DEMONSTRATIO MATHEMATICA Vol. XLVII No 1 2014 Le Thi Phuong Ngoc, Huynh Thi Hoang Dung, Pham Hong Danh, Nguyen Thanh Long 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 (E) fi pxq“ m ÿ k“1 n ÿ j“1 „ εa ijk Ψ ˆ x, fj pR ijk pxqq, ż X ijk pxq 0 fj ptqdt ˙ ` b ijk fj pS ijk pxqq  ` gi pxq, x 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:Ω ˆ R 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 pΩ ˆ R 2 ; Rq, we investigate the quadratic convergence of (E). Finally, in the case of Ψ P C N pΩ ˆ R 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. 1. Introduction In this paper, we consider the following system of nonlinear functional equations (1.1) f i pxq “ m ÿ k“1 n ÿ j “1 „ εa ijk Ψ ˆ x, f j pR ijk pxqq, ż X ijk pxq 0 f j ptqdt ˙ `b ijk f j pS ijk pxqq  `g i pxq, i “ 1,...,n, x P Ω “ r´b, bs, where a ijk ,b ijk are the given real constants; 2010 Mathematics Subject Classification : 39B72, 45F10. Key words and phrases : system of nonlinear functional equations, converges quadrati- cally, perturbed problem, asymptotic expansion. DOI: 10.2478/dema-2014-0008 c  Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology Brought to you by | University of North Dakota Authenticated Download Date | 5/24/15 9:48 AM