Best proximity point results in partially ordered metric spaces via simulation functions
- Bessem Samet^{1}Email author
https://doi.org/10.1186/s13663-015-0484-1
© Samet 2015
Received: 10 August 2015
Accepted: 7 December 2015
Published: 23 December 2015
Abstract
We obtain sufficient conditions for the existence and uniqueness of best proximity points for a new class of non-self mappings involving simulation functions in a metric space endowed with a partial order. Some interesting consequences including fixed point results via simulation functions are presented.
Keywords
best proximity point fixed point simulation functionMSC
90C26 47H10 06A061 Introduction
Recently, in [1] the authors introduced the class of simulation functions as follows.
Definition 1.1
- (i)
\(\xi(0,0)=0\);
- (ii)
\(\xi(t,s)< s-t\), for every \(t,s>0\);
- (iii)if \(\{a_{n}\}\) and \(\{b_{n}\}\) are two sequences in \((0,\infty )\), then$$\lim_{n\to\infty} a_{n}=\lim_{n\to\infty} b_{n}>0\quad \Longrightarrow\quad \limsup_{n\to\infty} \xi(a_{n},b_{n})< 0. $$
Various examples of simulation functions were presented in [1]. The class of such functions will be denoted by \(\mathcal{Z}\).
Definition 1.2
([1])
In [1], the authors established the following fixed point theorem that generalizes many previous results from the literature including the Banach fixed point theorem.
Theorem 1.3
([1])
Let \(T: X\to X\) be a given map, where X is a nonempty set equipped with a metric d such that \((X,d)\) is complete. Suppose that T is a \(\mathcal{Z}\)-contraction with respect to \(\xi\in\mathcal{Z}\). Then T has a unique fixed point. Moreover, for any \(x\in X\), the sequence \(\{T^{n}x\}\) converges to this fixed point.
For other results via simulation functions, we refer to [2–7].
Let \((X,d)\) be a metric space. Consider a mapping \(T: A\to B\), where A and B are nonempty subsets of X. If \(d(x,Tx)>0\) for every \(x\in A\), then the set of fixed points of T is empty. In this case, we are interested in finding a point \(x\in A\) such that \(d(x,Tx)\) is minimum in some sense.
Definition 1.4
Observe that if \(d(A,B)=0\), then a best proximity point of T is a fixed point of T.
The study of the existence of best proximity points is an interesting field of optimization and it attracted recently the attention of several researchers (see [1, 8–23] and the references therein).
- (C1)for every \(x_{1},x_{2},y_{1},y_{2}\in A\), we have$$y_{1}\preceq y_{2},\quad d(x_{1},Ty_{1})=d(x_{2},Ty_{2})=d(A,B) \quad \Longrightarrow\quad x_{1}\preceq x_{2}; $$
- (C2)for every \(x,y,u_{1},u_{2}\in A\), we havewhere$$x\preceq y, x\neq y,\quad d(u_{1},Tx)=d(u_{2},Ty)=d(A,B) \quad \Longrightarrow\quad \xi \bigl(d(u_{1},u_{2}),m(x,y) \bigr)\geq0, $$$$m(x,y)=\max \biggl\{ \frac{d(x,u_{1})d(y,u_{2})}{d(x,y)},d(x,y) \biggr\} . $$
Our aim in this paper is to study the existence and uniqueness of best proximity points for non-self mappings \(T: A\to B\) that belong to the class \(\mathcal{T}_{\xi}\), for some simulation function \(\xi\in\mathcal{Z}\).
2 Main results
Our first main result is the following.
Theorem 2.1
- (1)
\((X,d)\) is complete;
- (2)
A is closed with respect to the metric d;
- (3)
\(T(A_{0})\subseteq B_{0}\);
- (4)there exist \(x_{0},x_{1}\in A_{0}\) such that$$d(x_{1},Tx_{0})=d(A,B), \quad x_{0}\preceq x_{1}; $$
- (5)
T is continuous.
Proof
Next, we obtain a best proximity point result for mappings \(T\in \mathcal{T}_{\xi}\) that are not necessarily continuous.
Theorem 2.2
- (1)
\((X,d)\) is complete;
- (2)
\(A_{0}\) is closed;
- (3)
\(T(A_{0})\subseteq B_{0}\);
- (4)there exist \(x_{0},x_{1}\in A_{0}\) such that$$d(x_{1},Tx_{0})=d(A,B),\quad x_{0}\preceq x_{1}; $$
- (5)
A is \((d,\preceq)\)-regular.
Proof
Case 1. If \(|I|=\infty\).
Case 2. If \(|I|<\infty\).
Note that the assumptions in Theorems 2.1 and 2.2 do not guarantee the uniqueness of the best proximity point. The next example shows this fact.
Example 2.3
In the next theorem, we give a sufficient condition for the uniqueness of the best proximity point.
Theorem 2.4
Proof
Case 1. If \(z_{1}\) and \(z_{2}\) are comparable.
Case 2. If \(z_{1}\) and \(z_{2}\) are not comparable.
In the following corollaries we deduce some known and some new results in best proximity point theory via various choices of simulation functions.
- (F1)for every \(x_{1},x_{2},y_{1},y_{2}\in A\), we have$$y_{1}\preceq y_{2},\quad d(x_{1},Ty_{1})=d(x_{2},Ty_{2})=d(A,B) \quad \Longrightarrow\quad x_{1}\preceq x_{2}; $$
- (F2)for every \(x,y,u_{1},u_{2}\in A\), we havefor some constant \(k\in(0,1)\).$$\begin{aligned}& x\preceq y, x\neq y,\quad d(u_{1},Tx)=d(u_{2},Ty)=d(A,B) \\& \quad \Longrightarrow\quad d(u_{1},u_{2})\leq k \max \biggl\{ \frac {d(x,u_{1})d(y,u_{2})}{d(x,y)},d(x,y) \biggr\} , \end{aligned}$$
Take \(\xi(t,s)=ks-t\), for \(t,s\geq0\), we deduce from Theorems 2.1, 2.2 and 2.4 the following results.
Corollary 2.5
- (1)
\((X,d)\) is complete;
- (2)
A is closed with respect to the metric d;
- (3)
\(T(A_{0})\subseteq B_{0}\);
- (4)there exist \(x_{0},x_{1}\in A_{0}\) such that$$d(x_{1},Tx_{0})=d(A,B), \quad x_{0}\preceq x_{1}; $$
- (5)
T is continuous.
Corollary 2.6
- (1)
\((X,d)\) is complete;
- (2)
\(A_{0}\) is closed;
- (3)
\(T(A_{0})\subseteq B_{0}\);
- (4)there exist \(x_{0},x_{1}\in A_{0}\) such that$$d(x_{1},Tx_{0})=d(A,B), \quad x_{0}\preceq x_{1}; $$
- (5)
A is \((d,\preceq)\)-regular.
Corollary 2.7
- (G1)for every \(x_{1},x_{2},y_{1},y_{2}\in A\), we have$$y_{1}\preceq y_{2},\quad d(x_{1},Ty_{1})=d(x_{2},Ty_{2})=d(A,B) \quad \Longrightarrow \quad x_{1}\preceq x_{2}; $$
- (G2)for every \(x,y,u_{1},u_{2}\in A\), we havewhere \(\varphi:[0,\infty)\to[0,\infty)\) is lower semi-continuous function and \(\varphi^{-1}(\{0\})=\{0\}\).$$\begin{aligned}& x\preceq y, x\neq y,\quad d(u_{1},Tx)=d(u_{2},Ty)=d(A,B) \\& \quad \Longrightarrow \quad d(u_{1},u_{2})\leq\max \biggl\{ \frac {d(x,u_{1})d(y,u_{2})}{d(x,y)},d(x,y) \biggr\} \\& \hphantom{\quad \Longrightarrow \quad d(u_{1},u_{2})\leq{}}{}-\varphi \biggl(\max \biggl\{ \frac {d(x,u_{1})d(y,u_{2})}{d(x,y)},d(x,y) \biggr\} \biggr), \end{aligned}$$
Take \(\xi(t,s)=s-\varphi(s)-t\), for \(t,s\geq0\), we deduce from Theorems 2.1, 2.2 and 2.4 the following results obtained in [23].
