Approximating solutions of fractional differential equations via a new faster iteration scheme in hyperbolic spaces

Abstract

In this paper, we introduce a new four-step fixed point iteration scheme in the framework of hyperbolic spaces as defined by Kohlenbach. We establish strong convergence theorems for the proposed iteration scheme for contractive-like mappings having a fixed point. Furthermore, we present a detailed rate analysis and prove that explicit error majorants for our iteration decay faster than those for the Abbas–Nazir iteration under suitable control-parameter assumptions. For positive affine contractions, this comparison also applies to actual errors. We also establish a data dependence result for the new scheme. Finally, we illustrate the theoretical findings through numerical examples and an application to a fractional differential equation. The examples show fewer iteration cycles at stated tolerances; iteration counts, operator-evaluation costs, and discretization errors are distinguished.