- Research
- Open Access
Further generalized contraction mapping principle and best proximity theorem in metric spaces
- Yongfu Su^{1} and
- Jen-Chih Yao^{2, 3}Email author
https://doi.org/10.1186/s13663-015-0373-7
© Su and Yao 2015
- Received: 10 February 2015
- Accepted: 30 June 2015
- Published: 17 July 2015
Abstract
The aim of this paper is to prove a more generalized contraction mapping principle. By using this more generalized contraction mapping principle, a further generalized best proximity theorem was established. Some concrete results have been derived by using the above two theorems. The results of this paper improve many important results published recently in the literature.
Keywords
- contraction mapping principle
- complete metric spaces
- fixed point
- best proximity theorem
1 Introduction
In 1973, Geraghty [6] introduced the Geraghty-contraction and obtained the fixed point theorem.
Definition 1.1
Theorem 1.2
([6])
Let \((X, d)\) be a complete metric space and \(T : X \rightarrow X\) be a Geraghty-contraction. Then T has a unique fixed point \(x^{*}\) and for any given \(x_{0} \in X\), the iterative sequence \(T^{n}x_{0}\) converges to \(x^{*}\).
In 2012, Samet et al. [7] defined the notion of α-admissible mappings as follows.
Definition 1.3
([7])
Theorem 1.4
([7])
- (i)for all \(x, y \in X\), we havewhere \(\psi:[0,+\infty)\rightarrow[0,+\infty)\) is a nondecreasing function such that$$\alpha(x,y)d(Tx,Ty)\leq \psi(x,y), $$$$\sum_{n=1}^{+\infty}\psi^{n}(t)< + \infty,\quad \forall t>0; $$
- (ii)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
- (iii)
either T is continuous or for any sequence \(\{x_{n}\}\) in X with \(\alpha(x_{n},x_{n+1})\geq1\) for all \(n\geq0\) and \(x_{n}\rightarrow x\) as \(n\rightarrow+\infty\), then \(\alpha(x_{n},x)\geq1\).
In particular, existence of a fixed point for weak contractions and generalized contractions was extended to partially ordered metric spaces in [1, 8–19]. Among them, some involve altering distance functions. Such functions were introduced by Khan et al. in [20], where they presented some fixed point theorems with the help of such functions. We recall the definition of altering distance function.
Definition 1.5
- (a)
ψ is continuous and nondecreasing;
- (b)
\(\psi=0\) if and only if \(t=0\).
Recently, Harjani and Sadarangani proved some fixed point theorems for weak contraction and generalized contractions in partially ordered metric spaces by using the altering distance function in [11, 19] respectively. Their results improve the theorems of [8].
Theorem 1.6
([11])
Theorem 1.7
([19])
Subsequently, Amini-Harandi and Emami proved another fixed point theorem for contraction type maps in partially ordered metric spaces in [10]. The following class of functions is used in [10].
Theorem 1.8
([10])
In 2012, Yan et al. proved the following result.
Theorem 1.9
([1])
Several problems can be changed as equations of the form \(Tx = x\), where T is a given self-mapping defined on a subset of a metric space, a normed linear space, a topological vector space or some suitable space. However, if T is a non-self mapping from A to B, then the aforementioned equation does not necessarily admit a solution. In this case, it is worth consideration to find an approximate solution x in A such that the error \(d(x, Tx)\) is minimum, where d is the distance function. In view of the fact that \(d(x, Tx)\) is at least \(d(A,B)\), a best proximity point theorem (for short BPPT) guarantees the global minimization of \(d(x, Tx)\) by the requirement that an approximate solution x satisfies the condition \(d(x, Tx) = d(A,B)\). Such optimal approximate solutions are called best proximity points of the mapping T. Interestingly, best proximity point theorems also serve as a natural generalization of fixed point theorems since a best proximity point becomes a fixed point if the mapping under consideration is a self mapping. Research on the best proximity point is an important topic in the nonlinear functional analysis and applications (see [21–34]).
Let A, B be two nonempty subsets of a complete metric space and consider a mapping \(T:A\rightarrow B\). The best proximity point problem is whether we can find an element \(x_{0}\in A\) such that \(d(x_{0},Tx_{0})=\min\{d(x,Tx): x\in A\}\). Since \(d(x,Tx)\geq d(A,B)\) for any \(x\in A\), in fact, the optimal solution to this problem is the one for which the value \(d(A,B)\) is attained.
It is interesting to note that \(A_{0}\) and \(B_{0}\) are contained in the boundaries of A and B respectively provided A and B are closed subsets of a normed linear space such that \(d(A, B)>0\) [28, 29].
Definition 1.10
([33])
In [14], the authors proved that any pair \((A,B)\) of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property.
In [28], P-property was weakened to weak P-property and an example satisfying P-property but not weak P-property can be found there.
Definition 1.11
([28])
Example
([28])
Consider \((R^{2},d)\), where d is the Euclidean distance and the subsets \(A=\{(0,0)\}\) and \(B=\{y=1+\sqrt{1-x^{2}} \}\).
Definition 1.12
([34])
Theorem 1.13
([34])
The aim of this paper is to prove a further generalized contraction mapping principle. By using this further generalized contraction mapping principle, the authors prove a further generalized best proximity theorem. Some concrete results are derived by using the above two theorems. The results of this paper modify and improve many other important recent results.
2 Further generalized contraction mapping principle
In what follows, we prove the following theorem which generalizes many related results in the literature.
Theorem 2.1
Proof
Example 2.2
If we choose \(\psi_{5}(t)\), \(\phi_{5}(t)\) in Theorem 2.1, then we can get the following result.
Theorem 2.3
If we choose \(\psi_{4}(t)\), \(\phi_{4}(t)\) in Theorem 2.1, then we can get the following result.
Theorem 2.4
If we choose \(\psi_{3}(t)\), \(\phi_{3}(t)\) in Theorem 2.1, then we can get the following result.
Theorem 2.5
It is easy to prove the following conclusion and corollary.
Corollary 2.6
- (i)
\(\psi(0)=\phi(0)\);
- (ii)
\(\psi(t)>\phi(t)\), \(\forall t>0\);
- (iii)
ψ is lower semi-continuous, ϕ is upper semi-continuous.
Corollary 2.7
3 Further generalized best proximity point theorems
Before giving our main results, we first introduce the notion of \((\varphi,\psi)\)-P-property.
Definition 3.1
Theorem 3.2
Proof
Theorem 3.3
Proof
Let \(\varphi(t)=\psi(t)\) for all \(t \in [0,+\infty)\). Then the pair \((A,B)\) having the weak P-property implies that the pair \((A,B)\) has the \((\psi,\varphi)\)-P-property. Condition (3) of Theorem 3.3 implies conditions (3), (4) of Theorem 3.2 and (3.2) implies (3.1). By using Theorem 3.2 we get the conclusion of Theorem 3.3. □
If we choose \(\psi_{3}(t)\), \(\phi_{3}(t)\) in Theorem 3.3, then we can get the following result.
Theorem 3.4
If we choose \(\psi_{4}(t)\), \(\phi_{4}(t)\) in Theorem 3.3, then we can get the following result.
Theorem 3.5
Example 3.6
Declarations
Acknowledgements
The second author was partially supported by the grant MOST 103-2923-E-037-001-MY3.
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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