- Research
- Open Access
Some results on best proximity point on star-shaped sets in probabilistic Banach (Menger) spaces
- Hamid Shayanpour^{1}Email author,
- Maryam Shams^{1} and
- Asiyeh Nematizadeh^{1}
https://doi.org/10.1186/s13663-015-0487-y
© Shayanpour et al. 2016
- Received: 5 July 2015
- Accepted: 11 December 2015
- Published: 15 January 2016
Abstract
We first present the concepts of proximal contraction and proximal nonexpansive mappings on star-shaped sets in probabilistic Banach (Menger) spaces. We derive some results about the best proximity points for these mappings in probabilistic Banach (Menger) spaces. Next, we bring some examples that defend our main results.
Keywords
- proximal contraction
- proximal nonexpansive mappings
- best proximity point
- P-property
- star-shaped
MSC
- 41A65
- 47H10
- 47H09
- 46S50
1 Introduction and preliminaries
The equation \(Tx = x\) for a mapping \(T: A \rightarrow B \) may have no solution whenever \(A\cap B =\emptyset\), where A, B are two nonempty subsets in a metric space \((X, d)\). Under this condition, it is beneficial to determine a point \(a_{0} \in A \) such that \(d(a_{0} , Ta_{0}) \) is minimal. If \(d(a_{0} , Ta_{0})\) is the global minimum value of \(\operatorname{dist}(A,B)\), i.e., \(d(a_{0} , Ta_{0}) = \operatorname{dist}(A,B) = \min\lbrace d(a, b) : a \in A, b\in B\rbrace\), then \(a_{0}\) is called best proximity point of T.
Thereafter, this theorem has been generalized for continuous multivalued mappings by Reich [2, 3] and Sehgal and Singh [4].
Eldred et al. [5] showed that every relatively nonexpansive mapping has a proximal point under certain conditions. For further existence results of a best proximity point for several types of contractions, we refer to [6–25].
In 1942, a probabilistic metric (PM) space was introduced by Menger [26]. Schweizer and Sklar [27, 28] were two pioneers in the study of PM spaces.
PM spaces are very useful in probabilistic functional analysis, quantum particle physics, \(\epsilon^{\infty} \) theory, nonlinear analysis, and applications; see [29–33].
Indeed, the study of fixed point results in PM spaces is one of the most active research areas in fixed point theory. Sehgal and Bharucha-Reid [34] were two pioneers in this study. For further existence results of a fixed point and common fixed point in PM spaces, we refer, for example, to [35–37]. In 2014, Su and Zhang [38], proved some best proximity point theorems in PM spaces.
- (PM1)
\(F_{p,q}=\epsilon_{0}\) iff \(p=q\),
- (PM2)
\(F_{ p,q}=F_{q,p}\), and
- (PM3)
if \(F_{p,q}(t) = 1\) and \(F_{q,r}(s) = 1\), then \(F_{p,r}(t+s) = 1\)
For well-known definitions (such as t-norm, t-norm of H-type, probabilistic Menger space, complete probabilistic Menger space, probabilistic normed (PN) space, etc.) and known results, we refer to [27, 39].
First, we state some notation, definitions, and known results; afterward, we introduce concepts of proximal contraction, proximal nonexpansive, P-property, weak P-property, and semisharp proximinal pair in PM spaces. Throughout this paper, the minimum t-norm will be denoted by \(\Delta _{m}(a,b)=\min\{a,b\}\).
Lemma 1.1
([39])
Definition 1.2
Suppose that A is a nonempty subset of a probabilistic Menger space \((X, F,\Delta)\). Then the probabilistic diameter of A is the mapping \(D_{A}\) defined on \([0,\infty]\) by \(D_{A}(\infty)=1\) and \(D_{A}(x)=\lim_{t\rightarrow x^{-}}\varphi_{A}(t)\), where \(\varphi_{A} (t)=\inf\{F_{a,b}(t): a,b\in A\}\).
