- Research
- Open Access
Coupled best proximity point theorems for proximally g-Meir-Keeler type mappings in partially ordered metric spaces
- Ali Abkar^{1}Email author,
- Samaneh Ghods^{1} and
- Azizollah Azizi^{1}
https://doi.org/10.1186/s13663-015-0355-9
© Abkar et al. 2015
- Received: 25 March 2015
- Accepted: 17 June 2015
- Published: 8 July 2015
Abstract
In this paper we first introduce the notion of proximally g-Meir-Keeler type mappings, then we study the existence and uniqueness of coupled best proximity points for these mappings. This generalization is in line with Edelstein’s generalization of Meir-Keeler type mappings, as well as in line with the recent one used in (Eshaghi Gordji in Math. Probl. Eng. 2012:150363, 2012).
Keywords
- coupled best proximity point
- proximally g-Meir-Keeler type mapping
- proximal mixed strict g-monotone property
MSC
- 47H10
- 54H25
- 47J25
1 Introduction
The Banach contraction principle [1] is a classical and powerful tool in nonlinear analysis. This principle has been generalized in different directions by many authors (see, for example, [2–6] and the references therein). Afterward, Bhaskar and Lakshmikantham [7] introduced the notion of coupled fixed points of a given two-variable mapping F. They also established the uniqueness of coupled fixed point for the mapping F, and successfully applied their results to the problem of existence and uniqueness of solution for a periodic boundary value problem. Following their lines of research, some authors have extended these results in several directions (see, for instance, [8, 9]).
The well-known best approximation theorem, due to Fan [10], asserts that if A is a nonempty, compact and convex subset of a normed linear space X, and T is a continuous mapping from A to X, then there exists a point \(x\in A\) such that the distance of x to Tx is equal to the distance of Tx to A. Such a point x is called a best approximant of T in A. This result was in turn generalized by several authors (see, for example, [11–13], and the references therein).
Recently, Eshaghi Gordji et al. [14], defined the generalized g-Meir-Keeler type contractions and proved some coupled fixed point theorems under a generalized g-Meir-Keeler-contractive condition. In this way, they improved results of Bhaskar and Lakshmikantham [7]. We shall recall their definitions here.
Definition 1.1
[14]
They also defined the mixed strict g-monotone property.
Definition 1.2
[14]
Since we shall be dealing with coupled best proximity points, the following definitions will be needed.
Definition 1.3
Definition 1.4
It is easy to see that if \(A=B\) in Definition 1.4, then a coupled best proximity point is a coupled fixed point of F.
Pragadeeswarar et al. [22] defined a proximally coupled contraction as follows.
Definition 1.5
Motivated by the results of [14] and [22], in this paper we first introduce the notions of proximal mixed strict monotone property and proximally Meir-Keeler type functions and prove the existence and uniqueness of coupled best proximity point theorems for these mappings. This will be implemented in Section 2. In Section 3, we shall discuss some more generalizations of this notions. An example will be provided to illustrate our result.
2 Proximally Meir-Keeler type mappings
In this section we will define proximal mixed strict monotone property and proximally Meir-Keeler type mappings, and prove some theorems in this regard. In the next section we deal with some further generalizations of this topics. Therefore, we shall not provide any proof for the statements made in this section, instead we will comment on how these facts can be inferred from their counterparts in Section 3; indeed, the next section is devoted to proximally g-Meir-Keeler type mappings, as well as to proximal mixed strict g-monotone property, so that if we put \(g=\mathrm{identity}\), we get all the results stated in Section 2.
Definition 2.1
Definition 2.2
Proposition 2.3
Lemma 2.4
Theorem 2.5
- (a)
F is continuous;
- (b)
F has the proximal mixed strict monotone property on A such that \(F(A_{0},A_{0})\subseteq B_{0}\);
- (c)
F is a proximally Meir-Keeler type function;
- (d)
there exist elements \((x_{0},y_{0}), (x_{1},y_{1})\in A_{0} \times A_{0} \) such that \(d(x_{1},F(x_{0},y_{0}))=d(A,B)\) with \(x_{0}< x_{1}\), and \(d(y_{1},F(y_{0},x_{0}))=d(A,B)\) with \(y_{0}\geqslant y_{1}\).
Theorem 2.6
- (a)
F is continuous;
- (b)
F has the proximal mixed strict monotone property on A;
- (c)
F is a proximally Meir-Keeler type function;
- (d)
there exist elements \((x_{0},y_{0}), (x_{1},y_{1})\in A \times A \) such that \(x_{1}=F(x_{0},y_{0})\) with \(x_{0}< x_{1}\), and \(y_{1}=F(y_{0},x_{0})\) with \(y_{0}\geqslant y_{1}\).
