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Best proximity point theorems for probabilistic proximal cyclic contraction with applications in nonlinear programming
 Reza Saadati^{1}Email author
https://doi.org/10.1186/s1366301503305
© Saadati 2015
 Received: 31 January 2015
 Accepted: 17 May 2015
 Published: 6 June 2015
Abstract
In this paper, we derive a best proximity point theorem for nonselfmappings satisfied proximal cyclic contraction in PMspaces and this shows the existence of optimal approximate solutions of certain simultaneous fixed point equations in the event that there is no solution. As an application we consider a nonlinear programming problem. Our results extend and improve the recent results of (Sadiq Basha in Nonlinear Anal. 74(17):58445850, 2011).
Keywords
 optimal approximate solution
 fixed point
 best proximity point
 proximal contraction
 proximal cyclic contraction
1 Introduction
Best proximity point theorems are those results that provide sufficient conditions for the existence of a best proximity point and algorithms for finding best proximity points. It is interesting to note that best proximity point theorems generalized fixed point theorems in a natural fashion. Indeed, if the mapping under consideration is a selfmapping, a best proximity point becomes a fixed point.
One of the most interesting is the study of the extension of Banach contraction principle to the case of nonselfmappings. In fact, given nonempty closed subsets A and B of a complete PMspace \((X, F,*)\), a contraction nonselfmapping \(T : A \to B\) does not necessarily has a fixed point. Eventually, it is quite natural to find an element x such that \(F_{x,Tx}(t)\) is maximum for a given problem which implies that x and Tx are in close proximity to each other.
Many problems can be formulated as equations of the form \(Tx = x\), where T is a selfmapping in some suitable framework. Fixed point theory finds the existence of a solution to such generic equations and brings out the iterative algorithms to compute a solution to such equations.
However, in the case that T is nonselfmapping, the aforementioned equation does not necessarily have a solution. In such a case, it is worthy to determine an approximate solution x such that the error \(F_{x,Tx}(t)\) is maximum.
2 Preliminaries
Definition 2.1
([1])
 (a)
∗ is commutative and associative;
 (b)
∗ is continuous;
 (c)
\(a*1=a\) for all \(a\in[0,1]\);
 (d)
\(a*b \leq c*d \) whenever \(a\leq c\) and \(b\leq d\), and \(a,b,c,d\in[0,1]\).
Two typical examples of a continuous tnorm are \(a*b =ab\) and \(a*b =\min(a,b)\).
Definition 2.2
 (PM1)
\(F_{x,y}(t)=\varepsilon_{0}(t)\) for all \(t>0\) if and only if \(x=y\);
 (PM2)
\(F_{x,y}(t)=F_{y,x}(t)\);
 (PM3)
\(F_{x,z}( t+s) \geq F_{x,y}(t)*F_{y,z}(s)\) for all \(x,y,z\in X\) and \(t,s\geq0\).
For more details and examples of these spaces see also [3–5].
Definition 2.3
 (1)
A sequence \(\{x_{n}\}_{n}\) in X is said to be convergent to x in X if, for every \(\epsilon>0\) and \(\lambda>0\), there exists a positive integer N such that \(F_{x_{n},x}(\epsilon)>1\lambda\) whenever \(n\geq N\).
 (2)
A sequence \(\{x_{n}\}_{n}\) in X is called Cauchy sequence if, for every \(\epsilon>0\) and \(\lambda>0\), there exists a positive integer N such that \(F_{x_{n},x_{m}}(\epsilon)>1\lambda\) whenever \(n, m\geq N\).
 (3)
A PMspace \((X,F,*)\) is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.
Definition 2.4
The strong neighborhood system for X determines a Hausdorff topology for X.
Theorem 2.5
([1])
If \((X,F,*)\) is a PMspace and \(\{p_{n}\}\) and \(\{q_{n}\}\) are sequences such that \(p_{n}\to p\) and \(q_{n}\to q\), then \(\lim_{n\to\infty} F_{p_{n},q_{n}}(t)=F_{p,q}(t)\) for every continuity point t of \(F_{p,q}\).
Lemma 2.6
([2])

