Convergence Analysis of the HK-Iteration for Garcia-Falset Mappings in Banach Spaces with Application to Fractional Boundary Value Problems

Abstract

In this paper, a novel three-step iterative scheme, referred to as the HK-iteration, is introduced and analyzed for the approximation of fixed points of mappings that obey condition $(E)$ within Banach spaces. Theoretical outcomes ensuring both weak and strong convergence of the proposed scheme are established, accompanied by a detailed example of a mapping satisfying condition $(E)$. In addition, numerical experiments reveal that the HK-iteration attains a faster convergence rate and enhanced stability when compared with several existing iterative methods. To further illustrate its practical significance, the HK-iteration is applied to a class of fractional boundary value problems characterized by Caputo-type derivatives. The overall analysis demonstrates that the HK-iteration not only generalizes and connects many previously known iterative processes but also provides improved convergence efficiency and computational robustness.