This paper is devoted to the study of the following perturbed system of nonlinear functional equations (…) , where ε is a small parameter, aijk; bijk are the given real constants, Rijk; Sijk; Xijk : (…) are the given continuous functions and (…) 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 (…) ; we investigate the quadratic convergence of (E). Finally, in the case of (…) 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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In this paper, we discuss the generalized quasilinearization technique for a second order nonlinear differential equation with nonlinear three-point general boundary conditions. In fact, we obtain sequences of upper and lower solutions converging mono- tonically and quadratically to the unique solution of the nonlinear three-point boundary value problem.
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