Perjan, AndreiRusu, Galina2023-06-302023-06-302016PERJAN, Andrei, RUSU, Galina. Convergence estimates for some abstract linear second order differential equations with two small parameters. In: Asymptotic Analysis, 2016, nr. 3-4(97), pp. 337-349. ISSN 0921-7134. DOI: 10.3233/ASY-1613570921-7134https://doi.org/10.3233/ASY-161357https://msuir.usm.md/handle/123456789/10810In a real Hilbert space H we consider the following singularly perturbed Cauchy problem (Equation presented) where u0, u1 ∈ H, f : [0, T ] → H and ϵ, δ are two small parameters. We study the behavior of the solutions uϵδ to the problem (Pϵδ) in two different cases: (i) when ϵ → 0 and δ ≥ δ0 > 0; (ii) when ϵ → 0 and δ → 0. We obtain a priori estimates of the solutions to the perturbed problem, which are uniform with respect to the parameters, and a relationship between the solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior with respect to the parameters in the neighborhood of t = 0. We describe the boundary layer and the boundary layer function in both cases.ensingular perturbationCONVERGENCE ESTIMATES FOR SOME ABSTRACT LINEAR SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERSArticle