Unified relation-theoretic metrical fixed point theorems under an implicit contractive condition with an application

Md Ahmadullah,Javid Ali,Mohammad Imdad

doi:10.1186/s13663-016-0531-6

Md Ahmadullah, Javid Ali + Show 1 more

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https://doi.org/10.1186/s13663-016-0531-6

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Journal: Fixed Point Theory and Applications	Publication Date: Mar 25, 2016
Citations: 13	License type: CC BY 4.0

Affiliation: Aligarh Muslim University

Abstract

AbstractThe main purpose of this article is to establish relation-theoretic metrical fixed point theorems via an implicit contractive condition which is general enough to yield a multitude of corollaries corresponding to several well known contraction conditions (e.g. Banach (Fundam. Math. 3:133-181, 1922), Kannan (Am. Math. Mon. 76:405-408, 1969), Reich (Can. Math. Bull. 14:121-124, 1971), Bianchini (Boll. Unione Mat. Ital. 5:103-108, 1972), Chatterjea (C. R. Acad. Bulg. Sci. 25:727-730, 1972), Hardy and Rogers (Can. Math. Bull. 16:201-206, 1973), Ćirić (Proc. Am. Math. Soc. 45:267-273, 1974) and several others) wherein even such corollaries are new results on their own. As an example we utilize our main results, to prove a theorem on the existence and uniqueness of the solution of an integral equation besides providing an illustrative example.

Highlights

In, Banach formulated the classical contraction mapping principle in his Ph.D. thesis which was later published in Banach [ ]
The aim of this paper is to prove some unified metrical fixed point theorems employing an arbitrary binary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction conditions in one go besides yielding several new ones
Definition [ ] Let R be a binary relation defined on a non-empty set X

Summary

Introduction

In , Banach formulated the classical contraction mapping principle in his Ph.D. thesis which was later published in Banach [ ]. Alam and Imdad [ , ] established a new relationtheoretic version of the Banach contraction principle employing general binary relation which in turn generalizes several well known relevant order-theoretic fixed point theorems. The aim of this paper is to prove some unified metrical fixed point theorems employing an arbitrary binary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction conditions in one go besides yielding several new ones.

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Conclusion