A nonempty set A in a probabilistic Menger space is bounded if \(\lim_{x\rightarrow\infty}D_{A}(x)=1\). It is easy to see that \(F_{a,b}(t)\geq D_{A}(t)\) for all \(a,b\in A\) and \(t\geq0\).
Definition 1.3
Definition 1.4
Let \((X, F,\Delta)\) be a probabilistic Menger space, and \(A, B\subseteq X \). A mapping \(T:A\rightarrow B\) is said to be continuous at \(x\in A\) if for every sequence \((x_{n})\) in A that converges to x, the sequence \((Tx_{n})\) in B converges to Tx.
Remark 1.5
An immediate consequence of the definition of a PN space ([27], Section 15.1) is the following lemma.
Lemma 1.6
([27])
We call this probabilistic metric \(F^{\nu}\) on X the probabilistic metric induced by the probabilistic norm ν.
Definition 1.7
A PN space \((X,\nu, \Delta) \) is said to be a probabilistic Banach space if \((X,F^{\nu}, \Delta) \) is a complete probabilistic Menger space.
Remark 1.8
Definition 1.9
Let A be a nonempty subset of a PM space \((X,F)\). A mapping \(T:A\rightarrow X \) is called a contraction (nonexpansive) if \(F_{Tx,Ty}(t)\geq F_{x,y} (\frac{t}{\alpha} )\) (\(F_{Tx,Ty}(t)\geq F_{x,y} (t)\)) for some \(0<\alpha<1 \) and for all \(x,y\in A \) and \(t>0 \).
Definition 1.10
Definition 1.11
Clearly, if \(A_{0}\) (or \(B_{0}\)) is a nonempty subset, then A and B are nonempty subsets.
Definition 1.12
Example 1.13
Let \(X=[0,2] \), and \(T:X\rightarrow X\) be the mapping defined by \(Tx=\frac{1}{8}x \). If \(F_{x,y}(t)=\frac{t}{t+|x-y|} \), then it is easy to check that \(F_{X,X}(t)=1 \). If \(F_{u,Tx}(t)=1=F_{v,Ty}(t)\), then for \(\alpha=\frac{1}{8} \), we have \(F_{u,v}(t)= F_{x,y}(\frac{t}{\alpha})\), where \(u,v,x,y \in X\). Therefore, T is a proximal contraction.
Definition 1.14
Let X be a vector space, and A be a nonempty subset of X. Then the subset A is called a p-star-shaped set if there exists a point \(p \in A \) such that \(\alpha p + (1-\alpha)x \in A\) for all \(x\in A\), \(\alpha\in [0,1] \), and p is called the center of A.
Definition 1.15
Example 1.16
Example 1.17
Definition 1.18
Let \((X, F) \) be a PM space. A pair \((A,B) \) of nonempty subsets of X is called a semisharp proximinal pair if there exists at most one \((x_{0}, y_{0}) \in A \times B\) such that \(F_{x, y_{0}}(t) = F_{A,B}(t)=F_{x_{0}, y}(t) \) for all \((x, y) \in A \times B \).
It is easy to check that if a pair \((A,B) \) has the P-property, then the pair \((A,B) \) is a semisharp proximinal pair. Clearly, a semisharp proximinal pair \((A,B) \) does not necessarily have the P-property.
Example 1.19
Suppose that \(X =\mathbb{R} \), \(A=\{-10,10\} \), \(B=\{-2,2\} \), and \(F_{x,y}(t)=\frac{t}{t+|x-y|} \). It is easy to verify that \(F_{A,B}(t)=\frac{t}{t+8} \), \(A_{0}= A\), \(B_{0}= B\), and \(( A,B) \) is a semisharp proximinal pair but does not have the P-property.
