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 f i (x)=∑ k=1 m ∑ j=1 n ϵa ijk Ψx,f j (R ijk (x)),∫ 0 X ijk (x) f j (t)dt+b ijk f j (S ijk (x))+g i (x),(E) x∈Ω=[-b,b], 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→Ω, g i :Ω→ℝ, Ψ:Ω×ℝ 2 →ℝ are the given continuous functions and f i :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∈C 2 (Ω×ℝ 2 ;ℝ), we investigate the quadratic convergence of (E). Finally, in the case of Ψ∈C N (Ω×ℝ 2 ;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.