Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Abstract

Let B be a uniformly convex and uniformly smooth real Banach space with dual space B *. Let F : B → B * , K : B * → B be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u + KFu = 0. This theorem is a significant improvement of some important recent results which were proved in real Hilbert spaces under the assumption that F and K are maximal monotone continuous and bounded. The continuity and boundedness restrictions on K and F have been dispensed with, using our new method, even in the more general setting considered in our theorems. Finally, numerical experiments are presented to illustrate the convergence of the sequence of our algorithm.

A hybrid inertial algorithm for approximating solution of convex feasibility problems with applications

“…Many problems in applications can be transformed into the form of the inclusion (1.2). For example, problems arising from convex minimization, variational inequality, Hammerstein equations, and evolution equations can be transformed into the form of the inclusion (1.2) (see, e.g., Chidume et al [8,14], Rockafellar [37]).…”

An Inertial Algorithm for Solving Hammerstein Equations

“…A motivation for the study of Hammerstein-type integral equations arise from their connection with differential equations, in particular, elliptic boundary value problems see, e.g., [20,21] for concentrate examples.…”

“…Remark 3. For the purpose of numerical illustration, we shall compare our Algorithm (21) with Algorithm ( 22) of Chidume et al [10]. We give the theorem for completeness.…”

Accelerated derivative‐free method for nonlinear monotone equations with an application

“…An algorithm of inertial‐type is an iterative procedure in which subsequent points are obtained using the preceding two points. Many authors have shown numerically, that an algorithm with inertial extrapolation step (accelerated version) converges faster than the unaccelerated version (see, References 37‐40). In the literature, several basic algorithms for solving different types of problems have been incorporated with the inertial extrapolation step, resulting to the fast convergence of the sequence generated by the various algorithms (see, References 41‐46).…”