Singular Initial Value Problem for a System of Integro-Differential Equations

Abstract

Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem: $g_i(t)y_i^{\prime }(t)=a_i{y}_i(t)(1+f_i(t,\mathbf{y}(t),{\int }_{0^{+}}^t{K}_i(t,s,\mathbf{y}(t),\mathbf{y}(s))ds))$, $y_i(0^{+})=0$, $t\in (0,\hspace{0.17em}{t}_0]$, where $\mathbf{y}=(y_1,\hspace{0.17em}\dots ,\hspace{0.17em}{y}_n)$, $a_i>0$, $i=1,\hspace{0.17em}\dots ,\hspace{0.17em}n$ are constants and $t_0>0$. An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used. Particular attention is paid to construction of asymptotic expansions of solutions for certain classes of systems of integrodifferential equations in a right-hand neighbourhood of a singular point.