Best proximity point theorems for probabilistic proximal cyclic contraction with applications in nonlinear programming

Abstract

In this paper, we derive a best proximity point theorem for non-self-mappings satisfied proximal cyclic contraction in PM-spaces 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):5844-5850, 2011).

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 self-mapping, 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 non-self-mappings. In fact, given nonempty closed subsets A and B of a complete PM-space $(X, F,*)$, a contraction non-self-mapping $T : A \to B$ 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 self-mapping 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 non-self-mapping, 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

Throughout this paper, the space of all probability distribution functions (briefly, d.f.’s) is denoted by $\Delta^{+}$ = {$F:{ \mathbf{R}}\cup\{-\infty,+\infty\}\rightarrow[0,1]: F$ is left-continuous and non-decreasing on R, $F(0)=0$, and $F(+\infty)=1$} and the subset $D^{+} \subseteq\Delta^{+}$ is the set $D^{+}=\{F\in \Delta^{+}:l^{-}F(+\infty)=1\}$. Here $l^{-} f(x)$ denotes the left limit of the function f at the point x, $l^{-} f(x)=\lim_{t\to x^{-}}f(t)$. The space $\Delta^{+}$ is partially ordered by the usual point-wise ordering of functions, i.e., $F\leq G$ if and only if $F(t)\leq G(t)$ for all t in R. The maximal element for $\Delta^{+}$ in this order is the d.f. given by

$$\varepsilon_{0}(t) =\left \{ \textstyle\begin{array}{l@{\quad}l} 0, & t\leq0, \\ 1, & t>0 . \end{array}\displaystyle \right . $$

Definition 2.1

([1])

A mapping $*:[0,1]\times[0,1]\rightarrow[0,1]$ is a continuous t-norm if ∗ satisfies the following conditions:

• (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 t-norm are $a*b =ab$ and $a*b =\min(a,b)$.

A t-norm ∗ is said to be of Hadžić type if

$$ \forall\epsilon\in(0, 1)\ \exists \delta\in(0, 1)\mbox{:}\quad a >1-\delta \quad \Rightarrow\quad \overbrace{a*a * \cdots * a}^{n}> 1-\epsilon\quad (n\geq1) . $$

The t-norm minimum is a trivial example of a t-norm of Hadžić type, but there exists a t-norm of Hadžić type weaker than minimum (see [2]).

Definition 2.2

A probabilistic metric space (briefly, PM-space) is a triple $(X,F,*)$, where X is a nonempty set, ∗ is a continuous t-norm, and F is a mapping from $X\times X$ into $D^{+}$ such that, if $F_{x,y}$ denotes the value of F at the pair $(x,y)$, the following conditions hold: for all $x,y,z$ in X,

• (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

Let $(X,F,*)$ be a PM-space.

• (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 PM-space $(X,F,*)$ is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.

Definition 2.4

Let $(X,F,*)$ be a PM-space. For each p in X and $\lambda>0$, the strong λ-neighborhood of p is the set

$$ N_{p}(\lambda)=\bigl\{ q\in X:F_{p,q}(\lambda)>1-\lambda\bigr\} , $$

and the strong neighborhood system for X is the union $\bigcup_{p\in V}\mathcal{N}_{p}$ where $\mathcal{N}_{p}=\{N_{p}(\lambda): \lambda>0\}$.

The strong neighborhood system for X determines a Hausdorff topology for X.

Theorem 2.5

([1])

If $(X,F,*)$ is a PM-space 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])

Let $(X,F,*)$ be a Menger PM-space with ∗ of Hadžić-type and $\{x_{n}\}$ be a sequence in X such that, for some $k\in(0,1)$,

$$ F_{x_{n},x_{n+1}}(kt)\geq F_{x_{n-1},x_{n}}(t)\quad ( n\geq1, t>0). $$

Then $\{x_{n}\}$ is a Cauchy sequence.

Let A and B be two nonempty subsets of a PM-space and $t>0$, the following notions and notations are used in the sequel.

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

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

• $B_{0}:=\{ y\in B : F_{x, y}(t) = F_{A, B}(t)\mbox{ for some } x\in A \}$.

