ABSTRACT SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERS AND LIPSCHITZIAN NONLINEARITIES

SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERS AND LIPSCHITZIAN NONLINEARITIES In a real Hilbert space H we consider the following singularly perturbed Cauchy problem ε u′′ εδ(t) + δ u ′ εδ(t) + Auεδ(t) + B(uεδ(t)) = f(t), t ∈ (0, T ), uεδ(0) = u0, uεδ(0) = u1, where u0, u1 ∈ H, f : [0, T ] 7→ H, ε, δ are two small parameters, A is a linear self-adjoint operator and B is a nonlinear lipschitzian operator. We study the behavior of solutions uεδ in two di erent cases: ε → 0 and δ ≥ δ0 > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem.