2021/03/29 by Hoang, N. S.
#34G20 #37L05 #44J05 #47J35 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2103.15276
The stability of the solution to the equation (*)u = 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, q>0. We proved that the equilibrium solution v=0 to the equation v = F(t,v) is Lyapunov stable under persistently acting perturbation f(t) if supt≥ 0∫0t α(ξ)‖U(t,ξ)‖ dξ