Remark 1.20
Definition 1.21
In Section 2, we show some results on the best proximity points in probabilistic Banach (Menger) spaces. For example, if \((A,B) \) is a semisharp proximinal pair of a probabilistic Banach space \((X, \nu,\Delta_{m}) \) such that A is a p-star-shaped set, \(A_{0}\) is a nonempty compact set, B is a q-star-shaped set and \(\nu_{p - q}(t)=\nu_{A-B}(t)\) for all \(t>0\), then every proximal nonexpansive mapping \(T : A\rightarrow B \) with \(T(A_{0} )\subseteq B_{0}\) has a best proximity point. We also prove that if A is a nonempty, compact, and convex subset of a probabilistic Banach space \((X, \nu,\Delta_{m}) \) and \(T:A\rightarrow A \) is a nonexpansive mapping, then T has a fixed point. Finally, we give some examples which defend our main results.
2 Proximity point for proximal contraction and proximal nonexpansive mappings
We first give the following lemma and then we state the main results of this paper. We recall that if \(A_{0}\) (or \(B_{0}\)) is a nonempty subset, then A and B are nonempty subsets.
Lemma 2.1
Let \((X, F,\Delta) \) be a complete probabilistic Menger space such that Δ is a t-norm of H-type, and \(A,B \subseteq X \) be such that \(A_{0} \) is a nonempty closed set. If \(T : A\rightarrow B \) is a proximal contraction mapping such that \(T(A_{0} )\subseteq B_{0}\), then there exists a unique \(x\in A_{0} \) such that \(F_{x,Tx}(t)=F_{A,B}(t) \) for all \(t>0\).
Proof
Proposition 2.2
Proof
Corollary 2.3
Let the hypotheses of Lemma 2.1 be satisfied. Suppose that \(T : A\rightarrow B \) is a proximal contraction mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g: A\rightarrow A\) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \). Then there exists a unique \(x\in A_{0} \) such that \(F_{gx,Tx}(t)=F_{A,B}(t) \).
Proof
By Proposition 2.2, \(Tg^{-1}:G=g(A)\rightarrow B\) is proximal contraction, and \(Tg^{-1}(G_{0})=Tg^{-1}(A_{0})\subseteq T(A_{0})\subseteq B_{0}\). Now by Lemma 2.1 there exists a unique \(x'\in A_{0} \) such that \(F_{x',Tg^{-1}x'}(t)=F_{A,B}(t) \). Since \(A_{0}\subseteq g(A_{0})\), there exists \(x\in A_{0} \) such that \(x'=g(x)\), so that \(F_{g(x),Tx}(t)=F_{A,B}(t) \). Note that g is an injective mapping, therefore, by Lemma 2.1, x is unique, and hence the result follows. □
Theorem 2.4
Let \((X, \nu,\Delta_{m}) \) be a probabilistic Banach space, \(A,B \subseteq X \) be such that A is a convex set, \(A_{0}\) be a nonempty compact set, and B be a bounded convex set. Suppose that \(T : A\rightarrow B \) is a continuous affine and proximal nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g : A\rightarrow A \) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \). Then there exists an element \(x\in A_{0} \) such that \(\nu _{gx-Tx}(t)=\nu_{A-B}(t) \) for all \(t>0\).
Proof
Theorem 2.5
Let \((X, F,\Delta) \) be a complete probabilistic Menger space such that Δ is a t-norm of H-type, and \((A,B) \) be a pair of subsets of X with the weak P-property such that \(A_{0}\) is a nonempty closed set. If \(T:A\rightarrow B \) is a contraction mapping such that \(T(A_{0} )\subseteq B_{0}\), then there exists a unique x in A such that \(F_{x,Tx}(t)=F_{A,B}(t) \) for all \(t>0 \).
Clearly, the pair \((A, A) \) has the P-property, so we have the following result.
Corollary 2.6
Let \((X, F,\Delta) \) be a complete probabilistic Menger space such that Δ is a t-norm of H-type. Then every contraction self-mapping from each nonempty closed subset of X has a unique fixed point.
