关键词: fixed-point theorem , asymptotic expansion, jacketed solution, singular perturbation
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Abstract: In this paper,the fixed-point theorem is used to estimated an asymptotic solution of initial val- ue problems for a class of third nonlinear differential equations which has double initial-layer properties .   We obtain the uniformly valid asymptotic expansion of any orders including boundary layers .

Key words: fixed-point theorem , asymptotic expansion, jacketed solution, singular perturbation
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