Estimates of Integrally Bounded Solutions of Linear Differential Inequalities

We study integrally bounded solutions of the differential equation A(x)=z, where  A  is a linear differential operator of order l  defined on functions x:R→H (R=(−∞,∞), H () and H  is a finite-dimensional Euclidean space). The right-hand side z is an integrally bounded function on R ranging in H  and satisfying the inequality (ψ(t),z(t))≥δ|z(t)|, t∈R, δ>0. Conditions are given on the operator A  and the function ψ:R→H  that guarantee an inverse inequality of the following form for the solutions x  under consideration: ∫τ+1τ|x(l)(t)|dt≤c∫τ+2τ−1|x(t)|dt, where the constant  is independent of the choice of a real number t  and function x.