Perjan, AndreiRusu, Galina2023-03-132023-03-132022PERJAN, Andrei; RUSU, Galina. Convergence estimates for abstract second order differential equations with two small parameters and lipschitzian nonlinearities. In: Carpathian Journal of Mathematics. 2022, nr. 1(38), pp. 179-200. ISSN 1584-2851.1584-2851https://www.carpathian.cunbm.utcluj.ro/wp-content/uploads/2022_vol_38_1/carpathian_2022_38_1_179_200.pdfhttps://msuir.usm.md/handle/123456789/9018In a real Hilbert space H we consider the following singularly perturbed Cauchy problem { εu′′ εδ (t) + δ u′ εδ (t) + Auεδ (t) + B(uεδ (t)) = f (t), t ∈ (0, T ), uεδ (0) = u0, u′ εδ (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 A1/2 Lipschitzian operator. We study the behavior of solutions uεδ in two different cases: ε → 0 and δ ≥ δ0 > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem. We obtain some a priori estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of t = 0enArticle