Fixed-point Acceleration Algorithm of Total Asymptotically Nonexpansive Mappings

Fixed-point iterative methods play a fundamental role in nonlinear analysis and its applications. They provides a unified framework for establishing existence and uniqueness results for a wide class of nonlinear problems. In this paper, working in a real uniformly convex Banach space, we propose an accelerated four-step iterative scheme for approximating fixed-point of a class of mapping known as total asymptotically nonexpansive TAN-mappings. Weak and strong convergence results for proposed algorithm are established. The weak convergence analysis is obtained under Opial’s condition, while the strong convergence results require in addition a semi-compactness assumption on the underlying mapping. Furthermore, numerical simulations are presented to demonstrate the effectiveness of the proposed scheme and to compare its performance with existing iterative methods. The obtained results extend and refine several recent contributions in the literature. Finally, an application to two-dimensional equations is provided.