imply that ;
Then there exists and for any fixed , the sequence , defined by , converges to x. Further, for all .
Remark 3.2 Note that in the Theorem 3.1 is not required the continuity of the mapping . On the contrary, the continuity of is an hypothesis of the Theorem 3.2.
Our next theorem concerns a nonself-mapping 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 nonself-mapping.
is a rational proximal contraction of the first and second kinds;
Then there exists a unique . Further, for any fixed , the sequence , defined by , converges to x.
that is, . 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 . To avoid repetitions, we omit the details. □
It is easy to show that is a rational proximal contraction of the first and second kinds and . Then all the hypotheses of the Theorem 3.3 are satisfied and . Clearly, the Theorem 3.2 is not applicable in this case.
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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