On exponential stability of linear delay equations with oscillatory coefficients and kernels

New explicit exponential stability conditions are presented for the non-autonomous scalar linear functional differential equation $$ \dot{x}(t) + \sum_{k=1}^m a_k(t)x(h_k(t)) \int_{g(t)}^t K(t,s) x(s)ds=0, where $h_k(t)\leq t$, $g(t)\leq $a_k(\cdot)$ and kernel $K(\cdot,\cdot)$ oscillatory and, generally, discontinuous functions. The proofs based on establishing boundedness of solutions later using dichotomy equations stating that either homogeneous is exponentially stable or a non-homogeneous has an unbounded solution some bounded right-hand side. Explicit tests applied to models population dynamics, such as controlled Hutchinson Mackey-Glass equations. results illustrated with numerical examples, connection known discussed.