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
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
References
- Yan, F, Su, Y, Feng, Q: A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory Appl. 2012, 152 (2012) MathSciNetView ArticleGoogle Scholar
- Browder, FE: On the convergence of successive approximations for nonlinear functional equations. Proc. K. Ned. Akad. Wet., Ser. A, Indag. Math. 30, 27-35 (1968) MathSciNetView ArticleGoogle Scholar
- Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458-464 (1969) MathSciNetView ArticleMATHGoogle Scholar
- Jachymski, J: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125, 2327-2335 (1997) MathSciNetView ArticleMATHGoogle Scholar
- Jachymski, J, Jóźwik, I: Nonlinear contractive conditions: a comparison and related problems. In: Fixed Point Theory and Its Applications. Banach Center Publ., vol. 77, pp. 123-146. Polish Acad. Sci., Warsaw (2007) View ArticleGoogle Scholar
- Geraghty, M: On contractive mappings. Proc. Am. Math. Soc. 40, 604-608 (1973) MathSciNetView ArticleGoogle Scholar
- Samet, B, Vetro, C, Vetro, P: Fixed point theorem for α-ψ-contractive type mappings. Nonlinear Anal. 75, 2154-2165 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Dhutta, P, Choudhury, B: A generalization of contractions in partially ordered metric spaces. Appl. Anal. 87, 109-116 (2008) MathSciNetView ArticleGoogle Scholar
- Sastry, K, Babu, G: Some fixed point theorems by altering distance between the points. Indian J. Pure Appl. Math. 30, 641-647 (1999) MathSciNetMATHGoogle Scholar
- Amini-Harandi, A, Emami, H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 72, 2238-2242 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Harjani, J, Sadarangni, K: Fixed point theorems for weakly contraction mappings in partially ordered sets. Nonlinear Anal. 71, 3403-3410 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Burgic, D, Kalabusic, S, Kulenovic, M: Global attractivity results for mixed monotone mappings in partially ordered complete metric spaces. Fixed Point Theory Appl. 2009, Article ID 762478 (2009) MathSciNetView ArticleGoogle Scholar
- Ciric, L, Cakid, N, Rajovic, M, Uma, J: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008, Article ID 131294 (2008) View ArticleGoogle Scholar
- Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379-1393 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Lakshmikantham, V, Ciric, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70, 4341-4349 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Nieto, JJ, Rodriguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223-239 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Nieto, JJ, Rodriguez-López, R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. 23, 2205-2212 (2007) View ArticleMATHGoogle Scholar
- O’Regan, D, Petrusel, A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 341, 1241-1252 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Harjani, J, Sadarangni, K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 72, 1188-1197 (2010) MathSciNetView ArticleGoogle Scholar
- Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30(1), 1-9 (1984) MathSciNetView ArticleGoogle Scholar
- Sadiq Basha, S: Best proximity point theorems generalizing the contraction principle. Nonlinear Anal. 74, 5844-5850 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Mongkolkeha, C, Cho, YJ, Kumam, P: Best proximity points for Geraghty’s proximal contraction mappings. Fixed Point Theory Appl. 2013, 180 (2013) MathSciNetView ArticleGoogle Scholar
- Dugundji, J, Granas, A: Weakly contractive mappings and elementary domain invariance theorem. Bull. Greek Math. Soc. 19, 141-151 (1978) MathSciNetMATHGoogle Scholar
- Alghamdi, MA, Alghamdi, MA, Shahzad, S: Best proximity point results in geodesic metric spaces. Fixed Point Theory Appl. 2013, 164 (2013) View ArticleGoogle Scholar
- Nashine, HK, Kumam, P, Vetro, C: Best proximity point theorems for rational proximal contractions. Fixed Point Theory Appl. 2013, 95 (2013) MathSciNetView ArticleGoogle Scholar
- Karapinar, E: On best proximity point of ψ-Geraghty contractions. Fixed Point Theory Appl. 2013, 200 (2013) View ArticleGoogle Scholar
- Zhang, J, Su, Y, Cheng, Q: Best proximity point theorems for generalized contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2013, 83 (2013) View ArticleGoogle Scholar
- Zhang, J, Su, Y, Cheng, Q: A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl. 2013, 99 (2013) View ArticleGoogle Scholar
- Zhang, J, Su, Y: Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces. Fixed Point Theory Appl. 2014, 50 (2014) View ArticleGoogle Scholar
- Amini-Harandi, A, Hussain, N, Akbar, F: Best proximity point results for generalized contractions in metric spaces. Fixed Point Theory Appl. 2013, 164 (2013) MathSciNetView ArticleGoogle Scholar
- Kannan, R: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71-76 (1968) MathSciNetMATHGoogle Scholar
- Caballero, J, Harjani, J, Sadarangani, K: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231 MathSciNetGoogle Scholar
- Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851-862 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Sun, Y, Su, Y, Zhang, J: A new method for the research of best proximity point theorems of nonlinear mappings. Fixed Point Theory Appl. 2014, 116 (2014) MathSciNetView ArticleGoogle Scholar