Iterative Approximation for Mean Nonexpansive Mappings in Uniformly Convex Spaces with a Fractional Volterra Application

This paper investigates fixed point theory for mean nonexpansive mappings in $p$-uniformly convex metric spaces. It first establishes the existence of fixed points together with a demiclosedness principle in this setting. Building on these foundations, the two-step Karakaya iteration scheme is introduced, and a detailed convergence analysis is provided. In particular, both a $\Delta$-convergence theorem and a strong convergence theorem for mean nonexpansive mappings are proved. To illustrate the applicability of the results, new examples are constructed that clarify the scope of the assumptions. Furthermore, a numerical application to a nonlinear fractional Volterra integral equation within the framework of a $p$-uniformly convex metric space is presented. The existence of a Bochner solution is demonstrated and approximated using the Karakaya iteration scheme, with its numerical performance compared to that of the S-iteration and Thakur schemes.