# Article

 Title: Singular Dirichlet boundary value problems. II: Resonance case  (English) Author: O'Regan, Donal Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 Volume: 48 Issue: 2 Year: 1998 Pages: 269-289 Summary lang: English . Category: math . Summary: Existence results are established for the resonant problem $y^{\prime \prime }+\lambda _m \,a\,y=f(t,y)$ a.e. on $[0,1]$ with $y$ satisfying Dirichlet boundary conditions. The problem is singular since $f$ is a Carathéodory function, $a\in L_{{\mathrm loc}}^1(0,1)$ with $a>0$ a.e. on $[0,1]$ and $\int ^1_0 x(1-x)a(x)\,\mathrm{d}x <\infty$. MSC: 34B15 MSC: 34L30 idZBL: Zbl 0957.34016 idMR: MR1624319 . Date available: 2009-09-24T10:13:21Z Last updated: 2012-05-31 Stable URL: http://hdl.handle.net/10338.dmlcz/127416 . Reference: [1] Atkinson, F.V.: Discrete and continuous boundary problems.(1964), Academic Press, New York. Zbl 0117.05806, MR 0176141 Reference: [2] Bobisud, L.E., and O’Regan, D.: Positive solutions for a class of nonlinear singular boundary value problems at resonance.Jour. Math. Anal. Appl. 184 (1994), 263–284. MR 1278388 Reference: [3] Bobisud, L.E., O’Regan, D., and Royalty, W.D.: Singular boundary value problems.Appl. Anal. 23 (1986), 233–243. MR 0870490 Reference: [4] Everitt, W.N., Kwong, M.K., and Zettl, A.: Oscillations of eigenfunctions of weighted regular Sturm Liouville problems.J. London Math. Soc. 27 (1983), 106–120. MR 0686509 Reference: [5] Habets, P., and Zanolin, F.: Upper and lower solutions for a generalized Emden-Fowler equation.J. Math. Anal. Appl. 181 (1994), 684–700. MR 1264540 Reference: [6] Iannacci, R., and Nkashama, M.N.: Unbounded perturbations of forced second order ordinary differential equations at resonance.Jour. Diff. Eq. 69 (1987), 289–309. MR 0903389 Reference: [7] Mawhin, J.: Topological degree methods in nonlinear boundary value problems.AMS Regional Conf. Series in Math. 40, Providence, 1978. MR 0525202 Reference: [8] Mawhin, J., and Ward, J.R.: Nonuniform nonresonance conditions at the first two eigenvalues for periodic solutions of forced Liénard and Duffing equations.Rocky M.J. Math. 112 (1982), 643–654. Reference: [9] Naimark, M.A.: Linear differential operators, Part II.Ungar Publ. Co., London, 1968. Zbl 0227.34020, MR 0262880 Reference: [10] O’Regan, D.: Theory of singular boundary value problems.World Scientific Press, Singapore, 1994. Reference: [11] O’Regan, D.: Existence principles and theory for singular Dirichlet boundary value problems.Diff. Eqms. and Dynamical Systems 3 (1995), 289–304. MR 1386750 Reference: [12] O’Regan, D.: Singular Dirichlet boundary value problems I: Superlinear and nonresonance case.Nonlinear Analysis 29 (1997), 221–245. MR 1446226 .

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