Properties of solutions of functional-differential equations with measure (Q1778270)

This paper is concerned with the linear functional-differential equation \[ \dot x(t)=\int_a^td_sR(t,s)x(s)+F'(s),\quad t\in [a,b] , \tag{1} \] with aftereffect and with generalized input, where \(F\in BV[a,b]\), \(F'\) is the generalized derivative of \(F\), and \(x: [a,b]\to \mathbb R^N\) is the unknown vector function of bounded variation. Let \(K\) be a compact subset of \(\mathbb R^N\). Under some assumptions, several properties of the set of all solutions \(x(t)\) of equation (1) with \(x(a)\in K\) are proved. Some special classes of solutions of equation (1) are also obtained, too.