Iterative approximation of solutions of nonlinear equations of Hammerstein type

Abstract

Suppose X is a real q-uniformly smooth Banach space and F, K : X → X with D(K) = F(X) = X are accretive maps. Under various continuity assumptions on F and K such that 0 = u + KFu has a solution, iterative methods which converge strongly to such a solution are constructed. No invertibility assumption is imposed on K and the operators K and F need not be defined on compact subsets of X. Our method of proof is of independent interest.

Strong convergence theorem for approximation of solutions of equations of Hammerstein type

“…Remark 3.8. It is easy to see that the iterative scheme studied in this paper seems far simpler than the iterative scheme used by Chidume and Zegeye [21] (Theorem 1.2 above) and Chidume and Ofoedu [22] (Theorem 1.3 above) in the sense that the conditions  ∞ n=1 β 2 n < ∞ and  ∞ n=1 β n = +∞ on our iteration parameter are natural and not too strong compared to the conditions imposed on iteration parameters by these authors. The prototype for our iteration parameter is, for example, β n = 1 n+1 , n ≥ 1.…”

“…In 2005, Chidume and Zegeye [21] obtained an auxiliary operator, defined in a real Hilbert space in terms of K and F that is monotone whenever K and F are, and constructed an iterative procedure that converges strongly to the solution of Eq. (1.2).…”

“…In particular, they proved the following result for the approximation of the solution of a nonlinear integral equation of Hammerstein type in a real Hilbert space. Theorem 1.2 (Chidume and Zegeye, [21] …”

Visco-penalization of the sum of two monotone operators

“…The asymptotic behavior of approximating curves plays a central role in proving the convergence of parent discrete or continuous dynamical systems for solving (1.3), e.g., [1,5,11,12,21]. However, inclusions involving, as in (1.5), several set-valued operators are not easily dealt with and neither are the associated dynamical systems.…”

Hybrid approximation of solutions of integral equations of the Hammerstein type

“…Recently, Chidume and Zegeye [4] have introduced a method which contains an auxiliary operator, defined in an appropriate real Banach space in terms of K and F, which, under certain conditions, is uniformly monotone, whenever K and F, and whose zeros, are solutions of Eq. (1.6).…”