Best proximity points for generalized α-ϕ-Geraghty proximal contraction mappings and its applications
- Javad Hamzehnejadi^{1} and
- Rahmatollah Lashkaripour^{1}Email author
https://doi.org/10.1186/s13663-016-0561-0
© Hamzehnejadi and Lashkaripour 2016
Received: 26 November 2015
Accepted: 14 June 2016
Published: 29 June 2016
Abstract
In this paper, we introduce the new notion of generalized α-ϕ-Geraghty proximal contraction mappings and investigate the existence of the best proximity point for such mappings in complete metric spaces. The obtained results extend, generalize, and complement some known fixed and best proximity point results from the literature.
Keywords
MSC
1 Introduction
- (a)
ϕ is nondecreasing;
- (b)
ϕ is continuous;
- (c)
\(\phi(t) = 0 \Leftrightarrow t = 0\).
Definition 1.1
([8])
Definition 1.2
([10])
Let \((X, d)\) be a complete metric space, \(\alpha: X\times X\rightarrow[0,\infty)\) be a function, and let \(T:X\rightarrow X\) be a mapping. The sequence \(\{x_{n}\}\) is said to be α-regular if \(\alpha(x_{n}, x_{n+1}) \geq1\) for all \(n\in \mathbb{N}\) and \(x_{n}\rightarrow x\in X\) as \(n\rightarrow\infty\), implies that there exists a subsequence \(\{x_{n_{k}}\} \text{ of }\{x_{n}\} \) such that \(\alpha(x_{n_{k}}, x)\geq1 \text{ for all } k\).
The main result obtained in [8] is the following fixed point theorem.
Theorem 1.3
- (i)
T is generalized α-ϕ-Geraghty contraction type map;
- (ii)
T is triangular α-admissible;
- (iii)
there exists \(x_{1} \in X\) such that \(\alpha(x_{1}, Tx_{1}) \geq1\);
- (iv)
either, T is continuous, or any sequence \(\{x_{n}\}\) is α-regular;
We refer the reader to [11–13] for further examples.
In this work, we extend the concept of generalized α-ϕ-Geraghty contraction type mappings to generalized α-ϕ-Geraghty proximal contraction mappings to the case of non-self mappings. More precisely, we study the existence and uniqueness of best proximity points for generalized α-ϕ-Geraghty proximal contraction non-self-mappings. Several applications and interesting consequences of our obtained results are presented.
Definition 1.4
Definition 1.5
2 Main results
We start this section with the following definition.
Definition 2.1
In order to illustrate RJ-property, we present some examples.
Example 2.2
Example 2.3
Lemma 2.4
Let \(T: A \rightarrow B\) be a triangular α-proximal admissible mapping. Assume that \(\{x_{n}\}\) is a sequence in A such that \(\alpha(x_{n+1}, x_{n})\geq1\), for all \(n\in\mathbb{N}\). Then we have \(\alpha(x_{n}, x_{m})\geq1\) for all \(m, n \in\mathbb{ N}\) with \(n < m\).
Proof
Definition 2.5
Now we prove the following theorem, which extends, improves, and generalizes some earlier results in the literature on best proximity point theorems.
Theorem 2.6
- (i)
T is a generalized α-ϕ-Geraghty proximal contraction type mapping;
- (ii)
\(T(A_{0})\subseteq B_{0}\) and T is triangular α-proximal admissible;
- (iii)
T has the RJ-property;
- (iv)
if \(\{x_{n}\}\) is a sequence in A such that \(\alpha (x_{n}, x_{n+1}) \geq1\) for all n and \(x_{n}\rightarrow x\in A\) as \(n\rightarrow\infty\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n_{k}}, x)\geq1\) for all k;
- (v)there exist \(x_{0},x_{1}\in A\) such that$$d(x_{1},Tx_{0})=d(A,B)\quad \textit{and} \quad \alpha(x_{0},x_{1})\geq1. $$
Proof
If in Theorem 2.6 we take \(\phi(t) = t\text{ for all }t \geq0\), then we deduce the following corollary.
Corollary 2.7
- (i)T is a generalized α-Geraghty proximal contraction type mapping, that is,where \(M(x,y,u,v)=\max\{d(x,y),d(x,u),d(y,v)\}\), for any \(x,y,u,v\in A\).$$ \textstyle\begin{array}{lcl} \left . \textstyle\begin{array}{l} d (u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B) \end{array}\displaystyle \right \} \quad\Longrightarrow\quad \alpha(x,y)d(u,v) \leq\beta\bigl(M(x,y,u,v)\bigr)M(x,y,u,v), \end{array} $$
- (ii)
The conditions (ii), (iii), (iv) and (v) of Theorem 2.6 are satisfied.
By Example 2.2 a continuous map has the RJ-property and if all conditions of Theorem 2.6 are satisfied, then T has a best proximity point. In the next theorem, we prove that in Theorem 2.6, if mapping T is continuous, then condition (iv) is not needed.
Theorem 2.8
- (i)
The conditions (i), (ii) and (v) of Theorem 2.6 are satisfied;
- (ii)
T is continuous.
Proof
To illustrate our results given in Theorem 2.6, we present the following example, which shows that Theorem 2.6 is a proper generalization of Corollary 2.7.
Example 2.9
Note that \(x^{*}=(0,0)\) is the best proximity point of T.
Applying Example 2.3 and Theorem 2.6 we have the following corollary.
Corollary 2.10
- (i)
The conditions (i), (ii), (iv) and (v) of Theorem 2.6 are satisfied;
- (ii)
B is approximatively compact with respect to A.
3 Applications in fixed point theory
As applications of our results, we prove some new fixed point theorems as follows. We start with the following fixed point theorem which is proved by Karapinar in [8].
Theorem 3.1
- (i)
T is generalized α-ϕ-Geraghty contraction type map;
- (ii)
T is triangular α-admissible;
- (iii)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}) \geq1\);
- (iv)
either, T is continuous, or \(\{x_{n}\}\) is α-regular.
Proof
Remark 3.2
The fixed point of a generalized α-ϕ-Geraghty contraction mapping is unique if it satisfies the following condition:
(\(H_{1}\)) For all \(x, y\in \mathrm{Fix}(T)\), there exists \(z\in X\) such that \(\alpha(x, z) \geq1\) and \(\alpha(z, y)\geq1\).
Let \(\phi(t)=t\). Then we have the following definition and corollary.
Definition 3.3
Corollary 3.4
- (i)
T is generalized α-Geraghty contraction map;
- (ii)
T is triangular α-admissible;
- (iii)
there exists \(x_{0} \in X\) such that \(\alpha(x_{0}, Tx_{0}) \geq1\);
- (iv)
either, T is continuous, or \(\{x_{n}\}\) is α-regular.
Further, if for all \(x, y \in \mathrm{Fix}(T)\), there exists \(z\in X\) such that \(\alpha(x, z) \geq1\) and \(\alpha(z,y) \geq1\), and so fixed point of T is unique.
Declarations
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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