Iterative Approximation of Solutions of Hammerstein Integral Equations with Maximal Monotone Operators in Banach Spaces

Abstract

Let X be a uniformly convex and uniformly smooth real Banach space with dual space X * . Let F : X → X * and K : X * → X be bounded maximal monotone mappings. Suppose the Hammerstein equation u + KF u = 0 has a solution. An iteration sequence is constructed and proved to converge strongly to a solution of this equation.

"Solutions of Split Equality Hammerstein Type Equation Problems in Reflexive Real Banach Spaces"

“…A strong convergence theorem is established under the assumption that the associated mappings are uniformly continuous and monotone. The convergence of the method does not require the existence of a constant γ 0 , unlike the results in Chidume and Zegeye [15], Uba et al [36] and Bello et al [3]. A numerical example is also provided to clearly exhibit the behavior of the convergence of the proposed method.…”

“…Generally, as Hammerstein type equations are nonlinear, there is no closed way method to solve such type of equations. So, different authors have introduced different approximation methods for solving Hammerstein type equations (see, for instance, [10,12,14,15,16,18,20,21,25,35,36,40]). Chidume and Zegeye [12,15,16] were the first to propose and study iterative processes for approximating the solution of (1.5).…”

“…In 2016, Uba et al [36] proposed the following algorithm for solving (1.5) in a Banach space setting: Let X be a uniformly convex and uniformly smooth real Banach space and let F : X → X * , K : X * → X be maximal monotone and bounded mappings. For u 1 ∈ X, v 1 ∈ X * define the sequences {u n } and {v n } in X and X * , respectively, by…”

New Method for Computing zeros of monotone maps in Lebesgue spaces with applications to integral equations, fixed points, optimization, and variational inequality problems

“…Until now, no method, which finds the closed form solutions to these nonlinear equations, is known. Thus, iterative algorithms which estimate these solutions are of great interest (see e.g., [21,11,12,14,34,35,36] and also Chapter 13 of [9]).…”