» Convergence estimates for abstract 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

$\left\{ \begin{array}{l} \varepsilon u^{\prime\prime}_{\varepsilon\delta}(t)+\d\,u^{\prime}_{\varepsilon\delta}(t)+ A u_{\varepsilon\delta}(t)+ B\big(u_{\varepsilon\delta}(t)\big)=f(t), \quad t \in (0, T), \\ u_{\varepsilon\delta}(0)=u_{0},\quad u^{\prime}_{\varepsilon\delta}(0)=u_{1},\ \end{array} \right.$

where $u_0, u_1\in H$ , $f:[0,T]\mapsto H,$ $\varepsilon,$
$\delta$ are two small parameters, $A$ is a linear self-adjoint operator and $B$ is a nonlinear $A^{1/2}$ Lipschitzian operator.

We study the behavior of solutions $u_{\varepsilon\delta}$ in two different cases:
$\varepsilon\to 0$ and $\delta \geq \delta_0>0;$
$\varepsilon\to 0$ and $\delta \to 0,$ 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 $t=0.$