- Research
- Open Access
A new class of \(\mathcal{S}\)-contractions in complete metric spaces and \(\mathcal{G_{P}}\)-contractions in ordered metric spaces
- Somayya Komal^{1} and
- Poom Kumam^{1, 2, 3}Email author
https://doi.org/10.1186/s13663-016-0556-x
© Komal and Kumam 2016
- Received: 11 March 2016
- Accepted: 24 May 2016
- Published: 2 July 2016
Abstract
The basic purpose of this article is to define new so-called \(\mathcal{S}\)-contractions and discuss the presence of common best proximity point theorems for such contractions in the setting of Cauchy metric spaces. We also calculate some common optimal approximate solutions of some fixed point equations when there does not exist any common fixed point. We also introduce the notions of \(\mathcal{G_{P}}\)-functions and \(\mathcal {G_{P}}\)-contractions with the help of \(\mathcal{P}\)-functions and prove the existence of a unique best proximity point in partially order metric spaces. We give some examples that justify the validity of our results. These results extend and unify many existing results in the literature.
Keywords
- common best proximity point
- common fixed point
- best proximity point
- fixed point
- \(\mathcal{S}\)-contraction
- \(\mathcal{G_{P}}\)-function
- \(\mathcal{G_{P}}\)-contraction
- proximally commuting mappings
- P-property
MSC
- 58C30
- 47H10
1 Introduction
When discussing fixed points of various mappings satisfying certain conditions, we see that these maps have many applications and are important tools in various research activities. The Banach contraction principle [1] helps many mathematicians and researchers working in mathematics and mathematical sciences. Many important results of [1, 2], and [3] have become the source of motivation for many researchers and mathematicians that do research in the metric fixed point theory and best proximity point theory. When some self-mapping in a metric space, topological vector space, or any other appropriate space has no fixed points, then we are interested in the existence and uniqueness of some point that minimizes the distance between the origin and its corresponding image known as best proximity point. Best proximity point theorems for several types using different contraction maps are considered in [2, 4–10], and [11]. Best proximity point theorems establish a generalization of fixed points by considering self-mappings. Let sets \(A,B\neq\phi\) of \((X,d)\) with mappings \(S:A\rightarrow B\) and \(T:A\rightarrow B\) be such that the equations \(Tx=x\) and \(Sx=x\) have no common fixed point of the mappings S and T. In such a situation, when there does not exist any type of common solution, it is essential to find an element that is in close distance to Sx and Tx, and such an optimal approximate solution is known as the common best proximity point of the given non-self-mappings. If x is an element that gives the global minimum value for these two mappings S and T, then we write \(d(x,Sx)=d(x,Tx)=d(A,B)\).
Now, the main aim of this paper is to furnish new \(\mathcal {S}\)-contractions and to derive a common best proximity point theorem in the framework of metric spaces for the pair of non-self-mappings and to derive \(\mathcal{G_{P}}\)-contractions and functions to find optimal approximate solutions of certain contractive maps. We present some theorems and examples in favor of our work.
2 Preliminaries
In this section, we consider subsets \(A,B\neq\phi\) of a metric space X with metric d and collect some definitions and mathematical symbols, which will be used in this paper.
Definition 2.1
[8]
Definition 2.2
[8]
We know that best proximity point x for mapping T from A to B is defined as \(d(x,Tx)=d(A,B)\) and a common best proximity point is an element at which both functions S and T attain their global minimum since \(d(x,Sx)\geq d(A,B)\) and \(d(x,Tx)\geq d(A,B)\) for all x.
Definition 2.3
[8]
Definition 2.4
[8]
Definition 2.5
[7]
Theorem 2.1
[1]
Let \((X,d)\) be a complete metric space. Then every contraction mapping has a unique fixed point. It is known as the Banach contraction principle.
Definition 2.6
[12]
- 1.
\(\mathcal{P}(x,y)\geq0\) for every comparable \(x,y \in A\);
- 2.
for any sequences \(\{x_{n}\}\), \(\{y_{n}\}\) in X such that \(x_{n}\) and \(y_{n}\) are comparable at each \(n \in\mathbb{N}\), if \(\lim_{n\rightarrow\infty}x_{n}=x\) and \(\lim_{n\rightarrow\infty}y_{n}=y\), then \(\lim_{n\rightarrow\infty}\mathcal{P}(x_{n},y_{n})=\mathcal{P}(x,y)\);
- 3.
for any sequences \(\{x_{n}\}\), \(\{y_{n}\}\) in X such that \(x_{n}\) and \(y_{n}\) are comparable at each \(n \in\mathbb{N}\), if \(\lim_{n\rightarrow\infty}\mathcal{P}(x_{n},y_{n})=0\), then \(\lim_{n\rightarrow\infty}d(x_{n},y_{n})=0\).
- 4.
for any sequences \(\{x_{n}\}\), \(\{y_{n}\}\) in X such that \(x_{n}\) and \(y_{n}\) are comparable at each \(n \in\mathbb{N}\), if \(\lim_{n\rightarrow\infty}d(x_{n},y_{n})\) exists, then \(\lim_{n\rightarrow \infty}\mathcal{P}(x_{n},y_{n})\) also exists.
