Best proximity points of generalized almost ψ-Geraghty contractive non-self-mappings
© Aydi et al.; licensee Springer. 2014
Received: 11 November 2013
Accepted: 24 January 2014
Published: 11 February 2014
In this paper, we introduce the new notion of almost ψ-Geraghty contractive mappings and investigate the existence of a best proximity point for such mappings in complete metric spaces via the weak P-property. We provide an example to validate our best proximity point theorem. The obtained results extend, generalize, and complement some known fixed and best proximity point results from the literature.
MSC:47H10, 54H25, 46J10, 46J15.
Keywordsfixed point metric space best proximity point generalized almost ψ-Geraghty contractions
1 Introduction and preliminaries
Non-self-mappings are among the intriguing research directions in fixed point theory. This is evident from the increase of the number of publications related with such maps. A great deal of articles on the subject investigate the non-self-contraction mappings on metric spaces. Let be a metric space and A and B be nonempty subsets of X. A mapping is said to be 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 (Banach contraction principle)  if one takes where the induced metric space is complete.
In nonlinear analysis, the theory of fixed points is an essential instrument to solve the equation for a self-mapping T defined on a subset of an abstract space such as a metric space, a normed linear space or a topological vector space. Following the Banach contraction principle, most of the fixed point results have been proved for a self-mapping defined on an abstract space. It is quite natural to investigate the existence and uniqueness of a non-self-mapping which does not possess a fixed point. If a non-self-mapping has no fixed point, then the answer of the following question makes sense: Is there a point such that the distance between x and Tx is closest in some sense? Roughly speaking, best proximity theory investigates the existence and uniqueness of such a closest point x. We refer the reader to [2–9] and [10–32] for further discussion of best proximity.
It is clear that the notion of a fixed point coincided with the notion of a best proximity point when the underlying mapping is a self-mapping.
In , the authors presented sufficient conditions for the sets and to be nonempty.
The author defined contraction mappings via functions from this class and proved the following result.
Theorem 1.1 (Geraghty )
where , then T has a unique fixed point.
Recently, Caballero et al.  introduced the following contraction.
Definition 1.2 ()
Based on Definition 1.2, the authors  obtained the following result.
Theorem 1.2 (See )
Let be a pair of nonempty closet 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 .
The P-property mentioned in the theorem above has been introduced in .
It is easily seen that for any nonempty subset A of , the pair has the P-property. In , the author proved that any pair of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property.
Recently, Zhang et al.  defined the following notion, which is weaker than the P-property.
ψ is nondecreasing;
ψ is subadditive, that is, ;
ψ is continuous;
The notion of ψ-Geraghty contraction has been introduced very recently in , as an extension of Definition 1.2.
In , the author also proved the following best proximity point theorem.
Theorem 1.3 (See )
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 .
2 Main results
Our main results are based on the following definition which is a generalization of Definition 1.5.
Now, we state and prove our main theorem about existence and uniqueness of a best proximity point for a non-self-mapping satisfying a generalized almost ψ-Geraghty contraction.
Theorem 2.1 Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a generalized almost ψ-Geraghty contraction satisfying . Assume that the pair has the weak P-property. Then T has a unique best proximity point in A.
that is, . Therefore, . This implies that , which is a contradiction. Therefore, is a Cauchy sequence.
and so , which is a contradiction. Therefore, , that is, . In other words, is a best proximity point of T. This completes the proof of the existence of a best proximity point.
which is not possible, since ψ is nondecreasing. Therefore, we must have . This completes the proof. □
To illustrate our result given in Theorem 2.1, we present the following example, which shows that Theorem 2.1 is a proper generalization of Theorem 1.2.
Obviously, . Let be defined by . Notice that , and . Also, it is clear that the pair has the weak P-property.
We shall show that T is a generalized almost ψ-Geraghty contraction. Without loss of generality, consider the case where . Then we have and .
Thus, all hypotheses of Theorem 2.1 are satisfied, and is the unique best proximity point of the map T.
Then Theorem 1.2 (the main result of Caballero et al. ) is not applicable.
Similarly, we cannot apply Theorem 1.3 because T is not a ψ-Geraghty contraction. Let , and with . Then T does not satisfy (8).
If in Theorem 2.1 we take for all , then we deduce the following corollary.
Assume that the pair has the weak P-property. Then T has a unique best proximity point in A.
If further in the above corollary we take where , then we deduce another particular result.
Assume that the pair has the weak P-property. Then T has a unique best proximity point in A.
3 Application to fixed point theory
The case in Theorem 2.1 corresponds to a self-mapping and results in an existence and uniqueness theorem for a fixed point of the map T. We state this case in the next theorem.
Then T has a unique fixed point.
Finally, taking in Theorem 3.1, we get another fixed point result.
Then T has a unique fixed point.
Remark 3.1 The best proximity theorem given in this work, more precisely Theorem 2.1, is a quite general result. It is a generalization of Theorem 2.1 in , Theorem 8 in , and also Theorem 1.2 given in Section 1. In addition, Corollary 3.1 improves Theorem 1.1.
Remark 3.2 Very recently, Karapınar and Samet  proved that the function on the set X, where is also a metric on X. Therefore, some of the fixed theorems regarding contraction mappings defined via auxiliary functions from the set Ψ can be in fact deduced from the existing ones in the literature. However, our main result given in Theorem 2.1 is not a consequence of any existing theorems due to the fact that the contraction condition contains the term .
On the other hand, the definition of can be used to show that Theorem 3.1 follows from Corollary 3.1. Nevertheless, Corollary 3.1 and hence Theorem 3.1 are still new results.
