Iterating Fixed Point via Generalized Mann’s Iteration in Convex b-Metric Spaces with Application

Abstract

This manuscript investigates fixed point of single-valued Hardy-Roger’s type $F$ -contraction globally as well as locally in a convex $b$ -metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom (F3) of the $F$ -contraction is removed, and thus the mapping $F$ is relaxed. An important approach used in the article is, though a subset closed ball of a complete convex $b$ -metric space is not necessarily complete, the convergence of the Cauchy sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations.