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
MSC
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
References
- Fan, K: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 122, 234-240 (1969) View ArticleGoogle Scholar
- Reich, S: Approximate selections, best approximations, fixed points, and invariant sets. J. Math. Anal. Appl. 62(1), 104-113 (1978) MATHMathSciNetView ArticleGoogle Scholar
- Reich, S: Fixed points in locally convex spaces. Math. Z. 125, 17-31 (1972) MATHMathSciNetView ArticleGoogle Scholar
- Sehgal, VM, Singh, SP: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Am. Math. Soc. 102(3), 534-537 (1988) MATHMathSciNetGoogle Scholar
- Eldred, AA, Kirk, WA, Veeramani, P: Proximinal normal structure and relatively nonexpansive mappings. Stud. Math. 171(3), 283-293 (2005) MATHMathSciNetView ArticleGoogle Scholar
- Al-Thagafi, MA, Shahzad, N: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 70(3), 1209-1216 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Al-Thagafi, MA, Shahzad, N: Best proximity sets and equilibrium pairs for a finite family of multimaps. Fixed Point Theory Appl. 2008, Article ID 457069 (2008) MathSciNetView ArticleGoogle Scholar
- Al-Thagafi, MA, Shahzad, N: Convergence and existence results for best proximity points. Nonlinear Anal. 70(10), 3665-3671 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Eldred, AA, Veeramani, P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001-1006 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Di Bari, C, Suzuki, T, Vetro, C: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 69(11), 3790-3794 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Karpagam, S, Agrawal, S: Best proximity point theorems for p-cyclic Meir-Keeler contractions. Fixed Point Theory Appl. 2009, Article ID 197308 (2009) MathSciNetView ArticleGoogle Scholar
- Kim, WK, Kum, S, Lee, KH: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 68(8), 222-2216 (2008) MathSciNetView ArticleGoogle Scholar
- Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851-862 (2003) MATHMathSciNetView ArticleGoogle Scholar
- Mongkolkeha, C, Kumam, P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. (2012). doi:10.1007/s10957-012-9991-y MathSciNetGoogle Scholar
- Mongkolkeha, C, Kumam, P: Some common best proximity points for proximity commuting mappings. Optim. Lett. (2012). doi:10.1007/s11590-012-0525-1 Google Scholar
- Prolla, JB: Fixed-point theorems for set-valued mappings and existence of best approximants. Numer. Funct. Anal. Optim. 5(4), 449-455 (1983) MATHMathSciNetView ArticleGoogle Scholar
- Sadiq Basha, S, Veeramani, P: Best approximations and best proximity pairs. Acta Sci. Math. (Szeged) 63, 289-300 (1997) MathSciNetGoogle Scholar
- Sadiq Basha, S, Veeramani, P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103, 119-129 (2000) MATHMathSciNetView ArticleGoogle Scholar
- Sadiq Basha, S, Veeramani, P, Pai, DV: Best proximity pair theorems. Indian J. Pure Appl. Math. 32, 1237-1246 (2001) MATHMathSciNetGoogle Scholar
- Sehgal, VM, Singh, SP: A theorem on best approximations. Numer. Funct. Anal. Optim. 10(1-2), 181-184 (1989) MATHMathSciNetView ArticleGoogle Scholar
- Sintunavarat, W, Kumam, P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012, 93 (2012). doi:10.1186/1687-1812-2012-93 View ArticleGoogle Scholar
- Srinivasan, PS: Best proximity pair theorems. Acta Sci. Math. (Szeged) 67, 421-429 (2001) MATHMathSciNetGoogle Scholar
- Wlodarczyk, K, Plebaniak, R, Banach, A: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 70(9), 3332-3341 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Wlodarczyk, K, Plebaniak, R, Banach, A: Erratum to: ‘Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces’. Nonlinear Anal. 71, 3583-3586 (2009) MathSciNetGoogle Scholar
- Wlodarczyk, K, Plebaniak, R, Obczynski, C: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 72, 794-805 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Menger, K: Statistical metrics. Proc. Natl. Acad. Sci. USA 28, 535-537 (1942) MATHMathSciNetView ArticleGoogle Scholar
- Schweizer, B, Sklar, A: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland, New York (1983) MATHGoogle Scholar
- Schweizer, B, Sklar, A: Statistical metric spaces. Pac. J. Math. 10, 313-334 (1960) MATHMathSciNetView ArticleGoogle Scholar
- Chang, SS, Cho, YJ, Kang, SM: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science Publishers, New York (2001) MATHGoogle Scholar
- Chang, SS, Lee, BS, Cho, YJ, Chen, YQ, Kang, SM, Jung, JM: Generalized contraction mapping principle and differential equations in probabilistic metric spaces. Proc. Am. Math. Soc. 124, 2367-2376 (1996) MATHMathSciNetView ArticleGoogle Scholar
- El Naschie, MS: Fuzzy dodecahedron topology and E-infinity spacetimes as a model for quantum physics. Chaos Solitons Fractals 30(5), 1025-1033 (2006) View ArticleGoogle Scholar
- El Naschie, MS: On gauge invariance, dissipative quantum mechanics and self-adjoint sets. Chaos Solitons Fractals 32(2), 271-273 (2007) View ArticleGoogle Scholar
- El Naschie, MS: P-Adic analysis and the transfinite E8 exceptional Lie symmetry group unification. Chaos Solitons Fractals 38(3), 612-614 (2008) View ArticleGoogle Scholar
- Sehgal, VM, Bharucha-Reid, AT: Fixed points of contraction mappings on probabilistic metric spaces. Math. Syst. Theory 6, 97-102 (1972) MATHMathSciNetView ArticleGoogle Scholar
- Jungck, G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 9, 771-779 (1986) MATHMathSciNetView ArticleGoogle Scholar
- O’Regan, D, Saadati, R: Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 195, 86-93 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Razani, A, Shirdaryazdi, M: A common fixed point theorem of compatible maps in Menger space. Chaos Solitons Fractals 32, 26-34 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Su, Y, Zhang, J: Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces. Fixed Point Theory Appl. 2014, 170 (2014) View ArticleGoogle Scholar
- Hadžić, O, Pap, E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht (2001) Google Scholar