On best proximity point of ψ-Geraghty contractions
© Karapınar; licensee Springer 2013
Received: 12 March 2013
Accepted: 17 May 2013
Published: 24 July 2013
Very recently, Caballero, Harjani and Sadarangani (Fixed Point Theory Appl. 2012:231, 2012) observed some best proximity point results for Geraghty contractions by using the P-property. In this paper, we introduce the notion of ψ-Geraghty contractions and show the existence and uniqueness of the best proximity point of such contractions in the setting of a metric space. We state examples to illustrate our result.
MSC: 41A65, 90C30, 47H10.
Keywordsbest proximity point non-self mapping partial order metric space fixed point
1 Introduction and preliminaries
In nonlinear functional analysis, fixed point theory and best proximity point theory play a crucial role in the establishment of the existence of certain differential and integral equations. As a consequence, fixed point theory is very useful for various quantitative sciences that involve such equations. To list a few, certain branches of computer sciences, engineering and economics are well-known examples in which fixed point theory is used.
The most remarkable paper in this field was reported by Banach  in 1922. In this paper, Banach proved that every contraction in a complete metric space has a unique fixed point. Following this outstanding paper, many authors have extended, generalized and improved this remarkable fixed point theorem of Banach by changing either the conditions of the mappings or the construction of the space. In particular, one of the notable generalizations of Banach fixed point theorem was reported by Geraghty .
Theorem 1.1 (Geraghty )
then T has a unique fixed point.
It is very natural that some mappings, especially non-self-mappings defined on a complete metric space , do not necessarily possess a fixed point, that is, for all . In such situations, it is reasonable to search for the existence (and uniqueness) of a point such that is an approximation of an such that . In other words, one speculates to determine an approximate solution that is optimal in the sense that the distance between and is minimum. Here, the point is called a best proximity point.
This research subject has attracted attention of a number of authors; for example, see [2–23]. In this paper we generalize and improve certain results of Caballero et al. in . Notice also that in the best proximity point theory, we usually consider a non-self-mapping. In fixed point theory, almost all maps are self-mappings. For the sake of completeness, we recall some basic definitions and fundamental results on the best proximity theory.
Let be a metric space and A and B be nonempty subsets of a metric space . A mapping is called a k-contraction if there exists such that for any . It is clear that a k-contraction coincides with the celebrated Banach fixed point theorem if one takes , where A is a complete subset of X.
Definition 1.2 (See )
Theorem 1.2 (See )
Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a continuous Geraghty-contraction satisfying . Suppose that the pair has the P-property. Then there exists a unique in A such that .
In the following section, we improve the theorem above by using a distance function ψ in Definition 1.2. In particular, we introduce Definition 2.1 and broaden the scope of Theorem 1.2 to ψ-Geraghty-contractions.
2 Main results
ψ is nondecreasing;
ψ is subadditive, that is, ;
ψ is continuous;
We introduce the following contraction.
We are now ready to state and prove our main theorem.
Theorem 2.1 Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a ψ-Geraghty contraction satisfying . Suppose that the pair has the P-property. Then there exists a unique in A such that .
Regarding the properties of the function ψ, the limit above contradicts the assumption (14). Therefore, is a Cauchy sequence.
Since and A is a closed subset of the complete metric space , we can find such that .
a contradiction. Therefore, .
which is equivalent to saying that is the best proximity point of T. This completes the proof of the existence of a best proximity point.
a contradiction. This completes the proof. □
Notice that the pair satisfies the P-property for any nonempty subset A of X. Consequently, we have the following corollary.
Corollary 2.1 Let be a complete metric space and A be a nonempty closed subset of X. Let be a ψ-Geraghty-contraction. Then T has a unique fixed point.
Proof Apply Theorem 2.1 with . □
If we take we obtain Theorem 1.2 as a corollary of Theorem 2.1.
Corollary 2.2 Let be a complete metric space and A be a nonempty closed subset of X. Let be a Geraghty-contraction. Then T has a unique fixed point.
