FIXED POINT THEORY: NEW CONVERGENCE RESULTS AND GENERALIZED CONTRACTIVE MAPPINGS IN METRIC AND BANACH SPACES

Abstract

Abstract: This paper presents new convergence theorems and generalized contractive mapping principles within the unified framework of metric spaces, Banach spaces, and b-metric spaces. We introduce the notion of the interpolative Reich–Rus–Ćirić contraction of rational type, extend the Banach contraction principle to multi-valued mappings via the Nadler–Feng–Liu framework, and establish a novel three-step iterative scheme with superior convergence rate to existing Mann, Ishikawa, and S-iteration procedures. Three fixed point theorems are proved, with applications to nonlinear integral equations and fractional boundary value problems. Numerical experiments confirm the theoretical convergence rates. All results are new and subsume several recent contributions in the literature. Keywords: Fixed point; Banach contraction; interpolative contraction; b-metric space; iterative scheme; nonlinear integral equation; convergence rate