Theorem 2.7
- (a)
if \(\{x_{n}\}\) is a nondecreasing sequence in A such that such that \(x_{n}\rightarrow x\), then \(x_{n}< x\) and if \(\{y_{n}\}\) is a nonincreasing sequence in A such that \(y_{n}\rightarrow y\), then \(y_{n}\geqslant y\);
- (b)
F has the proximal mixed strict monotone property on A such that \(F(A_{0},A_{0})\subseteq B_{0}\);
- (c)
F is a proximally Meir-Keeler type function;
- (d)
there exist elements \((x_{0},y_{0}), (x_{1},y_{1})\in A_{0} \times A_{0} \) such that \(d(x_{1},F(x_{0},y_{0}))=d(A,B)\) with \(x_{0}< x_{1}\), and \(d(y_{1},F(y_{0},x_{0}))=d(A,B)\) with \(y_{0}\geqslant y_{1}\).
Remark 2.8
Theorems 2.5 and 2.6 hold true, if we replace the continuity of F by the following.
If \(\{x_{n}\}\) is a nondecreasing sequence in A such that \(x_{n}\rightarrow x\), then \(x_{n}< x\) and if \(\{y_{n}\}\) is a nonincreasing sequence in A such that \(y_{n}\rightarrow y\), then \(y_{n}\geqslant y\).
Theorem 2.9
Suppose that all the hypotheses of Theorem 2.5 hold and, further, for all \((x,y),(x^{\ast},y^{\ast}) \in A_{0} \times A_{0}\), there exists \((u,v) \in A_{0} \times A_{0}\) such that \((u,v) \) is comparable with \((x,y)\), \((x^{\ast},y^{\ast})\). Then there exists a unique \((x,y) \in A \times A\) such that \(d(x,F(x,y))=d(A,B)\) and \(d(y,F(y,x))=d(A,B)\).
Example 2.10
Note also that the other hypotheses of Theorem 2.9 are satisfied, then there exists a unique point \((3,3) \in A \times A \) such that \(d(3,F(3,3))=6=d(A,B)\).
3 Proximally g-Meir-Keeler type mappings
Let X be a nonempty set. We recall that an element \((x,y) \in X \times X\) is called a coupled coincidence point of two mappings \(F: X \times X \rightarrow X \) and \(g: X \rightarrow X \) provided that \(F(x,y)= g(x) \) and \(F(y,x)= g(y)\) for all \(x,y \in X\). Also, we say that F and g are commutative if \(g(F(x,y))= F(g(x),g(y))\) for all \(x,y \in X\).
We now present the following definitions.
Definition 3.1
Definition 3.2
Proposition 3.3
Proof
Suppose (1) is satisfied. For every \(\epsilon> 0\), we set \(\epsilon_{0}=\epsilon/2\) and \(\delta(\epsilon_{0})=\epsilon_{0}\). Now, if \(\epsilon_{0} \leq\frac{k}{2} [d(g(x_{1}),g(x_{2})) + d(g(y_{1}),g(y_{2}))] < \epsilon+ \delta(\epsilon)\), it follows from (1) that \(d(g(u_{1}),g(u_{2})) \leq\frac{k}{2} [d(g(x_{1}),g(x_{2})) + d(g(y_{1}),g(y_{2}))] < \epsilon_{0} + \delta(\epsilon_{0})= \epsilon\), i.e., \(d(g(u_{1}),g(u_{2}))<\epsilon\). □
Lemma 3.5
Proof
Lemma 3.7
Proof
Lemma 3.8
Proof
The proof is similar to that of Lemma 3.7, so we omit the details. □
Theorem 3.9
- (a)
F and g are continuous;
- (b)
F has the proximal mixed strict g-monotone property on A such that \(g(A_{0}) = A_{0} \), \(F(A_{0},A_{0})\subseteq B_{0} \);
- (c)
F is a proximally g-Meir-Keeler type function;
- (d)
there exist elements \((x_{0},y_{0}), (x_{1},y_{1})\in A_{0} \times A_{0} \) such that \(d(g(x_{1}),F(g(x_{0}),g(y_{0})))=d(A,B)\) with \(g(x_{0})< g(x_{1})\), and \(d(g(y_{1}),F(g(y_{0}),g(x_{0})))=d(A,B)\) with \(g(y_{0})\geq g(y_{1})\).
Proof
Let \((x_{0},y_{0}),(x_{1},y_{1}) \in A_{0} \times A_{0} \) be such that \(d(g(x_{1}),F(g(x_{0}),g(y_{0})))=d(A,B)\) with \(g(x_{0})< g(x_{1})\), and \(d(g(y_{1}),F(g(y_{0}),g(x_{0})))=d(A,B)\) with \(g(y_{0})\geq g(y_{1})\).
Since \(F(A_{0},A_{0})\subseteq B_{0}\), \(g(A_{0}) = A_{0}\), there exists an element \((x_{2},y_{2}) \in A_{0} \times A_{0} \) such that \(d(g(x_{2}),F(g(x_{1}),g(y_{1})))=d(A,B)\) and \(d(g(y_{2}),F(g(y_{1}),g(x_{1})))=d(A,B)\).
Hence from Lemma 3.7 and Lemma 3.8 we obtain \(g(x_{1})< g(x_{2})\) and \(g(y_{1})> g(y_{2})\).
