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.
Highlights
Introduction and PreliminariesThe present paper is a relation-theoretic fixed point result of a self-mapping on a metric space which is assumed to satisfy a weak contraction inequality
Further we note that the above theorems combine two trends of research in metric fixed point theory, one in which contractive conditions are weakened in a particular way by using weak inequalities and the other in which restricting the relevant pairs of points with respect to which contractive conditions are to be satisfied through the introduction of appropriate relations
The present work is in line with research exploring fixed point properties of various types of contractions under restrictions imposed by relations introduced in the original space on which the function is defined
Summary
The present paper is a relation-theoretic fixed point result of a self-mapping on a metric space which is assumed to satisfy a weak contraction inequality. In the present context we prove a fixed point result for a self-mapping satisfying a weak contraction inequality for choices of points which are related by some relation having specific properties both in relation to the metric spaces and the function defined on it. R-preserving sequence: [31] Let X be any nonempty set with a binary relation R on it. Path of length k: [33] Let X be a nonempty set with a binary relation R on it. T-Transitive relation: [26] Let X be a nonempty set with a self-mapping T on it.
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