Seminorm-Wise Convergence of a Tri-Inertial Split-Averaged λ-Iteration for Berinde-Type Contractions in Locally Convex SpacesTERERATION AND APPLICATIONS

Abstract

We develop new fixed point theorems in complete locally convex spaces by introducing a novel three-step inertial algorithm called the \emph{Tri-Inertial Split-Averaged $\lambda$-iteration} (TISA-$\lambda$). This scheme incorporates inertial directions, relaxation factors, and perturbation terms, and generalizes classical iterative methods such as Picard, Mann, Ishikawa, and Nesterov acceleration. Our main results establish convergence theorems for Berinde-type contractions formulated in terms of seminorms, and we extend the theory to common fixed points of compatible mappings. Full proofs of the auxiliary lemmas and propositions are provided. Applications are given to nonlinear Volterra integral equations and operator equations in Fr\'echet spaces. Detailed examples illustrate the effectiveness of the method.