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 = (-infinity,infinity) 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 (psi(t), z(t)) >= delta|z(t)|, t is an element of R, delta > 0. Conditions are given on the operator A and the function psi: R -> H that guarantee an inverse inequality of the following form for the solutions x under consideration:integral (tau+1 )(tau)|x((l))(t)|dt <= c integral(tau+2)(tau-1)|x(t)| dt,where the constant c is independent of the choice of a real number tau and function x.