Definition 2.7

Let $(X,F,*)$ be a PM-space. Given non-self-mappings $S:A\rightarrow B$ and $T:B\rightarrow A$, the pair $(S, T)$ is said to form a proximal cyclic contraction if there exists a non-negative number $\alpha<1$ such that

$$ \left .\textstyle\begin{array}{l} F_{u, Sx}(t) = F_{A, B}(t), \\ F_{v, Ty}(t) = F_{A, B}(t) \end{array}\displaystyle \right \} \quad \Longrightarrow\quad F_{u, v}( t) \geq\min \biggl\{ F_{x, y} \biggl(\frac {t}{\alpha} \biggr),F_{A, B} (t ) \biggr\} $$

for all u, x in A and v, y in B and $t>0$.

Note that, if S is a self-mapping that is a contraction, then the pair the pair $(S,S)$ forms a proximal cyclic contraction.

Definition 2.8

Let $(X,F,*)$ be a PM-space. A mapping $S:A\rightarrow B$ is said to be a proximal contraction of the first kind if there exists a non-negative number $\alpha<1$ such that

$$ \left .\textstyle\begin{array}{l} F_{u_{1}, Sx_{1}}(t) = F_{A, B}(t), \\ F_{u_{2}, Sx_{2}}(t) = F_{A, B}(t) \end{array}\displaystyle \right \}\quad \Longrightarrow\quad F_{u_{1}, u_{2}}(\alpha t) \geq F_{x_{1}, x_{2}}(t) $$

for all $u_{1}$, $u_{2}$, $x_{1}$, $x_{2}$ in A and $t>0$.

Definition 2.9

Let $(X,F,*)$ be a PM-space. A mapping $S:A\rightarrow B$ is said to be a proximal contraction of the second kind if there exists a non-negative number $\alpha<1$ such that

$$ \left .\textstyle\begin{array}{l} F_{u_{1}, Sx_{1}}(t) = F_{A, B}(t), \\ F_{u_{2}, Sx_{2}}(t) = F_{A, B}(t) \end{array}\displaystyle \right \}\quad \Longrightarrow \quad F_{Su_{1}, Su_{2}}( \alpha t) \geq F_{Sx_{1}, Sx_{2}}(t) $$

for all $u_{1}$, $u_{2}$, $x_{1}$, $x_{2}$ in A and $t>0$.

Definition 2.10

Let $(X,F,*)$ be a PM-space. Given a mapping $S:A\rightarrow B$ and an isometry $g:A\rightarrow A$, the mapping S is said to preserve isometric distance with respect to g if

$$F_{Sgx_{1}, Sgx_{2}}(t) = F_{Sx_{1}, Sx_{2}}(t) $$

for all $x_{1}$ and $x_{2}$ in A. and $t>0$.

Definition 2.11

Let $(X,F,*)$ be a PM-space. An element x in A is said to be a best proximity point of the mapping $S:A\rightarrow B$ if it satisfies the condition that

$$F_{x, Sx}(t) =F_{A, B}(t) $$

for all x in A and $t>0$.

It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping.

Definition 2.12

Let $(X,F,*)$ be a PM-space. 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 non-void 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 non-self proximal contractions of the first kind. Our results extend and improve some results of [6].

Theorem 3.1

Let A and B be non-void closed subsets of a complete PM-space $(X,F,*)$ with ∗ of Hadžić-type such that $A_{0}$ and $B_{0}$ are non-void. Let $S:A\rightarrow B$, $T:B\rightarrow A$ and $g:A\cup B\rightarrow A\cup B$ satisfy the following conditions:

• (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 $B_{0} \subseteq g(B_{0})$.

