Asymptotic Representations of Solutions of n× n Systems of Ordinary Differential Equations with a Large Parameter

The paper considers n × n systems of ordinary differential equations of the form y'-By-C(·, λ)y=λ Ay, y=y(x), x ∈ [0, 1], where A=diag₁(x), , aₙ(x)\ , B=ⱼₖ(x), k=1ⁿ , and C= ⱼₖ(x, λ), k=1ⁿ . All functions in these matrices are complex-valued and integrable over x ∈ [0, 1] , and \|cⱼₖ(·, λ)\|L₁ → 0 as λ → ∞ . The theorems proved in the paper generalize the results of the classical Birkhoff–Tamarkin–Langer theory concerning asymptotic representations of fundamental solutions in sectors and half-strips of the complex plane as λ → ∞ . The focus is on the minimality of the smoothness requirements on the coefficients.