Boundary value problem with an inner point for the singularly perturbed semilinear differential equations. (English) Zbl 1224.34037

Vrábel’, Róbert

Math. Bohem. 136, No. 1, 1-8 (2011).

Summary: We investigate the problem of existence and asymptotic behavior of solutions to the nonlinear equation \[ \varepsilon y''+ky=f(t,y),\;t\in \langle a,b \rangle,\;k<0,\;0<\varepsilon \ll 1 \] satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions and sophisticated estimations.

MSC:

34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34E20	Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34E05	Asymptotic expansions of solutions to ordinary differential equations

Keywords:

singular perturbation; boundary value problem; upper solution; lower solution