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Open Access Published by De Gruyter Open Access March 7, 2014

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

  • Le Thi Phuong Ngoc EMAIL logo , Huynh Thi Hoang Dung EMAIL logo , Pham Hong Danh EMAIL logo and Nguyen Thanh Long EMAIL logo
From the journal Demonstratio Mathematica

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

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Published Online: 2014-03-07
Published in Print: 2014-03-1

This content is open access.

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