Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type

C E Chidume,N Djitté

doi:10.5402/2012/169751

Abstract

Suppose that H is a real Hilbert space and F,K:H→H are bounded monotone maps with D(K)=D(F)=H. Let u* denote a solution of the Hammerstein equation u+KFu=0. An explicit iteration process is shown to converge strongly to u*. No invertibility or continuity assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our result is a significant improvement on the Galerkin method of Brézis and Browder.

Highlights

Let X be a real normed linear space with dual X∗
A map G with domain D G in a normed linear space X is said to be strongly accretive if there exists a constant k > 0 such that for every x, y ∈ D G, there exists jq x − y ∈ Jq x − y such that
We show that there exists a unique sequence zn xn, yn ∈ E such that θn xn − u1 Fxn − yn 0, 3.15 θn yn − v1 Kyn xn 0, 3.16 and xn → x∗, yn → y∗, with x∗ KFx∗ 0 and y∗ Fx∗

Summary

Introduction

Let X be a real normed linear space with dual X∗. For q > 1, we denote by Jq the generalized duality mapping from X to 2X∗ defined byJq x : f ∗ ∈ X∗ : x, f ∗ x · f∗ , f∗x q−1 , 1.1 where ·, · denotes the generalized duality pairing. In the special case in which the operators are defined on subsets D of X which are compact or more generally, angle-bounded see e.g., Pascali and Sburlan 5 for definition , Brezis and Browder 7 have proved the strong convergence of a suitably defined Galerkin approximation to a solution of 1.8 see Brezis and Browder 9. Assume that F and K are monotones and satisfy the range condition.

Results

Conclusion