Relation-Theoretic Fixed Point Theorems for Generalized Weakly Contractive Mappings

Abstract

In recent times there have been two prominent trends in metric fixed point theory. One is the use of weak contractive inequalities and the other is the use of binary relations. Combining the two trends, in this paper we establish a relation-theoretic fixed point result for a mapping which is defined on a metric space with an arbitrary binary relation and satisfies a weak contractive inequality for any pair of points whenever the pair of points is related by a given relation. The uniqueness is obtained by assuming some extra conditions. The metric space is assumed to be R-complete. We use R-continuity of functions. The property of local T-transitivity of the relation R is used in the main theorem. There is an illustrative example. An existing fixed point result is generalized through the present work. We use a method in the proof of our main theorem which is a blending of relation-theoretic and analytic approaches.

Fixed Point Theorems for Nonexpansive Mappings under Binary Relations

“…In 2015, Alam and Imdad [17] established a novel version of the Banach contraction principle, employing an amorphous binary relation. In recent years, various metrical fixed theorems were proved under different types of contractivity conditions, employing certain binary relations (e.g., [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]). In such results, the involved contraction conditions remain relatively weaker than the usual contraction conditions, as these are required to hold merely for those elements which are related in the underlying binary relation.…”

Coincidence Point Results on Relation Theoretic F

“…In the development of the metric fixed-point theory, one of the main pillar is the Banach contraction principle [1], which states that every contraction on a complete metric space has a unique fixed point. Due to its extensive application potential, this concept has been observed in various forms over the years (see [2][3][4][5][6][7][8][9]).…”