Convergence estimates for abstract second order differential equations with two small parameters and monotone nonlinearities

Andrei Perjan, Galina Rusu



In a real Hilbert space $H$ we consider the following perturbed Cauchy problem $$ \begin{cases} \varepsilonu''_{\varepsilon\delta}(t)+ \deltau'_{\varepsilon\delta}(t)+Au_{\varepsilon\delta}(t)+B(u_{\varepsilon\delta}(t))= f(t),\quad t\in(0,T),\\ u_{\varepsilon\delta}(0)=u_0,\quad u'_{\varepsilon\delta}(0)=u_1, \end{cases} \leqno(\rom{P}_{\varepsilon\delta}) $$% where $u_0, u_1\in H$, $f\colon [0,T]\mapsto H$ and $\varepsilon,$ $\delta$ are two small parameters, $A$ is a linear self-adjoint operator, $B$ is a locally Lipschitz and monotone operator. We study the behavior of solutions $u_{\varepsilon\delta}$ to the problem (P$_{\varepsilon\delta}$) in two different cases: \begin{enumerate} \item[(i)] when $\varepsilon\to 0$ and $\delta \geq \delta_0> 0 ;$ \item[(ii)] when $\varepsilon\to 0$ and $\delta \to 0.$ \end{enumerate} We obtain some {\it 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 neighborhood of $t=0.$ We show the boundary layer and boundary layer function in both cases.


Singular perturbation; abstract second order Cauchy problem; boundary layer function; a priori estimate

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V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010.

M. Cakir, Uniform Second-Order Difference Method for a Singularly Perturbed ThreePoint Boundary Value Problem, Adv. Difference Equ. (2010), DOI:10.1155/2010/102484.

H.O. Fattorini, The hyperbolic singular perturbation problem: an operator approach, J. Differential Equations 70 (1987), no. 1, 1–41.

M. Ghisi and M. Gobbino, Global-in-time uniform convergence for linear M.hyperbolicparabolic singular perturbations, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 4, 1161–1170.

M. Gobbino, Singular perturbation hyperbolic-parabolic for degenerate nonlinear equations of Kirchhoff type, Nonlinear Anal. 44 (2001), no. 3, 361–374.

R. E. O’Malley Jr, Two parameter singular perturbation problems for second order equations, J. Math. Mech. 16 (1967), 1143–1164.

Gh. Moroşanu, Nonlinear Evolution Equations and Applications, Ed. Acad. Române, Bucureşti, 1988.

B. Najman, Convergence estimate for second order Cauchy problems with a small parameter, Czechoslovak Math. J. 48 (1998), no. 123, 737–745.

A. Perjan, Linear singular perturbations of hyperbolic-parabolic type, Bul. Acad. Ştiinţ. Repub. Mold. Mat. 2 (2003), no. 42, 95–112.

A. Perjan and G. Rusu, Convergence estimates for abstract second-order singularly perturbed Cauchy problems with Lipschitz nonlinearities, Asymptot. Anal. 74 (2011), no. 3–4, 135–165.

A. Perjan and G. Rusu, Convergence estimates for abstract second order singularly perturbed Cauchy problems with monotone nonlinearities, Ann. Acad. Rom. Sci. Ser. Math. Appl. 4 (2012), no. 2, 128–182.

A. Perjan and G. Rusu, Convergence estimates for abstract second-order singularly perturbed Cauchy problems with Lipschitz nonlinearities, Asymptot. Anal. 97 (2016), no. 3–4, 337–349.

A. Perjan and G. Rusu, Singularly perturbed problems for abstract differential equations of second order in Hilbert spaces, New Trends in Differential Equations, Control Theory and Optimization, (V. Barbu, C. Lefte and I.I. Vrabie, eds.), Word Scientific, 2016, 277–293.

E. O’Rodin, L.M. Pickett and G.I. Shishkin, Singularly perturbed problems modeling reaction-convection-diffusion processes, Comp. Methods Appl. Math. 3 (2003), no. 3, 424–442.

M.M. Vainberg, The variational method and the method of monotone operators, Nauka, Moscow, 1972. (Russian)

W.K. Zahra and A.M. El Mhlawy, Numerical solution of two-parameter singularly perturbed boundary value problems via exponential spline, Journal of King Saud University Science 25 (2013), 201–208.


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