Asymptotic behavior of solutions of perturbed linear systems

Abstract

The existence of solutions of the system y ′ + A y = f ( t , y ) y’ + Ay = f(t,y) having the form y ( t ) = Z ( t ) a ( t ) y(t) = Z(t)a(t) is proved, where Z ( t ) Z(t) satisfies Z ′ + A Z = 0 Z’ + AZ = 0 and the vector a ( t ) a(t) has limit α \alpha as t increases. Estimates for the rate of convergence to zero of a ( t ) − α a(t) - \alpha and of y ( t ) − Z ( t ) α y(t) - Z(t)\alpha are obtained.