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The Dirichlet problem for a quasilinear singularly perturbed second order system. (English) Zbl 0829.34043

Consider the Dirichlet problem for the singular perturbed second order quasilinear system of the form (1) \(\varepsilon d^2 y/dt^2 = A(y,t) dy/dt + g(y,t)\), \(0 < t < 1\); (2) \(y(0, \varepsilon) = \alpha (\varepsilon)\), \(y (1, \varepsilon) = \beta (\varepsilon)\), \(0 < \varepsilon \ll 1\); where \(y,g, \alpha\) and \(\beta\) are all \(n\)- dimensional vectors and \(A\) is an \(n \times n\) nonsingular matrix. D. R. Smith [Rocky Mt. J. Math. 18, No. 1, 67-103 (1988; Zbl 0667.34074)] and H. Zhu and W. Lin [J. East China Norm. Univ., Nat. Sci. Ed. 1992, No. 4, 15-24 (1992; Zbl 0773.34044)] have both considered the above given problem under the assumption that there exists an \(n\)-dimensional continuously differentiable vector-valued function \(f(y,t)\) such that the Jacobin \(\nabla_y f(y,t) \equiv A(y,t)\). This is a strong restriction. Is it necessary? The purpose of this paper is to eliminate the above mentioned assumption with no other additional condition imposed so as to get the same result. In this paper, we first transform the problem (1), (2) into its equivalent boundary value problem of the singularly perturbed system in the critical case (also called singular singularly perturbed boundary value problem); next, construct a formal asymptotic solution of the problem by a boundary layer function method; finally, make the remainder estimate of the solution with Banach- Picard fixed point theorem and Riccati transformation so as to prove that the resulting formal asymptotic solution is a real one of the problem.

MSC:

34E15 Singular perturbations for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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