Weak reciprocal continuity and fixed point theorems

Abstract

The aim of the present paper is to introduce the notion of weak reciprocal continuity and obtain fixed point theorems by employing the new notion. The new notion is a proper generalization of reciprocal continuity and is applicable to compatible mappings as well as noncompatible mappings. Our results generalize several fixed point theorems.

A Generalized Common Fixed Point Theorem for Two Families of Self-Maps

“…Then Theorem 1.1 is generalized through an extension of property (EA) to a pair of sequences of self-maps and the notions of weakly compatible self-maps and implicit relation. Interestingly, this will also be a generalization of recent results obtained in [27], [32] and [33].…”

“…Self-maps f and r on X are weakly reciprocally continuous at z ∈ X if for any sequence x n ∞ n=1 ⊂ X with the choice (2.6), we have lim n→∞ f rx n = f z or lim n→∞ rf x n = rz. With these ideas, Pant et al [27] proved: Theorem 2.1. Let f and r be weakly reciprocally continuous self-maps on X satisfying the inclusion f (X) ⊂ r(X) and the inequality d(f x, f y) ≤ ad(rx, ry) + bd(f x, rx) + cd(ry, f y) for all x, y ∈ X, (2.9)…”

“…Self-maps f and r on X are reciprocally continuous at z ∈ X if for any sequence x n ∞ n=1 ⊂ X with the choice (2.6), we have lim n→∞ f rx n = f z and lim n→∞ rf x n = rz. This was further weakened by Pant et al [27] as follows: Definition 2.6. Self-maps f and r on X are weakly reciprocally continuous at z ∈ X if for any sequence x n ∞ n=1 ⊂ X with the choice (2.6), we have lim n→∞ f rx n = f z or lim n→∞ rf x n = rz.…”

“…Brief developments in metrical fixed point theory are covered and a significant generalization of recent results obtained in [18], [27], [32] and [33] is established through an extension of the property (EA) to two sequences of self-maps using the notions of weak compatibility and implicit relation. …”

Coincidence and Fixed Point for Weakly Reciprocally Continuous Single-Valued and Multi-Valued Maps

“…Generalizing reciprocal continuity, Pant et al [10] recently introduced Weak Reciprocal Continuity (w.r.c.) for a pair of single-valued maps as follows:…”