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 (, ) y= Ay, y=y (x), x 0, 1, where A=diag\a₁ (x), , aₙ (x) \, B=\b₉₊ (x) \₉, ₊=₁ⁿ, and C= \c₉₊ (x, ) \₉, ₊=₁ⁿ. All functions in these matrices are complex-valued and integrable over x 0, 1, and \|c₉₊ (, ) \|₋䃑 0 as. 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. The focus is on the minimality of the smoothness requirements on the coefficients.