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Best proximity point theorems for probabilistic proximal cyclic contraction with applications in nonlinear programming
Fixed Point Theory and Applications volume 2015, Article number: 79 (2015)
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
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
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
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 \(k\in(0,1)\),
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
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
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
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
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
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
Further, for any fixed element \(x_{0}\) in \(A_{0}\), the sequence \(\{x_{n}\}\), defined by
converges to the element x. For any fixed element \(y_{0}\) in \(B_{0}\), the sequence \(\{y_{n}\}\), defined by
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
in which
for \(a_{i}\in(0,1]\) and
where \(z_{n+1}\in A\) satisfies the condition that
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
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
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
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,
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
for \(t>0\). Because g is an isometry and T is a proximal contraction of the first kind, it follows that
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
for \(t>0\).
Letting \(n\rightarrow\infty\), since \(F_{x, y}(t)\leq F_{x, y}(t/\alpha)\) we have,
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
for \(t>0\). Because S is a proximal contraction of the first kind,
for \(t>0\). Letting \(n\rightarrow\infty\), we have the result that \(u=gx\). Thus, it follows that
for \(t>0\). Similarly, it can be shown that \(v=gy\) and hence
for \(t>0\). To prove the uniqueness, let us suppose that there exist elements \(x^{*}\) in A and \(y^{*}\) in B such that
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
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
and
where \(z_{n+1}\in A\) satisfies the condition that
for \(t>0\). Since S is a proximal contraction of the first kind,
Given \(\delta\in(0,1)\), for all \(n\geq N\) we have
for \(t>0\). Since \(\delta\in(0,1)\) was arbitrary, we have
for \(t>0\). Now,
for \(t>0\). Then
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
when \(t>0\) and
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
Then it is easy to see that
\(A_{0} = \{0\}\) 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
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
for \(t>0\). Further, for any fixed element \(x_{0}\) in \(A_{0}\), the sequence \(\{x_{n}\}\), defined by
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
and
where \(z_{n+1}\in A\) satisfies the condition that
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
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,
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,
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
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 \(B_{0}\),
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,
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
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
when \(t>0\), and
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
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
Further, for any fixed element \(x_{0}\) in \(A_{0}\), the sequence \(\{x_{n}\}\), defined by
converges to the best proximity point x of S.
4 Application
A solution to the nonlinear programming problem
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 \(t>0\).
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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.
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The author carried out the proof. The author conceived of the study and participated in its design and coordination. The author read and approved the final manuscript.
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Saadati, R. Best proximity point theorems for probabilistic proximal cyclic contraction with applications in nonlinear programming. Fixed Point Theory Appl 2015, 79 (2015). https://doi.org/10.1186/s13663-015-0330-5
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DOI: https://doi.org/10.1186/s13663-015-0330-5