Strictly Pseudo-Contractive Volterra Integral Operators in Banach Spaces: Weak and Strong Convergence of the Mann Iteration with a Numerical Analysis Extension

We study nonlinear Volterra integral equations in uniformly convex Banach spaces $L^p(\Omega)$ $(1<p<\infty)$ using the language of strictly pseudo-contractive operators defined via the normalized duality mapping $J_p$. Under a one-sided Lipschitz condition on the kernel (which is strictly weaker than a global Lipschitz condition), we prove that the associated Volterra operator $T$ is $k$-strictly pseudo-contractive. Using known fixed point theory for such mappings, we establish that the Mann iteration converges weakly to a solution of the integral equation. Strong convergence is obtained under an additional demicompactness assumption, which is verified for a large class of kernels. In the second part, we propose a numerical analysis extension: semi-discretisation in time (method of lines) transforms the Volterra equation into a system of ordinary differential equations. We then apply the Mann iteration as an inexact solver for the resulting nonlinear system and prove convergence of the fully discrete scheme under realistic smoothness assumptions. This provides a rigorous foundation for computational methods. The results are new for Volterra equations with one-sided Lipschitz kernels and open the door to reliable numerical simulations in $L^p$ spaces.