Approximation theorems by summability and iteration methods in banach spaces

In this thesis , new convergence results for fixed point and equilibrium problems are established by combining the Cesàro mean with the Kirk iteration method within the framework of nonlinear ergodic theory in Banach and Hilbert spaces. First, the convergence behavior of the Kirk iteration method with the Cesàro mean is investigated for asymptotically nonexpansive mappings, and new approximation theorems are established for this iterative scheme. Subsequently, the Kirk iteration, the Cesàro mean, and the hybrid projection technique are integrated to propose new Kirk–Cesàro hybrid algorithms for approximating common solutions of fixed point and equilibrium problems. In the final part of the dissertation, the proposed algorithms are applied to split equilibrium and fixed point problems. Accordingly, new iterative algorithms combining the shrinking projection method with the Cesàro mean are developed for finite families of asymptotically nonexpansive mappings. The strong convergence properties of the proposed algorithms are analyzed in detail, and new strong convergence theorems extending the existing results in the literature are established.