Then there exist a unique element x in A and a unique element y in B satisfying the conditions that

$$\begin{aligned}& F_{gx, Sx}(t) = F_{A, B}(t), \\& F_{gy, Ty}(t) = F_{A, B}(t), \\& F_{x, y}(t) = F_{A, B}(t). \end{aligned}$$

Further, for any fixed element $x_{0}$ in $A_{0}$, the sequence $\{x_{n}\}$, defined by

$$F_{gx_{n+1}, Sx_{n}}(t) = F_{A, B}(t), $$

converges to the element x. For any fixed element $y_{0}$ in $B_{0}$, the sequence $\{y_{n}\}$, defined by

$$F_{gy_{n+1}, Ty_{n}}(t) = F_{A, B}(t), $$

converges to the element y.

On the other hand, a sequence $\{u_{n}\}$ of elements in A converges to x if there is a sequence $\{\epsilon_{n}\}$ of positive numbers for which

$$\lim_{n\rightarrow\infty} \prod_{i=0}^{n}(1- \epsilon_{i} )=1 $$

in which

$$\prod_{i=0}^{n} a_{i}=a_{0}* \cdots * a_{n} $$

for $a_{i}\in(0,1]$ and

$$F_{u_{n+1}, z_{n+1}}(t) \geq 1-\epsilon_{n}, $$

where $z_{n+1}\in A$ satisfies the condition that

$$F_{z_{n+1}, Su_{n}}(t) = F_{A, B}(t) $$

for $t>0$.

Proof

Let $x_{0}$ be an element in $A_{0}$. In view of the facts that $S(A_{0})$ is contained in $B_{0}$ and that $A_{0}$ is contained in $g(A_{0})$, it is ascertained that there is an element $x_{1}$ in $A_{0}$ such that

$$F_{gx_{1}, Sx_{0}}(t) = F_{A, B}(t) $$

for $t>0$. Again, since $S(A_{0})$ is contained in $B_{0}$, and $A_{0}$ is contained in $g(A_{0})$, there exists an element $x_{2}$ in $A_{0}$ such that

$$F_{gx_{2}, Sx_{1}}(t) = F_{A, B}(t) $$

for $t>0$. One can proceed further in a similar fashion to find $x_{n}$ in $A_{0}$. Having chosen $x_{n}$, one can determine an element $x_{n+1}$ in $A_{0}$ such that

$$F_{gx_{n+1}, Sx_{n}}(t) = F_{A, B}(t), $$

because of the facts that $S(A_{0})$ is contained in $B_{0}$ and that $A_{0}$ is contained in $g(A_{0})$. In light of the facts that g is an isometry and that S is a proximal contraction of the first kind,

$$F_{x_{n}, x_{n+1}}(\alpha t) = F_{gx_{n}, gx_{n+1}}(\alpha t) \geq F_{x_{n-1}, x_{n}}(t) $$

for $t>0$. Therefore, by Lemma 2.6, $\{x_{n}\}$ is a Cauchy sequence and hence converges to some element x in A. Similarly, in view of the facts that $T(B_{0})$ is contained in $A_{0}$ and that $B_{0}$ is contained in $g(B_{0})$, it is guaranteed that there is a sequence $\{y_{n}\}$ of elements in $B_{0}$ such that

$$F_{gy_{n+1}, Ty_{n}}(t) = F_{A, B}(t) $$

for $t>0$. Because g is an isometry and T is a proximal contraction of the first kind, it follows that

$$F_{y_{n}, y_{n+1}}(\alpha t) =F_{gy_{n}, gy_{n+1}}(\alpha t) \geq F_{y_{n-1}, y_{n}}(t) $$

for $t>0$. Therefore, by Lemma 2.6, $\{y_{n}\}$ is a Cauchy sequence and hence converges to some element y in B. Since the pair $(S, T)$ forms a proximal cyclic contraction and g is an isometry, it follows that

$$F_{x_{n+1}, y_{n+1}}(t) = F_{gx_{n+1}, gy_{n+1}}(t) \geq\min \biggl\{ F_{x_{n}, y_{n}} \biggl(\frac{t}{\alpha} \biggr),F_{A, B} (t ) \biggr\} $$

for $t>0$.