The authors thank to the referees for their careful reading and valuable comments and remarks which contributed to the improvement of the article.
- Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.MATHGoogle Scholar
- Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Anal. 2009, 70: 3665–3671. 10.1016/j.na.2008.07.022View ArticleMathSciNetMATHGoogle Scholar
- Anuradha J, Veeramani P: Proximal pointwise contraction. Topol. Appl. 2009, 156: 2942–2948. 10.1016/j.topol.2009.01.017View ArticleMathSciNetMATHGoogle Scholar
- Basha SS, Veeramani P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103: 119–129. 10.1006/jath.1999.3415View ArticleMathSciNetMATHGoogle Scholar
- Bilgili N, Karapınar E, Sadarangani K: A generalization for the best proximity point of Geraghty-contractions. J. Inequal. Appl. 2013. 10.1186/1029-242X-2013-286Google Scholar
- Caballero J, Harjani J, Sadarangani K: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012. 10.1186/1687-1812-2012-231Google Scholar
- Jleli M, Samet B: Best proximity points for α - ψ -proximal contractive type mappings and applications. Bull. Sci. Math. 2013, 137: 977–995. 10.1016/j.bulsci.2013.02.003View ArticleMathSciNetMATHGoogle Scholar
- Eldred AA, Veeramani P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 2006, 323: 1001–1006. 10.1016/j.jmaa.2005.10.081View ArticleMathSciNetMATHGoogle 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. 10.1155/2010/510974Google Scholar
- Karapınar E: Best proximity points of cyclic mappings. Appl. Math. Lett. 2012, 25: 1761–1766. 10.1016/j.aml.2012.02.008View ArticleMathSciNetMATHGoogle Scholar
- Karapınar, E: On best proximity point of ψ-Geraghty contractions. PreprintGoogle Scholar
- Karapınar E, Erhan IM: Best proximity point on different type contractions. Appl. Math. Inform. Sci. 2011, 3: 342–353.Google Scholar
- Karapınar E: Best proximity points of Kannan type cylic weak ϕ -contractions in ordered metric spaces. An. Univ. “Ovidius” Constanţa, Ser. Mat. 2012, 20: 51–64.MATHGoogle Scholar
- Karapınar E: On best proximity point of ψ -Geraghty contractions. Fixed Point Theory Appl. 2013. 10.1186/1687-1812-2013-200Google Scholar
- Karapınar E, Samet B: A note on ‘ ψ -Geraghty type contractions’. Fixed Point Theory Appl. 2014. 10.1186/1687-1812-2014-26Google Scholar
- Karapınar E, Pragadeeswarar V, Marudai M: Best proximity point for generalized proximal weak contractions in complete metric space. J. Appl. Math. 2013., 2013: Article ID 150941Google 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-120026380View ArticleMathSciNetMATHGoogle Scholar
- Kumam P, Aydi H, Karapınar E, Sintunavarat W: Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorem. Fixed Point Theory Appl. 2013. 10.1186/1687-1812-2013-242Google 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.2008.03.045View ArticleMathSciNetMATHGoogle Scholar
- Mongholkeha C, Cho YJ, Kumam P: Best proximity points for Geraghty’s proximal contraction mappings. Fixed Point Theory Appl. 2013. 10.1186/1687-1812-2013-180Google Scholar
- Mongholkeha C, Cho YJ, Kumam P: Best proximity points for generalized proximal C -contraction mappings in metric spaces with partial orders. J. Inequal. Appl. 2013. 10.1186/1029-242X-2013-94Google Scholar
- Mongholkeha C, Kumam P: Best proximity points for asymptotic proximal pointwise weaker Meir-Keeler type ψ -contraction mappings. J. Egypt. Math. Soc. 2013, 21(2):87–90. 10.1016/j.joems.2012.12.002View ArticleGoogle Scholar
- Mongholkeha C, Kumam P: Some common best proximity points for proximity commuting mappings. Optim. Lett. 2013, 7(8):1825–1836. 10.1007/s11590-012-0525-1View ArticleMathSciNetGoogle Scholar
- Mongholkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 2012, 155: 215–226. 10.1007/s10957-012-9991-yView ArticleMathSciNetGoogle Scholar
- Nashine HK, Vetro C, Kumam P: Best proximity point theorems for rationasl proximal contractions, for proximity commuting mappings. Fixed Point Theory Appl. 2013. 10.1186/1687-1812-2013-95Google 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
- Raj VS, Veeramani P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 2009, 10: 21–28.View ArticleMathSciNetMATHGoogle Scholar
- Raj VS: A best proximity theorem for weakly contractive non-self mappings. Nonlinear Anal. 2011, 74: 4804–4808. 10.1016/j.na.2011.04.052View ArticleMathSciNetMATHGoogle Scholar
- Raj, VS: Banach’s contraction principle for non-self mappings. PreprintGoogle Scholar
- Samet B: Some results on best proximity points. J. Optim. Theory Appl. 2013. 10.1007/s10957-013-0269-9Google Scholar
- Shahzad N, Basha SS, Jeyaraj R: Common best proximity points: global optimal solutions. J. Optim. Theory Appl. 2011, 148: 69–78. 10.1007/s10957-010-9745-7View ArticleMathSciNetMATHGoogle Scholar
- Srinivasan PS: Best proximity pair theorems. Acta Sci. Math. 2001, 67: 421–429.MATHGoogle Scholar
- Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5View ArticleMathSciNetMATHGoogle Scholar
- Zhang J, Su Y, Cheng Q: A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl. 2013. 10.1186/1687-1812-2013-99Google 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.