Analytical solutions of Cauchy problem for certain linear systems of functional-differential equations of neutral type
The author gives conditions of existence of analytical solutions of the Cauchy problem (1) \(\dot x(t) + c(t) \dot x (g(t)) = A^0 (t)x(t) + \sum^p_{i = 1} A^i (t) x(h_i (t) + F(t)\), \(t \in J \leq J^* = [0, t_*]\), \(x(0) = x_0\); \(g(t) = \sum^\infty_{n = 1} g_n t^n\), \(h_i (t) = \sum^\infty_{n = 1} h_{i,n} t^n\); \(0 \leq [g(t), h_i (t)] < t\), \(t > 0\), \(i = \overline {1,p}\). Here \(x(t)\), \(F(t) : J \to R^k\); \(A^i (t) : J \to R^{k \times k}\), \(i = \overline {0,p}\); \(c(t)\): \(J \to R\), \(x_0 \in R^k\). The mentioned functions are assumed analytical on corresponding sets \((A^i (t) = \sum^\infty_{n = 0} A^i_n t^n\), \(A^i_n \in R^{k \times k}\), \(i = \overline {0,p}\), \(t \in J^*\), etc.). The solutions are determined in the form \(x(t) = \sum^\infty_{n = 0} x_nt^n\), \(x_n \in R^k\).
scientific article; zbMATH DE number 794602