Tests for boundedness and exponential stability of linear integro-differential equations with unbounded delays

Linear delayed integro-differential scalar equations with unbounded delays $$ \dot{x}(t)+a_0(t)x(t)+ \sum_{k=1}^m a_k(t)x(h_k(t)) +\sum_{l=1}^n\int_{g_l(t)}^t G_l(t,s)x(s) ds+f(t)=0 $$ are considered on a semi-axis $[t_0,\infty)$, $t_0\in\mathbb{R}$, where $x(t)=\varphi(t)$ if $t\leq t_0$ and $\varphi \colon (-\infty,t_0]\to\mathbb{R}$. It is assumed that coefficients $a_k\colon [t_0,\infty)\to \mathbb{R}$, delays $h_k, g_l\colon [t_0,\infty)\to{\mathbb{R}}$ satisfying $t_*\le h_k(t)\le t$, $t_*\le g_l(t)\le t$ for a $t_*\le t_0$, integral kernels $G_l\colon [t_0,\infty)\times [t_*,\infty)\to{\mathbb{R}}$ and non-homogeneity $f\colon [t_0,\infty)\to{\mathbb{R}}$ are Lebesgue measurable functions (of one or two variables respectively). The paper derives new explicit criteria on the boundedness and exponential stability of solutions without imposing the usual restrictions of all delays and coefficients being bounded. Investigation of exponential stability is reduced to a problem of the boundedness of all solutions to an auxiliary equation. Corollaries, specific for scalar equations, are deduced from the main results. Illustrative examples are considered as well.