Theorem 2.7
Let \((X, \nu,\Delta_{m}) \) be a probabilistic Banach space, and \((A,B) \) be a semisharp proximinal pair of X such that A is a p-star-shaped set, \(A_{0}\) be a nonempty compact set, B be a q-star-shaped set, and let \(\nu_{p - q}(t)=\nu_{A-B}(t)\) for all \(t>0\). If \(T : A\rightarrow B \) is a proximal nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\), then there exists an element \(x\in A_{0} \) such that \(\nu_{x - Tx}(t)=\nu_{A-B}(t)\) for all \(t>0\).
Proof
Theorem 2.8
Let \((X,\nu,\Delta_{m}) \) be a probabilistic Banach space, \((A,B) \) be a semisharp proximinal pair of X with the weak P-property such that A is a p-star-shaped set, \(A_{0} \) be a nonempty compact set, B be a q-star-shaped set, and let \(\nu_{p - q}(t)=\nu_{A-B}(t)\) for all \(t>0\). If \(T:A\rightarrow B \) is a nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\), then T has a best proximity point in \(A_{0} \).
Proposition 2.9
Proof
The result follows by using a similar argument as in the proof of Proposition 2.2. □
The following theorem is an immediate consequence of Theorem 2.7 and Proposition 2.9.
Theorem 2.10
Let \((X, \nu,\Delta_{m}) \) be a probabilistic Banach space, \((A,B) \) be a semisharp proximinal pair of X such that A is a p-star-shaped set, \(A_{0}\) be a nonempty compact set, B be a q-star-shaped set, and let \(\nu_{p - q}(t)=\nu_{A-B}(t)\) for all \(t>0\). If \(T : A\rightarrow B \) is a proximal nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g : A\rightarrow A \) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \), then there exists an element \(x\in A_{0} \) such that \(\nu_{gx - Tx}(t)=\nu_{A-B}(t)\) for all \(t>0\).
Corollary 2.11
Let \((X, \nu,\Delta_{m}) \) be a probabilistic Banach space, and let \((A,B) \) be a pair of convex subsets of X with the P-property such that \(A_{0}\) is a nonempty compact set. If \(T : A\rightarrow B \) is a nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g : A\rightarrow A \) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \), then there exists an element \(x\in A_{0} \) such that \(\nu _{gx-Tx}(t)=\nu_{A-B}(t) \) for all \(t>0\).
In Corollary 2.11, if \(g(x)=x\), then we have the following corollary.
Corollary 2.12
With the hypotheses of the previous corollary, if \(T:A\rightarrow B \) is a nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\), then T has a best proximity point.
In Corollary 2.12, if \(A=B\), then we have the following corollary.
Corollary 2.13
If A is a nonempty, compact, and convex subset of a probabilistic Banach space \((X, \nu,\Delta_{m}) \) and \(T:A\rightarrow A \) is a nonexpansive mapping, then T has a fixed point.
In the following, we give some examples that defend our main results.
Example 2.14
Example 2.15
The following example shows that the weak P-property of the pair \((A, B) \) cannot be removed from Theorem 2.5.
Example 2.16
Let \(X =\mathbb{R} \), \(A =\lbrace-10,10\rbrace\), \(B = \lbrace-2, 2\rbrace\), and \(F_{p,q}(t)=\frac{t}{t+|p-q|} \). Clearly, \((X, F,\Delta_{m}) \) is a complete probabilistic Menger space. Then \(A_{0} = A \), \(B_{0} = B \), and \(F_{A,B}(t)=\frac{t}{t+8} \). Let \(T : A \rightarrow B\) be a mapping given by \(T (-10) = 2 \) and \(T (10) = -2 \). It is easy to see that for \(\alpha=\frac{1}{5} \), T is a contraction mapping with \(T (A_{0})\subseteq B_{0}\). The mapping T does not have any best proximity point because \(F_{x,Tx}(t)=\frac{t}{t+12} < \frac{t}{t+8}= F_{A,B}(t)\) for all \(x \in A \). It should be noted that the pair \((A, B) \) does not have the weak P-property.
Example 2.17
Example 2.18
Example 2.19
Declarations
Acknowledgements
The authors would like to express their sincere appreciation to the Shahrekord University and the Center of Excellence for Mathematics for financial support.
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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