Letting $n\rightarrow\infty$, since $F_{x, y}(t)\leq F_{x, y}(t/\alpha)$ we have,

$$F_{x, y}(t) = F_{A, B}(t) $$

for $t>0$. Thus, it can be concluded that x is a member of $A_{0}$ and that y is a member of $B_{0}$. Since $S(A_{0})$ is contained in $B_{0}$, and $T(B_{0})$ is contained in $A_{0}$, there exist an element u in A and an element v in B such that

$$\begin{aligned}& F_{u, Sx}(t) = F_{A, B}(t), \\& F_{v, Ty}(t) = F_{A, B}(t) \end{aligned}$$

for $t>0$. Because S is a proximal contraction of the first kind,

$$F_{u, gx_{n+1}}(\alpha t) \geq F_{x, x_{n}}(t) $$

for $t>0$. Letting $n\rightarrow\infty$, we have the result that $u=gx$. Thus, it follows that

$$F_{gx, Sx}(t) = F_{A, B}(t) $$

for $t>0$. Similarly, it can be shown that $v=gy$ and hence

$$F_{gy, Ty}(t) = F_{A, B}(t) $$

for $t>0$. To prove the uniqueness, let us suppose that there exist elements $x^{*}$ in A and $y^{*}$ in B such that

$$\begin{aligned}& F_{gx^{*}, Sx^{*}}(t) = F_{A, B}(t), \\& F_{gy^{*}, Ty^{*}}(t) = F_{A, B}(t) \end{aligned}$$

for $t>0$. Since g is an isometry, and the non-self-mappings S and T are proximal contractions of the first kind, it follows that

$$\begin{aligned}& F_{x, x^{*}}(\alpha t) = F_{gx, gx^{*}}(\alpha t) \geq F_{x, x^{*}}(t), \\& F_{y, y^{*}}(\alpha t) = F_{gy, gy^{*}}(\alpha t) \geq F_{y, y^{*}}(t) \end{aligned}$$

for $t>0$. Therefore, x and $x^{*}$ are identical, and y and $y^{*}$ are identical.

On the other hand, let $\{u_{n}\}_{n=0}^{\infty}$ in which $u_{0}=x_{0}$ be a sequence of elements in A and $\{\epsilon_{n}\}$ a sequence $(0,1)$ such that

$$\lim_{n\rightarrow\infty} \prod_{i=0}^{n}(1- \epsilon_{i} )=1 $$

and

$$F_{u_{n+1}, z_{n+1}}(t) \geq1-\epsilon_{n}, $$

where $z_{n+1}\in A$ satisfies the condition that

$$F_{z_{n+1}, Su_{n}}(t) = F_{A, B}(t) $$

for $t>0$. Since S is a proximal contraction of the first kind,

$$F_{x_{n+1}, z_{n+1}}(\alpha t) \geq F_{x_{n}, u_{n}}(t). $$

Given $\delta\in(0,1)$, for all $n\geq N$ we have

$$\begin{aligned} F_{x_{n+1}, u_{n+1}}( t+\delta) \geq& F_{x_{n+1}, z_{n+1}}( t) *F_{z_{n+1}, u_{n+1}}( \delta) \\ \geq& F_{x_{n}, u_{n}} \biggl(\frac{t}{\alpha} \biggr) *(1- \epsilon_{n}) \\ \geq& F_{x_{n}, u_{n}} \biggl(\frac{t}{\alpha^{2}} \biggr)*(1- \epsilon _{n-1}) *(1- \epsilon_{n}) \\ \ge&\cdots\ge F_{x_{0}, u_{0}} \biggl(\frac{t}{\alpha^{n+1}} \biggr)*\prod _{i=0}^{n}(1- \epsilon_{i}) \end{aligned}$$

for $t>0$. Since $\delta\in(0,1)$ was arbitrary, we have

$$ F_{x_{n+1}, u_{n+1}}( t)\ge\prod_{i=0}^{n}(1- \epsilon_{i}) $$

for $t>0$. Now,

$$\begin{aligned} F_{u_{n+1}, x}(2t) \ge& F_{u_{n+1}, x_{n+1}}(t)*F_{x_{n+1}, x}(t) \\ \ge&\prod_{i=0}^{n}(1- \epsilon_{i}) *F_{x_{n+1}, x}(t) \end{aligned}$$

for $t>0$. Then

$$\lim_{n\rightarrow\infty}F_{u_{n+1}, x}(2t) \to1 $$

for $t>0$, and it can be concluded that $\{u_{n}\}$ converges to x. This completes the proof of the theorem. □

The following example illustrates the preceding generalized best proximity point theorem.

