Asymptotic Representations of Solutions of \(n\times n\) Systems of Ordinary Differential Equations with a Large Parameter

The paper considers \(n \times n\) systems of ordinary differential equations of the form \(y'-By-C(\cdot, \lambda)y=\lambda Ay, \qquad y=y(x), \quad x \in [0, 1],\) where \(A=\operatorname{diag}\{a_1(x), \dots, a_n(x)\}\) , \(B=\{b_{jk}(x)\}_{j, k=1}^n\) , and \(C= \{c_{jk}(x, \lambda)\}_{j, k=1}^n\) . All functions in these matrices are complex-valued and integrable over \(x \in [0, 1]\) , and \(\|c_{jk}(\cdot, \lambda)\|_{L_1} \to 0\) as \(\lambda \to \infty\) . 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 \(\lambda \to \infty\) . The focus is on the minimality of the smoothness requirements on the coefficients.