Linear Approximation And Asymptotic Expansion Associated With The System Of Nonlinear Functional Equations

Abstract

This paper is devoted to the study of the following perturbed system of nonlinear functional equations x ∊Ω=[-b,b], i = 1,…., n; where ε is a small parameter, aijk; bijk are the given real constants, Rijk, Sijk , Xijk : Ω → Ω ,gi → Ω →ℝ , Ψ: Ω x ℝ2→ ℝ are the given continuous functions and ƒi :Ω →ℝ are unknown functions. First, by using the Banach fixed point theorem, we find sufficient conditions for the unique existence and stability of a solution of (E). Next, in the case of Ψ ∊ C2(Ω x ℝ2; ℝ); we investigate the quadratic convergence of (E). Finally, in the case of Ψ ∊ CN(Ω x ℝ2; ℝ) and ε sufficiently small, we establish an asymptotic expansion of the solution of (E) up to order N + 1 in ε. In order to illustrate the results obtained, some examples are also given

Keywords

• system of nonlinear functional equations
• converges quadratically
• perturbed problem
• asymptotic expansion