Example 3.2

Consider the complete PM-space $({\mathbf{R}},F,\mathrm{min})$ where

$$F_{x,y}(t)=\frac{t}{t+|x-y|}, $$

when $t>0$ and

$$F_{x,y}(t)=0, $$

when $t\le0$ for x, y in R.

Let $A = [-1,0]$ and $B = [0,1]$.

Let $S:A\rightarrow B$, $T:B\rightarrow A$, and $g:A\cup B\rightarrow A\cup B$ be defined as

$$\begin{aligned}& S(x) = \frac{-x}{2}, \\& T(y) = \frac{-y}{2}, \\& g(x) = -x. \end{aligned}$$

Then it is easy to see that

$$F_{A, B}(t)=1, $$

$A_{0} = \{0\}$ and $B_{0} = \{0\}$. The mapping g is an isometry and the non-self-mappings S and T are proximal contractions of the first kind, and the pair $(S, T)$ forms a proximal cyclic contraction. The other hypotheses of Theorem 3.1 are also satisfied. Further, it is easy to observe that the element 0 in A and B satisfy the conditions in the conclusion of the preceding result.

If g is assumed to be the identity mapping, then Theorem 3.1 yields the following best proximity point result.

Corollary 3.3

Let A and B be non-void closed subsets of a complete PM-space $(X,F,*)$ with ∗ of Hadžić-type such that $A_{0}$ and $B_{0}$ are non-void. Let $S:A\rightarrow B$ and $T:B\rightarrow A$ satisfy the following conditions:

• (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.

Then there exist a unique element x in A and a unique element y in B satisfying the conditions that

$$\begin{aligned}& F_{x, Sx}(t) = F_{A, B}(t) , \\& F_{y, Ty}(t) = F_{A, B}(t) , \\& F_{x, y}(t) = F_{A, B}(t) \end{aligned}$$

for $t>0$.

Theorem 3.4

Let A and B be non-void closed subsets of a complete PM-space $(X,F,*)$ with ∗ of Hadžić-type such that $A_{0}$ and $B_{0}$ are non-void. Let $S:A\rightarrow B$ and $g:A\rightarrow A$ satisfy the following conditions:

• (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 $g(A_{0})$.

Then there exists a unique element x in A such that

$$F_{gx, Sx}(t) = F_{A, B}(t) $$

for $t>0$. Further, for any fixed element $x_{0}$ in $A_{0}$, the sequence $\{x_{n}\}$, defined by

$$F_{gx_{n+1}, Sx_{n}}(t) = F_{A, B}(t), $$

converges to the element x for $t>0$.

On the other hand, a sequence $\{u_{n}\}$ of elements in A converges to x if there is a sequence $\{\epsilon_{n}\}$ of positive numbers for which

$$\lim_{n\rightarrow\infty} \prod_{i=0}^{n}(1- \epsilon_{i} )=1 $$

and

$$F_{u_{n+1}, z_{n+1}}(t) \geq 1-\epsilon_{n}, $$

where $z_{n+1}\in A$ satisfies the condition that

$$F_{z_{n+1}, Su_{n}}(t) = F_{A, B}(t) $$

for $t>0$.

Proof

Proceeding as in Theorem 3.1, it is possible to find a sequence $\{x_{n}\}$ of elements in $A_{0}$ such that

$$F_{gx_{n+1}, Sx_{n}}(t) = F_{A, B}(t) $$

for $t>0$ and for all non-negative integral values of n, because of the facts that $S(A_{0})$ is contained in $B_{0}$ and that $A_{0}$ is contained in $g(A_{0})$. Due to the facts that S is a proximal contraction of the first kind and g is an isometry,

$$F_{x_{n}, x_{n+1}}( \alpha t) = F_{gx_{n}, gx_{n+1}}( \alpha t) \geq F_{x_{n-1}, x_{n}}(t) $$

