 Research
 Open Access
Best proximity point theorems for reckoning optimal approximate solutions
 S Sadiq Basha^{1},
 N Shahzad^{2}Email author and
 R Jeyaraj^{3}
https://doi.org/10.1186/168718122012202
© Basha et al.; licensee Springer 2012
 Received: 27 June 2012
 Accepted: 17 October 2012
 Published: 12 November 2012
Abstract
Given a nonself mapping from A to B, where A and B are subsets of a metric space, in order to compute an optimal approximate solution of the equation $Sx=x$, a best proximity point theorem probes into the global minimization of the error function $x\u27f6d(x,Sx)$ corresponding to approximate solutions of the equation $Sx=x$. This paper presents a best proximity point theorem for generalized contractions, thereby furnishing optimal approximate solutions, called best proximity points, to some nonlinear equations. Also, an iterative algorithm is presented to compute such optimal approximate solutions.
MSC:90C26, 90C30, 41A65, 46B20, 47H10, 54H25.
Keywords
 global optimization
 optimal approximate solution
 best proximity point
 fixed point
 contraction
 generalized contraction
 nonexpansive mapping
1 Introduction
Best proximity point theory involves an intertwining of approximation and global optimization. Indeed, it explores the existence and computation of an optimal approximate solution of nonlinear equations of the form $Sx=x$, where S is a nonself mapping in some framework. Such equations are confronted when we attempt the mathematical formulation of several problems. Given a nonself mapping $S:A\u27f6B$, where A and B are nonempty subsets of a metric space, the equation $Sx=x$ does not necessarily have a solution because of the fact that a solution of the preceding equation constrains the equality between an element in the domain and an element in the range of the mapping. In such circumstances, one raises the following questions:

Is it possible to find an optimal approximate solution with the least possible error?

If an approximate solution exists, is there any iterative algorithm to compute such a solution?

Can one have more than one approximate solution with the least possible error?
Best proximity point theory is an outgrowth of attempts in many directions to answer previously posed questions for various families of nonself mappings. In fact, a best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution ${x}^{\ast}$ that is optimal in the sense that the error $d({x}^{\ast},S{x}^{\ast})$ assumes the global minimum value $d(A,B)$. Such an optimal approximate solution is known as a best proximity point of the mapping S. It is straightforward to observe that a best proximity point becomes a solution of the equation in the special case that the domain of the mapping intersects the codomain of the mapping. In essence, a best proximity point theorem delves into the global minimization of the error function $x\u27f6d(x,Sx)$ corresponding to the approximate solutions of the equation $Sx=x$. Many interesting best proximity point theorems for various classes of nonself mappings in different frameworks and best approximation theorems have been elicited in [1–17] and [18–40]. The main objective of this article is to present, in the framework of complete metric spaces, a best proximity point theorem for a new family of nonself mappings, known as generalized contractions, thereby computing an optimal approximate solution to the equation $Sx=x$, where S is a generalized contraction. Further, some results in the literature are realizable as special cases from the preceding result.
2 Preliminaries
Throughout this section, we assume that A and B are nonempty subsets of a metric space. We recall the following notions that will be used in the sequel.
Definition 2.1 ([41])
for all ${x}_{1}$, ${x}_{2}$ in A.
It is apparent that every generalized contraction is a contractive mapping and hence it must be continuous.
Due to the fact that $d(x,Sx)\ge d(A,B)$ for all x in A, the global minimum of the error function $x\u27f6d(x,Sx)$ corresponding to approximate solutions of the equation $Sx=x$ is attained at any best proximity point. Moreover, if the mapping under consideration is a selfmapping, a best proximity point reduces to a fixed point.
Definition 2.3 ([21])
It is quite easy to observe that the minmax condition is less restrictive so that several classes of pairs of mappings meet this requirement.
Definition 2.4 ([21])
 (a)
a cyclic inequality pair if $d(A,B)<d(x,y)\u27f9d(Sx,Ty)\ne d(x,y)$
 (b)
a cyclic contractive pair if $d(A,B)<d(x,y)\u27f9d(Sx,Ty)<d(x,y)$
 (c)
a cyclic expansive pair if $d(A,B)<d(x,y)\u27f9d(Sx,Ty)>d(x,y)$
for all $x\in A$ and $y\in B$.
It is remarked that cyclic inequality pairs, cyclic contractive pairs, and cyclic expansive pairs satisfy the minmax condition.
for all x in A and y in B.
It is straightforward to see that every generalized cyclic contraction forms a cyclic contractive pair and hence satisfies the minmax condition.
3 Generalized contractions
We are now ready to establish the following interesting best proximity point theorem for nonself generalized contractions.
 (a)
S is a generalized contraction.
 (b)
T is a nonexpansive mapping.
 (c)
The pair $(S,T)$ satisfies the minmax condition.
Further, if S has two distinct best proximity points, then $d(A,B)$ does not vanish and hence the sets A and B should be disjoint.
Letting $k\u27f6\mathrm{\infty}$, we deduce that b vanishes, which is incompatible with our assumption. Therefore, it can be concluded that ${b}_{n}\u27f60$ as $n\u27f6\mathrm{\infty}$.
Thus, $d(A,B)>0$ and hence the sets A and B are disjoint. This completes the proof of the theorem. □
We illustrate the preceding best proximity point theorem by means of the following example.
Let $A:=\{{f}_{n}:n=0,\pm 1,\pm 2,\pm 3,\dots \}$ and $B:=\{f:f\in A\}$.
Then, it is easy to see that S is a generalized contraction and T is a nonexpansive mapping. Also, the pair $(S,T)$ satisfies the minmax condition. However, $(S,T)$ is neither a cyclic contractive pair nor a cyclic expansive pair. Finally, we can note that the element ${x}^{\ast}={f}_{0}$ in A is a best proximity point of the mapping S and the element ${y}^{\ast}={f}_{0}$ in B is a best proximity point of the mapping T such that $d({x}^{\ast},{y}^{\ast})=d(A,B)=2$.
One can easily see that best proximity point Theorem 3.1 subsumes the following result.
 (a)
S is a generalized contraction.
 (b)
T is a nonexpansive mapping.
 (c)
$(S,T)$ is a generalized cyclic contraction.
Moreover, if S has two distinct best proximity points, then $d(A,B)>0$ and hence the sets A and B must be disjoint.
Best proximity point Theorem 3.1 subsumes the following fixed point theorem, due to Krasnoselskii [41], which in turn extends the most interesting and wellknown contraction principle.
Corollary 3.4 Let X be a complete metric space. If the selfmapping $T:X\u27f6X$ is a generalized contraction, then it has a unique fixed point ${x}^{\ast}$, and for every x in X, the sequence $\{{T}^{n}(x)\}$ converges to ${x}^{\ast}$.
The following best proximity point theorem, due to Basha [19], which extends the contraction principle to the case of nonself mappings, is a special case of Theorem 3.1.
 (a)
S is a contraction.
 (b)
T is a nonexpansive mapping.
 (c)
$(S,T)$ is a cyclic contractive pair.
Moreover, if S has two distinct best proximity points, then $d(A,B)>0$ and hence the sets A and B must be disjoint.
Declarations
Authors’ Affiliations
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