APPROXIMATION SOLVABILITY OF HAMMERSTEIN EQUATION
Irina A. Leca · 2005
Let X be a real reflexive Banach space and X ∗ its dual space. Let K: D(K) ⊂ X → X ∗ be a linear operator and F: D(F) ⊂ X ∗ → X be a nonlinear one with R(F) ⊂ D(K) andf ∈ X ∗. We study the abstract equation of Hammerstein type u + KFu = f and we present an approximation solvability method by using the class of perturbation of type (C). 1. Mappings of type (C) In this paper, we introduce a class of mappings of type (C) which is fit for study of the approximation solvability of the equation u + KFu = f. Throughout this paper, X is a reflexive Banach space. ”→ ” and”⇀ ” denote strong and weak convergence. In the framework of monotone-like mapping, we introduce: Definition 1. Let A: D(A) ⊂ X → X ∗. A is called mapping of type (C) if for any {un} ⊂D(A) such that un ⇀u0 and lim n→ ∞ (A(un),un − u0) ≤ 0,it follows that A(un) → A(u0), as n →∞. In order to determine the relationship of the class of the operators of type (C) with another mappings of monotone type we recall some definitions from [5], [7]: Key Words: mapping of type (C), mapping of type (S+), equation of Hammerstein type, regularizing equation. 91 92 Irina A. Leca 1) A is called mapping of type (S +) if for any sequence {un} ⊂D(A) converging weakly to u0 in X, for which lim n→ ∞ (A(un) − A(u0),un − u0) ≤ 0is in fact strongly convergent in X. 2) A is said to be quasi-monotone if each sequence {un} ⊂D(A) with un ⇀u0 in X, it follows that lim n→ ∞ (A(un),un − u0) ≥ 0. 3) A is called angle-bounded with the constant a ≥ 0if|(A(u),v) − (A(v),u) | ≤