Proof Apply Theorem 2.1 with and . □
In order to illustrate our results, we present the following example.
and and .
Since , the pair has the P-property.
Notice that and and .
where is defined as .
More precisely, the point is the best proximity point of T.
which yields that , a contradiction. Therefore condition (6) holds for all . Hence, the conditions of Theorem 2.1 hold and T has a unique best proximity point. Here, is the best proximity point of T.
The author expresses his gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions.
- Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.Google Scholar
- Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5MathSciNetView ArticleGoogle Scholar
- Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Anal. 2009, 70: 3665–3671. 10.1016/j.na.2008.07.022MathSciNetView ArticleGoogle Scholar
- Anuradha J, Veeramani P: Proximal pointwise contraction. Topol. Appl. 2009, 156: 2942–2948. 10.1016/j.topol.2009.01.017MathSciNetView ArticleGoogle Scholar
- Basha SS, Veeramani P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103: 119–129. 10.1006/jath.1999.3415MathSciNetView ArticleGoogle Scholar
- Caballero J, Harjani J, Sadarangani K: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 231Google Scholar
- Jleli, M, Samet, B: Best proximity points for α-ψ-proximal contractive type mappings and applications. Bull. Sci. Math. (in press). doi:10.1016/j.bulsci.2013.02.003. 10.1016/j.bulsci.2013.02.003
- Eldred AA, Veeramani P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 2006, 323: 1001–1006. 10.1016/j.jmaa.2005.10.081MathSciNetView ArticleGoogle Scholar
- De la Sen M: Fixed point and best proximity theorems under two classes of integral-type contractive conditions in uniform metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 510974Google Scholar
- Karapınar E: Best proximity points of cyclic mappings. Appl. Math. Lett. 2012, 25(11):1761–1766. 10.1016/j.aml.2012.02.008MathSciNetView ArticleGoogle Scholar
- Karapınar E, Erhan IM: Best proximity point on different type contractions. Appl. Math. Inf. Sci. 2011, 3(3):342–353.Google Scholar
- Karapınar E: Best proximity points of Kannan type cyclic weak ϕ -contractions in ordered metric spaces. An. Univ. Ovidius Constanţa 2012, 20(3):51–64.Google Scholar
- Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24: 851–862. 10.1081/NFA-120026380MathSciNetView ArticleGoogle Scholar
- Markin J, Shahzad N: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces. Nonlinear Anal. 2009, 70: 2435–2441. 10.1016/j.na.2008.03.045MathSciNetView ArticleGoogle Scholar
- Pragadeeswarar V, Marudai M: Best proximity points: approximation and optimization in partially ordered metric spaces. Optim. Lett. 2012. 10.1007/s11590-012-0529-xGoogle Scholar
- Mongkolkeha C, Cho YJ, Kumam P: Best proximity points for generalized proximal C -contraction mappings in metric spaces with partial orders. J. Inequal. Appl. 2013., 2013: Article ID 94Google Scholar
- Raj VS, Veeramani P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 2009, 10: 21–28.MathSciNetView ArticleGoogle Scholar
- Raj VS: A best proximity theorem for weakly contractive non-self mappings. Nonlinear Anal. 2011, 74: 4804–4808. 10.1016/j.na.2011.04.052MathSciNetView ArticleGoogle Scholar
- Raj, VS: Banach’s contraction principle for non-self mappings. Preprint
- Samet B: Some results on best proximity points. J. Optim. Theory Appl. 2013. 10.1007/s10957-013-0269-9Google Scholar
- Sintunavarat W, Kumam P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 93Google Scholar
- Shahzad N, Basha SS, Jeyaraj R: Common best proximity points: global optimal solutions. J. Optim. Theory Appl. 2011, 148: 69–78. 10.1007/s10957-010-9745-7MathSciNetView ArticleGoogle Scholar
- Srinivasan PS: Best proximity pair theorems. Acta Sci. Math. 2001, 67: 421–429.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.