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Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with a small parameter standing before the derivative, in the case of a singular matrix A ( t ) standing in front of the unknown function. In the present paper, the orthogonal projectors onto k e r A ( t ) and k e r A ( t ) ′ (the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction.

Projector Approach to Constructing Asymptotic Solution of Initial Value Problems for Singularly Perturbed Systems in Critical Case