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

Abstract

The 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.