Fixed-point Acceleration Algorithm of Total Asymptotically Nonexpansive Mappings

<ns5:p>The fixed-point iterative methods play a fundamental role in nonlinear functional analysis and its applications. In this paper, we consider a real uniformly convex Banach space and introduce a modified accelerated four-step iterative scheme for approximating fixed-point of total asymptotically nonexpansive mappings. The proposed approach is formulated within a general total asymptotically nonexpansive mappings framework and does not require compactness assumptions at the level of weak convergence. Weak convergence of the algorithm is established under Opial’s condition, whereas strong convergence is obtained by assuming, in addition, the semi-compactness of the underlying mapping. Moreover, a rigorous comparative convergence analysis is presented to examine the convergence rate of the proposed scheme relative to the two-step iterative process. It is worth noting that the iterative sequence types under study have different natures and vary style. The theoretical results are supported by numerical experiments illustrating the convergence dynamics of both methods. Finally, the applicability of the framework is demonstrated by applying it to a nonlinear two-dimensional Volterra integral equation.</ns5:p>