LINEAR APPROXIMATION ASSOCIATED WITH THE SYSTEM OF NONLINEAR FUNCTIONAL- INTEGRAL

Abstract

We consider the following system of nonlinear functional- integral equations (*) ff_i (x)=∑_(k=1)^m▒∑_(j=1)^n▒a_ijk (x,∫_0^(X_ijk (x))▒〖f_j (t)dt〗)+∑_(k=1)^m▒∑_(j=1)^n▒b_ijk f_j (S_ijk (x))+g_i (x), Vx; i = 1,..., n, where ε is a small parameter,  =[a,b] or  is a non-compact interval of IR, a_ijk, b_ijk are the given real constants; g_i: → IR, X_ijk, S_ijk :  , and  :  x IR → IR are the given continuous functions and f_i: → IR are unknown functions. By using the Banach fixed point theorem, we prove the system (*) has a unique solution. If   C^2( x IR; IR) and max ∑_(i=1)^n▒∑_(k=1)^m▒max|b_ijk | <1 we obtain the quadratic convergence of the system (*). Moreover, we also obtain some results concerning the existence of C1 - solutions of a system(*).