Modified Z-algorithms for reckoning fixed points with application to nonlinear integral equations

In this paper, we investigate the existence of common fixed points for C − α nonexpansive mappings. We propose a novel four-step iterative scheme, referred to as the Z -iteration, which is specifically developed for handling pairs of such mappings. Using this algorithm, we establish several weak and strong convergence results that guarantee the existence of common fixed points. To substantiate the theoretical results, we present constructive examples. Moreover, the practical utility of the proposed method is demonstrated by applying it to approximate solutions of a specific class of nonlinear integral equations in Banach spaces, with a representative example provided to validate its effectiveness.