A study of asymptotically non-expansive mapping iteration and weakly convergent approximation methods based on Banach spaces

The theory of immovable points is an important branch of research in general functional analysis, which has wide and deep applications in the disciplines of differential equations, integral equations, numerical analysis, response theory, cybernetics, and optimisation. In this paper, based on the existence theorem of immovable points, we study the convergence of asymptotically non-expansive mappings under a mixed sequence of iterations (with mean error terms) in Banach spaces with consistent Gateaux differentiable parametrization and consistent regular structure. Let E be a consistent convex Banach space, K a nonempty convex subset of E, and : P E → K a non-expansive contraction map on E to K. Let S₁, S₂ : K→K be two asymptotically non-expansive self-mappings and T₁, T₂ : K → E, be two asymptotically non-expansive nonself-mappings. The weakly convergent approximation theorem for the iteration of asymptotically non-expansive mappings is that the Banach space E has the Frechet differentiable paradigm, the Kadec-Klee property, the Opial’s condition, and Definition {x_n} weakly converges to a common immovable point of S₁, S₂, T₁ and T₂.