New Approach of Generalized α-Nonexpansive Mappings via HK-Iteration Process with Applications to Fractional Differential Equation and Boundary Value Problems

Abstract

This paper introduces the HK-Iteration process and investigates its application to generalized α-nonexpansive mappings in Banach spaces. The proposed scheme not only unifies but also extends several classical iterative processes, thereby offering a broader framework within fixed point theory. Convergence analysis is carried out in detail: weak convergence is established by employing Opial's property, while strong convergence is obtained under the assumptions of uniform convexity together with condition (I). To demonstrate the practical utility of the process, we further apply the HK-Iteration to a boundary value problem associated with a fractional differential equation involving the Caputo derivative, where Greens function is used to reformulate the problem in operator form. A numerical example is provided, indicating that the HK-Iteration achieves faster convergence and improved accuracy compared with existing iterative schemes, thereby underscoring both the originality and the practical significance of the method.