\(F_{A, B}(t) :=\sup\{F_{x, y}(t) : x\in A, y\in B \}\),

\(A_{0} :=\{ x\in A : F_{x, y}(t) = F_{A, B}(t)\mbox{ for some }y\in B \}\),

\(B_{0}:=\{ y\in B : F_{x, y}(t) = F_{A, B}(t)\mbox{ for some } x\in A \}\).
Definition 2.7
Note that, if S is a selfmapping that is a contraction, then the pair the pair \((S,S)\) forms a proximal cyclic contraction.
Definition 2.8
Definition 2.9
Definition 2.10
Definition 2.11
It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a selfmapping.
Definition 2.12
Let \((X,F,*)\) be a PMspace. B is said to be approximatively compact with respect to A if every sequence \(\{y_{n}\}\) of B satisfying the condition that for all \(t>0\), \(F_{x, y_{n}}(t)\rightarrow F_{x, B}(t)\) for some x in A has a convergent subsequence.
It is easy to observe that every set is approximatively compact with respect to itself, and that every compact set is approximatively compact. Moreover, \(A_{0}\) and \(B_{0}\) are nonvoid if A is compact and B is approximatively compact with respect to A.
3 Proximal contractions
The following main result is a generalized best proximity point theorem for nonself proximal contractions of the first kind. Our results extend and improve some results of [6].
Theorem 3.1
 (a)
S and T are proximal contractions of the first kind.
 (b)
\(S(A_{0}) \subseteq B_{0}\) and \(T(B_{0}) \subseteq A_{0}\).
 (c)
The pair \((S, T)\) forms a proximal cyclic contraction.
 (d)
g is an isometry.
 (e)
\(A_{0} \subseteq g(A_{0})\) and \(B_{0} \subseteq g(B_{0})\).
Proof
The following example illustrates the preceding generalized best proximity point theorem.
Example 3.2
Let \(A = [1,0]\) and \(B = [0,1]\).
If g is assumed to be the identity mapping, then Theorem 3.1 yields the following best proximity point result.
Corollary 3.3
 (a)
S and T are proximal contractions of the first kind.
 (b)
\(S(A_{0}) \subseteq B_{0}\) and \(T(B_{0}) \subseteq A_{0}\).
 (c)
The pair \((S, T)\) forms a proximal cyclic contraction.
Theorem 3.4
 (a)
S is a proximal contraction of the first and second kind.
 (b)
\(S(A_{0})\) is contained in \(B_{0}\).
 (c)
g is an isometry.
 (d)
S preserves isometric distance with respect to g.
 (e)
\(A_{0}\) is contained in \(g(A_{0})\).
Proof
The preceding generalized best proximity point theorem is illustrated by the following example.
Example 3.5
Further, let \(g:A\rightarrow A\) be defined as \(gx=x\). Then g is an isometry, S preserves the isometric distance with respect to g, and \(A_{0}\subseteq g(A_{0})\). It can also be observed that \(F_{g(1), S(1)}(t) = F_{A,B}(t)\) for \(t>0\).
If g is assumed to be the identity mapping, then Theorem 3.4 yields the following best proximity point theorem.
Corollary 3.6
 (a)
S is a proximal contraction of the first and second kind.
 (b)
\(S(A_{0})\) is contained in \(B_{0}\).
4 Application
Considering the fact that \(F_{x,Tx}(t)\) is at least \(F_{A,B}(t)\) for all x in A, a solution x to the aforementioned nonlinear programming problem becomes an approximate solution with the lowest possible error to the corresponding equation \(Tx = x\) if it satisfies the condition that \(F_{x,Tx}(t)=F_{A,B}(t)\) for \(t>0\).
Declarations
Acknowledgements
The author is thankful to the three anonymous referees for giving valuable comments and suggestions, which helped to improve the final version of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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