Definition 2.7
[12]
3 \(\mathcal{S}\)-Functions and \(\mathcal{S}\)-contractions
Definition 3.1
- 1.
if there exists another mapping \(F:A\rightarrow B\) in \((X,d)\), then \(d(Fx,Fy) < d(\mathcal{S}x,\mathcal{S}y)\) with \(F(A_{0})\subseteq \mathcal{S}(A_{0})\);
- 2.
for any \(A,B \subseteq(X,d)\), if \(A_{0}\) and \(B_{0}\) are nonempty, then \(\mathcal{S}(A_{0}) \subseteq B_{0}\);
- 3.
for any sequence \(\{x_{m}\}\) in A, if \(\lim_{m\rightarrow\infty }x_{m}=x \in A\), then \(\lim_{m\rightarrow\infty}\mathcal {S}x_{m}=\mathcal{S}x \in B\), where \(A\subseteq X\) and \(m\in\mathbb{N}\).
Definition 3.2
We denote by \(\mathcal{F}\) the collection of all mappings \(\beta: [0,\infty) \rightarrow[0,1)\) such that \(\beta(t_{n}) \to1\) implies \(t_{n} \rightarrow0\) as \(n \to\infty\). The following theorem is based on the existence of a unique common best proximity point for non-self-maps and also furnishes fixed point results in Cauchy metric spaces.
Example 3.1
Theorem 3.1
Proof
Our main result asserts that if we take \(\beta(t)=k \in[0,1)\) and take self-mappings in \(A=B=X\) in Theorem 3.1, by the definition of \(\mathcal {S}\)-functions and \(\mathcal{S}\)-contractions we get the following fixed point result of [7] and [8].
Corollary 3.1
- 1.There is a nonnegative real number \(k <1\) such thatfor any \(x_{n}\) and \(x_{n+1}\) in A.$$d(Tx_{n},Tx_{n+1})\leq k d(Sx_{n},Sx_{n+1}) $$
- 2.
S and T commute and are continuous.
- 3.
\(T(X)\subseteq S(X)\).
Further, if we take \(\beta(t)=k \in[0,1)\) and add two extra conditions in Theorem 3.1, then we get the main result of [7]:
Corollary 3.2
- 1.There is a nonnegative real number \(\beta<1\) such thatfor any \(x_{n}\) and \(x_{n+1}\) in A.$$d(Tx_{n},Tx_{n+1})\leq\beta d(Sx_{n},Sx_{n+1}) $$
- 2.
S and T commute proximally, swapped proximally and continuous.
- 3.
\(T(A_{0})\subseteq B_{0}\), \(S(A_{0})\).
- 4.
A is approximatively compact with respect to B.
Proof
By adding hypotheses (4) and the condition that S and T swapped proximally in our Theorem 3.1 we obtain the above well-known result. □
Example 3.2
4 \(\mathcal{G_{P}}\)-Functions and \(\mathcal{G_{P}}\)-contractions
Motivated by [12], we define here the generalized \(\mathcal {P}\)-functions and contractions.
Definition 4.1
- 1.
\(\mathcal{A}(x,y)\geq0\) for every comparable \(x,y \in A\);
- 2.
for any sequences \(\{x_{n}\}\), \(\{y_{n}\}\) in A such that \(x_{n}\) and \(y_{n}\) are comparable at each \(n \in\mathbb{N}\), if \(\lim_{n\rightarrow\infty}x_{n}=x\) and \(\lim_{n\rightarrow\infty}y_{n}=y\), then \(\lim_{n\rightarrow\infty}\mathcal{A}(x_{n},y_{n})=\mathcal{A}(x,y)\);
- 3.
for any sequences \(\{x_{n}\}\), \(\{y_{n}\}\) in A such that \(x_{n}\) and \(y_{n}\) are comparable at each \(n \in\mathbb{N}\), if \(\lim_{n\rightarrow\infty}\mathcal{A}(x_{n},y_{n})=0\), then \(\lim_{n\rightarrow\infty}d(x_{n},y_{n})=0\).
- 4.
for any sequences \(\{x_{n}\}\), \(\{y_{n}\}\) in A such that \(x_{n}\) and \(y_{n}\) are comparable at each \(n \in\mathbb{N}\), if \(\lim_{n\rightarrow\infty}d(x_{n},y_{n})\) exists, then \(\lim_{n\rightarrow \infty}\mathcal{A}(x_{n},y_{n})\) also exists.
Definition 4.2
Theorem 4.1
- 1.
f is a continuous generalized \(\mathcal{P}\)-contraction w.r.t. ⪯ with \(f(A_{0}) \subseteq B_{0}\);
- 2.
the pair \((A, B)\) has the P-property.
Proof
Remark 4.1
By taking \(A=B=X\) in the last theorem we obtain the result of [12].
Example 4.1
Consider \(X=\mathbb{R}^{2}\). Let \(A=\{1\}\times [0,\infty)\) and \(B=\{0\}\times[0,\infty)\), and take \(A_{0}=A \) and \(B_{0}=B\). Here, \(d(A,B)=3\) and \(A, B\neq\phi\) are closed subsets of X,
5 Conclusions
In this article, the authors introduced the new notions of \(\mathcal {S}\)-contractions and \(\mathcal{G_{P}}\)-contractions. These contractions and the results in this paper introduced new techniques for finding optimal approximate and global optimal approximate solutions in ordered metric spaces.
Declarations
Acknowledgements
Somayya Komal was supported by the Petchra Pra Jom Klao Doctoral Scholarship Academic for Ph.D. Program at KMUTT.
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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