Approximation of solutions to nonregular nonlinear equations by attractors of dynamic systems in a Banach space

Abstract

1. PROBLEM DEFINITION Consider the equation F (x) = 0, x ∈ X, (1.1) where F : X → X is a nonlinear operator, acting in a Banach space X. Assume that the space X is reflexive and has a Gateaux differentiable norm. The simplest examples of such spaces are lp, Lp with 1 < p < ∞, as well as the Hilbert spaces ([1], pp. 35–37). Denote by x∗ a solution to equation (1.1) and put ΩR(x) = {z ∈ X : ‖z − x‖ < R}. Assume that the operator F is Frechet differentiable and ‖F (x1)− F (x2)‖ ≤ L‖x1 − x2‖ ∀x1, x2 ∈ ΩR(x). (1.2) Hereinafter the symbol ‖ · ‖ stands for the norms of various spaces. The derivative F ′(x) is not necessarily boundedly invertible, so the initial problem (1.1) is, generally, ill-posed. Heeding the fact that F (x) is defined inaccurately, a stable approximation of solutions to the problem is based on regularization methods (see, e. g., [2], pp. 126–128; [3], pp. 9–11; [4], pp. 15–18). Assume that instead of the exact operator F in (1.1) we know only its approximation F : X → X; let the latter be Frechet differentiable and satisfy inequality (1.2) (with the same constant L) and the conditions ‖F (x∗)‖ ≤ δ; ‖F ′(x)− F ′(x)‖ ≤ δ ∀x ∈ ΩR(x). (1.3)