Asymptotic integration of singularly perturbed differential algebraic equations with turning points. II (Q2058606)

The present paper is a continuation of [\textit{A. M. Samoilenko} and \textit{P. F. Samusenko}, Ukr. Math. J. 72, No. 12, 1928--1943 (2021; Zbl 1472.34110); translation from Ukr. Mat. Zh. 72, No. 12, 1669--1681 (2021)], where the authors derived the fundamental system of solutions of system \[ \varepsilon B(t,\varepsilon)\frac{dx}{dt} = A(t,\varepsilon)x, \quad t\in [0;T],\ x\in\mathbb{R}^n \] with the turning point on the interval \(\left[k_0\varepsilon^{\frac{p}{p+1}}, t_0\right],\) \(t_0\leq T,\) and where \(0 < \varepsilon \ll1,\) and \(p<n.\) In this second part of the paper, the authors constructed solutions of the system above on the interval \(\left[0, k'_0\varepsilon^{\frac{p}{p+1}}\right],\) \(k'_0 > k_0,\) and join the obtained asymptotic expansions. The proposed technique consisting of the external decomposition, internal decomposition and joining of these asymptotic expansions is demonstrated on the example.