NUMERICAL STABILITY OF A CLASS ( OF SYSTEMS ) OF NONLINEAR EQUATIONS
In this article we consider stability of nonlinear equations which have the following form: Ax + F (x) = b, (1) where F is any function, A is a linear operator, b is given and x is an unknown vector. We give (under some assumptions about function F and operator A) a generalization of inequality: ‖X1 − X2‖ ‖X1‖ ≤ ‖A‖ ∥∥A−1∥∥ ‖b1 − b2‖ ‖b1‖ (2) (equation (2) estimates the relative error of the solution when the linear equation Ax = b1 becomes the equation Ax = b2) and a generalization of inequality: ‖X1 − X2‖ ‖X1‖ ≤ ∥∥∥A−1 1 ∥∥∥ ‖A1‖ (‖b1 − b2‖ ‖b1‖ + ‖A1‖ ∥∥∥A−1 2 ∥∥∥ ‖b2‖ ‖b1‖ · ‖A1 − A2‖ ‖A1‖ ) (3) (equation (3) estimates the relative error of the solution when the linear equation A1x = b1 becomes the equation A2x = b2). 1. Basic results Teorem 1. Let V be a normed space, let the linear operator A : V → V be invertible and bounded, let the inverse operator of the operator A be also bounded, let b1, b2 ∈ V and let the functions F1, F2 : V → V and the set S ⊆ V have the following properties: 1. the function F1 is Lipschitz on S, i.e., (∃L > 0) (∀x1, x2 ∈ S) ‖F1 (x1) − F1 (x2)‖ ≤ L ‖x1 − x2‖ , and the constant L is such that the inequality 1 − L∥∥A−1∥∥ > 0, holds; 2. (∃M > 0) (∀x ∈ S) ‖F1 (x)‖ ≤ M ‖x‖; and 3. (∃ε ≥ 0) (∀x ∈ S) ‖F1 (x) − F2 (x)‖ ≤ ε. AMS Subject Classification: 65J15