Corollary 3.11
- (a)
F and g are continuous;
- (b)
F has the proximal mixed strict g-monotone property on A such that \(F(g(A),g(A))\subseteq g(A)\);
- (c)
F is a proximally g-Meir-Keeler type function;
- (d)
there exist elements \((x_{0},y_{0}), (x_{1},y_{1})\in A \times A\) such that \(g(x_{1})=F(g(x_{0}),g(y_{0}))\) with \(g(x_{0})< g(x_{1})\), and \(g(y_{1})=F(g(y_{0}),g(x_{0}))\) with \(g(y_{0})\geqslant g(y_{1})\).
Theorem 3.12
- (a)
g is continuous;
- (b)
F has the proximal mixed strict g-monotone property on A such that \(F(A_{0},A_{0})\subseteq B_{0}\), \(g(A_{0}) = A_{0}\);
- (c)
F is a proximally g-Meir-Keeler type function;
- (d)
there exist elements \((x_{0},y_{0}), (x_{1},y_{1})\in A_{0} \times A_{0} \) such that \(d(g(x_{1}),F(g(x_{0}),g(y_{0})))=d(A,B)\) with \(g(x_{0})< g(x_{1})\), and \(d(g(y_{1}),F(g(y_{0}),g(x_{0})))=d(A,B)\) with \(g(y_{0})\geqslant g(y_{1})\);
- (e)
if \(\{x_{n}\}\) is a nondecreasing sequence in A such that \(x_{n}\rightarrow x\), then \(x_{n}< x\) and if \(\{y_{n}\}\) is a nonincreasing sequence in A such that \(y_{n}\rightarrow y\), then \(y_{n}\geqslant y\).
Proof
From (e), we get \(g(x_{n})< g(x)\), and \(g(y_{n})\geqslant g(y)\). Since \(F(A_{0},A_{0})\subseteq B_{0}\), it follows that \(F(g(x),g(y))\) and \(F(g(y),g(x))\) are in \(B_{0}\). Therefore, there exists \((x^{\star}_{1},y^{\star}_{1})\in A_{0} \times A_{0} \) such that \(d(x^{\star}_{1},F(g(x),g(y)))=d(A,B)\) and \(d(y^{\star }_{1},F(g(y),g(x)))=d(A,B)\).
Remark 3.13
Corollary 3.11 holds true if we replace the continuity of F by the following statement.
If \(\{x_{n}\}\) is a nondecreasing sequence in A such that \(x_{n}\rightarrow x\), then \(x_{n}< x\) and if \(\{y_{n}\}\) is a nonincreasing sequence in A such that \(y_{n}\rightarrow y\), then \(y_{n}\geqslant y\).
Theorem 3.14
Suppose that all the hypotheses of Theorem 3.9 hold and, further, for all \((x,y),(x^{\ast},y^{\ast}) \in A_{0} \times A_{0} \), there exists \((u,v) \in A_{0} \times A_{0} \) such that \((u,v) \) is comparable to \((x,y)\), \((x^{\ast},y^{\ast})\) (with respect to the ordering in \(A \times A\)). Then there exists a unique \((x,y) \in A \times A\) such that \(d(g(x),F(g(x),g(y)))=d(A,B)\) and \(d(g(y),F(g(y),g(x)))=d(A,B)\).
Proof
First, let \((g(x),g(y))\) be comparable to \((g(x^{\ast}),g(y^{\ast}))\) with respect to the ordering in \(A \times A\).
Second, let \((g(x),g(y))\) is not comparable to \((g(x^{\ast}),g(y^{\ast}))\), then there exists \((g(u_{1}),g(v_{1})) \) in \(A_{0} \times A_{0} \) which is comparable to \((g(x),g(y))\) and \((g(x^{\ast}),g(y^{\ast}))\). Since \(F(A_{0},A_{0})\subseteq B_{0}\) and \(g(A_{0})=A_{0}\), there exists \((g(u_{2}),g(v_{2}))\in A_{0} \times A_{0}\) such that \(d(g(u_{2}),F(g(u_{1}),g(v_{1})))=d(A,B)\) and \(d(g(v_{2}),F(g(v_{1}),g(u_{1})))=d(A,B)\).
We shall illustrate the above theorem by the following example.
Example 3.15
Let X be the real numbers endowed with usual metric and consider the usual ordering \((x,y)\leq(u,v)\Longleftrightarrow x\leq u \), \(y\leq v\). Suppose that \(A=[2,+\infty)\) and \(B=(-\infty,-2]\). Then A, B are nonempty closed subsets of X. Also, we have \(d(A,B)=4\) and \(A_{0}=\lbrace2 \rbrace\) and \(B_{0}= \lbrace-2 \rbrace\).
Note also that the other hypotheses of Theorem 3.14 are satisfied, then there exists a unique point \((2,2) \in A \times A\) such that \(d(g(2),F(g(2),g(2)))=4=d(A,B)\).
Declarations
Acknowledgements
This research was done while the first author was on a sabbatical leave at State University of New York at Albany (SUNYA); he wishes to thank SUNYA for its hospitality and Imam Khomeini International University for the 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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