The Bohl-Perron theorem and the inverse theorem about asymptotic stability for hybrid linear systems with aftereffect

The abstract hybrid system of functional differential equations is given. One part of the equation for variable functional differential, according to another of the variables is the difference one, the second part of the equation for variable differential, according to another of the variables is functional differential one. There is a system of two equations with two unknowns. Apply W-method N.V. Azbelev's to two equations. Two model equations were studied: one is a system of functional differential equations, and the second is a system of differential equations. We studied the solutions spaces. The Bohl-Perron theorem on asymptotic stability for a hybrid system of functional differential equations is obtained. The inverse theorem is formulated.