for $t>0$. Therefore, by Lemma 2.6, $\{x_{n}\}$ is a Cauchy sequence and hence converges to some element x in A. Because of the facts that S is a proximal contraction of the second kind and preserves the isometric distance with respect to g,

$$F_{Sx_{n}, Sx_{n+1}}(\alpha t) = F_{Sgx_{n}, Sgx_{n+1}}(\alpha t) \geq F_{Sx_{n-1}, Sx_{n}}(t) $$

for $t>0$. Therefore, by Lemma 2.6, $\{Sx_{n}\}$ is a Cauchy sequence and hence converges to some element y in B. Thus, it can be concluded that

$$F_{gx, y}(t) = \lim_{n\rightarrow\infty}F_{gx_{n+1}, Sx_{n}}(t) = F_{A, B}(t). $$

Eventually, gx is an element of $A_{0}$. Because of the fact that $A_{0}$ is contained in $g(A_{0})$, $gx = gz$ for some member z in $A_{0}$. Owing to the fact that g is an isometry, $F_{x, z}(t) = F_{gx, gz}(t) = 1$. Consequently, the elements x and z must be identical, and hence x becomes an element of $A_{0}$. Because $S(A_{0})$ is contained $B_{0}$,

$$F_{u, Sx}(t) = F_{A, B}(t) $$

for $t>0$, for some element u in A. On account of the fact that the mapping S is a proximal contraction of the first kind,

$$F_{u, gx_{n+1}}( \alpha t) \geq F_{x, x_{n}}(t) $$

for $t>0$. As a result, the sequence $\{g(x_{n})\}$ must converge to u. However, because of the continuity of g, the sequence $\{g(x_{n})\}$ converges to gx as well. Therefore, u and gx must be identical. Thus, we have the result that

$$F_{gx, Sx}(t) = F_{z, Sx}(t) = F_{A, B}(t) $$

for $t>0$. The uniqueness and the remaining part of the proof follow as in Theorem 3.1. This completes the proof of the theorem. □

The preceding generalized best proximity point theorem is illustrated by the following example.

Example 3.5

Consider the complete PM-space $({\mathbf{R}},F,\mathrm{min})$ where

$$F_{x,y}(t)=\frac{t}{t+|x-y|}, $$

when $t>0$, and

$$F_{x,y}(t)=0, $$

when $t\le0$ for $x,y\in{\mathbf{R}}$.

Let $A = [-1, 1]$ and $B = [-3, -2] \cup[2, 3]$. Then $F_{A,B}(t)=\frac{t}{t+1}$, $A_{0} = \{-1, 1\}$, and $B_{0} = \{-2, 2\}$. Let $S:A\rightarrow B$ be defined as

$$Sx = \left \{ \textstyle\begin{array}{l@{\quad}l} 2 & \mbox{if }x \mbox{ is rational}, \\ 3 & \mbox{otherwise}. \end{array}\displaystyle \right . $$

Then S is a proximal contraction of the first and second kind, and $S(A_{0}) \subseteq B_{0}$.

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

Let A and B be non-void closed subsets of a complete PM-space $(X,F,*)$ with ∗ of Hadžić-type such that $A_{0}$ and $B_{0}$ are non-void. Let $S:A\rightarrow B$ satisfy the following conditions:

• (a) S is a proximal contraction of the first and second kind.

• (b) $S(A_{0})$ is contained in $B_{0}$.

Then there exists a unique element x in A such that

$$F_{x, Sx}(t) = F_{A, B}(t). $$

Further, for any fixed element $x_{0}$ in $A_{0}$, the sequence $\{x_{n}\}$, defined by

$$F_{x_{n+1}, Sx_{n}}(t) = F_{A, B}(t), $$

converges to the best proximity point x of S.

4 Application

A solution to the nonlinear programming problem

$$\max_{x\in A}F_{x,Tx}(t) $$

is fundamentally an ideal optimal approximate solution to the equation $Tx = x$ which is shifting to have a solution when T is supposed to be a non-self-mapping.

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 $t>0$.