Stability of solutions to abstract evolution equations in Banach spaces under nonclassical assumptions

Abstract

The stability of the solution to the equation (∗)u̇ = F (t, u) + f(t), t ≥ 0, u(0) = u0 is studied. Here F (t, u) is a nonlinear operator in a Banach space X for any fixed t ≥ 0 and F (t, 0) = 0, ∀t ≥ 0. We assume that the Fréchet derivative of F (t, u) is Hölder continuous of order q > 0 with respect to u for any fixed t ≥ 0, i.e., ‖F ′ u(t, w) − F ′ u(t, v)‖ ≤ α(t)‖v − w‖ , q > 0. We proved that the equilibrium solution v = 0 to the equation v̇ = F (t, v) is Lyapunov stable under persistently acting perturbation f(t) if supt≥0 ∫ t 0 α(ξ)‖U(t, ξ)‖ dξ < ∞ and supt≥0 ‖U(t)‖ < ∞. Here, U(t) := U(t, 0) and U(t, ξ) is the solution to the equation d dt U(t, ξ) = F ′ u(t, 0)U(t, ξ), t ≥ ξ, U(ξ, ξ) = I, where I is the identity operator in X . Sufficient conditions for the solution u(t) to equation (*) to be bounded and for limt→∞ u(t) = 0 are proposed and justified. Stability of solutions to equations with unbounded operators in Hilbert spaces is also studied.