Comparable nonlinear contractions in ordered metric spaces

Abstract

In this article, we generalize some frequently used metrical notions such as: completeness, closedness, continuity, gcontinuity and compatibility to order-theoretic setting especially in ordered metric spaces and utilize these relatively weaker notions to prove some existence and uniqueness results on coincidence points for g-comparable mappings satisfying Boyd-Wong type nonlinear contractivity conditions. We also furnish some illustrative examples to demonstrate our results. Finally, as an application of our certain newly proved results, we establish the existence and uniqueness of solution of an integral equation.

Relation Contractive Selfmaps Involving Cauchy Sequences

“…The uniqueness of the fixed point can be guaranteed from several additional properties of the relation ℜ (cf. [1,4,5,7,8,9,10,11]) which here we do not take under examination.…”

“…Fixed point theorems involving contraction conditions under preserving relations are known in literature (cf. [1,2,3,4,5]). These theorems involve usual sequences of successive approximations in complete metric spaces.…”