Corollary 2.8
- (1)
\((X,d)\) is complete;
- (2)
A is closed with respect to the metric d;
- (3)
\(T(A_{0})\subseteq B_{0}\);
- (4)there exist \(x_{0},x_{1}\in A_{0}\) such that$$d(x_{1},Tx_{0})=d(A,B), \quad x_{0}\preceq x_{1}; $$
- (5)
T is continuous.
Corollary 2.9
- (1)
\((X,d)\) is complete;
- (2)
\(A_{0}\) is closed;
- (3)
\(T(A_{0})\subseteq B_{0}\);
- (4)there exist \(x_{0},x_{1}\in A_{0}\) such that$$d(x_{1},Tx_{0})=d(A,B), \quad x_{0}\preceq x_{1}; $$
- (5)
A is \((d,\preceq)\)-regular.
Corollary 2.10
- (H1)for every \(x_{1},x_{2},y_{1},y_{2}\in A\), we have$$y_{1}\preceq y_{2}, \quad d(x_{1},Ty_{1})=d(x_{2},Ty_{2})=d(A,B) \quad \Longrightarrow\quad x_{1}\preceq x_{2}; $$
- (H2)for every \(x,y,u_{1},u_{2}\in A\), we havewhere \(\varphi:[0,\infty)\to[0,1)\) is a function such that \(\limsup_{t\to r^{+}}\varphi(t)<1\), for all \(r>0\).$$\begin{aligned}& x\preceq y, x\neq y,\quad d(u_{1},Tx)=d(u_{2},Ty)=d(A,B) \\& \quad \Longrightarrow \quad d(u_{1},u_{2})\leq\varphi \biggl(\max \biggl\{ \frac {d(x,u_{1})d(y,u_{2})}{d(x,y)},d(x,y) \biggr\} \biggr) \\& \hphantom{\quad \Longrightarrow \quad d(u_{1},u_{2})\leq{}}{}\times\max \biggl\{ \frac {d(x,u_{1})d(y,u_{2})}{d(x,y)},d(x,y) \biggr\} , \end{aligned}$$
Take \(\xi(t,s)=s\varphi(s)-t\), for \(t,s\geq0\), we deduce from Theorems 2.1, 2.2 and 2.4 the following results.
Corollary 2.11
- (1)
\((X,d)\) is complete;
- (2)
A is closed with respect to the metric d;
- (3)
\(T(A_{0})\subseteq B_{0}\);
- (4)there exist \(x_{0},x_{1}\in A_{0}\) such that$$d(x_{1},Tx_{0})=d(A,B),\quad x_{0}\preceq x_{1}; $$
- (5)
T is continuous.
Corollary 2.12
- (1)
\((X,d)\) is complete;
- (2)
\(A_{0}\) is closed;
- (3)
\(T(A_{0})\subseteq B_{0}\);
- (4)there exist \(x_{0},x_{1}\in A_{0}\) such that$$d(x_{1},Tx_{0})=d(A,B),\quad x_{0}\preceq x_{1}; $$
- (5)
A is \((d,\preceq)\)-regular.
Corollary 2.13
Finally, take \(A=B=X\) in Theorems 2.1, 2.2 and 2.4, we obtain the following fixed point theorems.
- (I)for every \(x,y\in X\), we have$$x\preceq y \quad \Longrightarrow\quad Tx\preceq Ty; $$
- (II)for every \(x,y\in X\), we have$$x\preceq y, x\neq y \quad \Longrightarrow\quad \xi \biggl(d(Tx,Ty),\max \biggl\{ \frac{d(x,Tx)d(y,Ty)}{d(x,y)},d(x,y) \biggr\} \biggr)\geq0. $$
Corollary 2.14
- (1)
\((X,d)\) is complete;
- (2)
there exists some \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\);
- (3)
T is continuous.
Corollary 2.15
- (1)
\((X,d)\) is complete;
- (2)
there exists some \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\);
- (3)
X is \((d,\preceq)\)-regular.
Declarations
Acknowledgements
The author extends his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Prolific Research group (PRG-1436-10).
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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