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# Article

 Title: Boundary value problem with an inner point for the singularly perturbed semilinear differential equations (English) Author: Vrábeľ, Róbert Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 Volume: 136 Issue: 1 Year: 2011 Pages: 1-8 Summary lang: English . Category: math . Summary: In this paper we investigate the problem of existence and asymptotic behavior of solutions for the nonlinear boundary value problem $$\epsilon y''+ky=f(t,y),\quad t\in \langle a,b \rangle , \ k<0,\ 0<\epsilon \ll 1$$ satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions and delicate estimations. (English) Keyword: singular perturbation Keyword: boundary value problem Keyword: upper solution Keyword: lower solution MSC: 34B16 idZBL: Zbl 1224.34037 idMR: MR2807703 . Date available: 2011-03-31T11:19:12Z Last updated: 2013-07-31 Stable URL: http://hdl.handle.net/10338.dmlcz/141442 . Reference: [1] Coster, C. De, Habets, P.: Two-Point Boundary Value Problems: Lower and Upper Solutions.Volume 205 (Mathematics in Science and Engineering), Elsevier (2006). Reference: [2] Mawhin, J.: Points fixes, points critiques et problemes aux limites.Semin. Math. Sup. no. 92, Presses Univ. Montreal (1985). Zbl 0561.34001, MR 0789982 Reference: [3] Šeda, V.: On some non-linear boundary value problems for ordinary differential equations.Arch. Math., Brno {\it 25} (1989), 207-222. MR 1188065 Reference: [4] Vrábeľ, R.: Asymptotic behavior of T-periodic solutions of singularly perturbed second-order differential equation.Math. Bohem. 121 (1996), 73-76. MR 1388177 Reference: [5] Vrábeľ, R.: Semilinear singular perturbation.Nonlin. Anal. Theory Methods Appl. 25 (1995), 17-26. MR 1331985, 10.1016/0362-546X(94)00123-Y .

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