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

Abstract

Abstract. This paper is devoted to the study of the following perturbed system of nonlinear functional equationsx P Ω " r´b, bs, i " 1, . . . , n, where ε is a small parameter, a ijk , b ijk are the given real constants, R ijk , S ijk , X ijk : Ω Ñ Ω, gi : Ω Ñ R, Ψ : ΩˆR 2 Ñ R are the given continuous functions and fi : Ω Ñ 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 P C 2 pΩˆR 2 ; Rq, we investigate the quadratic convergence of (E). Finally, in the case of Ψ P C N pΩˆR 2 ; Rq 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.

Solvability, stability, smoothness and compactness of the set of solutions for anonlinear functional integral equation

“…Furthermore, the quadratic convergence and an asymptotic expansion of solutions were also investigated. Motivated by the ideas and the techniques used in [14] - [16], recently, similar (and more general) results were obtained in [19] for the following system of nonlinear functional equations…”