Convergence estimates for abstract second-order singularly perturbed Cauchy problems with Lipschitzian nonlinearities

We study the behavior of solutions to the problem

ε(u″_ε(t)+A₁u_ε(t))+u′_ε(t)+A₀u_ε(t)+B(u_ε(t))=f_ε(t), t∈(0,T],

u_ε(0)=u_0ε, u′_ε(0)=u_1ε,

in the Hilbert space H as ε→0, where A₁, A₀ are two linear self-adjoint operators and B is a Lipschitzian operator.