# Article

 Title: Resonance and multiplicity in periodic boundary value problems with singularity  (English) Author: Rachůnková, Irena Author: Tvrdý, Milan Author: Vrkoč, Ivo Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 Volume: 128 Issue: 1 Year: 2003 Pages: 45-70 Summary lang: English . Category: math . Summary: The paper deals with the boundary value problem $u^{\prime \prime }+k\,u=g(u)+e(t),\quad u(0)=u(2\pi ),\,\,u^{\prime }(0)=u^{\prime }(2\pi ),$ where $k\in \mathbb{R}$, $g\:I\mapsto \mathbb{R}$ is continuous, $e\in \mathbb{L}J$ and $\lim _{x\rightarrow 0+}\int _x^1g(s)\,\hspace{0.56905pt}\text{d}s=\infty .$ In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems. Keyword: second order nonlinear ordinary differential equation Keyword: periodic problem Keyword: lower and upper functions MSC: 34B15 MSC: 34C25 idZBL: Zbl 1023.34015 idMR: MR1973424 . Date available: 2009-09-24T22:06:52Z Last updated: 2012-06-18 Stable URL: http://hdl.handle.net/10338.dmlcz/133937 . Reference: [1] M. del Pino, R. Manásevich, A. Montero: $T$-periodic solutions for some second order differential equations with singularities.Proc. Royal Soc. Edinburgh 120A (1992), 231–244. MR 1159183 Reference: [2] A. Fonda: Periodic solutions of scalar second order differential equations with a singularity.Mémoire de la Classe de Sciences, Acad. Roy. Belgique 8-IV (1993), 1–39. Zbl 0792.34040, MR 1259048 Reference: [3] A. Fonda, R. Manásevich, F. Zanolin: Subharmonic solutions for some second-order differential equations with singularities.SIAM J. Math. Anal. 24 (1993), 1294–1311. MR 1234017 Reference: [4] W. Ge, J. Mawhin: Positive solutions to boundary value problems for second-order ordinary differential equations with singular nonlinearities.Result. Math. 34 (1998), 108–119. MR 1635588 Reference: [5] P. Habets, L. Sanchez: Periodic solutions of some Liénard equations with singularities.Proc. Amer. Math. Soc. 109 (1990), 1035–1044. MR 1009991 Reference: [6] A. C. Lazer, S. Solimini: On periodic solutions of nonlinear differential equations with singularities.Proc. Amer. Math. Soc. 99 (1987), 109–114. MR 0866438 Reference: [7] J. Mawhin: Topological degree and boundary value problems for nonlinear differential equations.Topological Methods for Ordinary Differential Equations. Lect. Notes Math., M. Furi (ed.) vol. 1537, Springer, Berlin, 1993, pp. 73–142. Zbl 0798.34025, MR 1226930 Reference: [8] D. S. Mitrinović, J. E. Pečarić, A. M. Fink: Inequalities Involving Functions and Their Integrals and Derivatives.Kluwer, Dordrecht, 1991. MR 1190927 Reference: [9] P. Omari, W. Ye: Necessary and sufficient conditions for the existence of periodic solutions of second order ordinary differential equations with singular nonlinearities.Differ. Integral Equ. 8 (1995), 1843–1858. Zbl 0831.34048, MR 1347985 Reference: [10] I. Rachůnková: On the existence of more positive solutions of periodic BVP with singularity.Applicable Anal. 79 (2001), 257–275. MR 1880535 Reference: [11] I. Rachůnková, M. Tvrdý: Nonlinear systems of differential inequalities and solvability of certain nonlinear second order boundary value problems.J. Inequal. Appl. 6 (2001), 199–226. MR 1835526 Reference: [12] I. Rachůnková, M. Tvrdý: Construction of lower and upper functions and their application to regular and singular boundary value problems.Nonlinear Analysis, T.M.A. 47 (2001), 3937–3948. MR 1972337 Reference: [13] I. Rachůnková, M. Tvrdý: Localization of nonsmooth lower and upper functions for periodic boundary value problems.Math. Bohem. 127 (2002), 531–545. MR 1942639 Reference: [14] I. Rachůnková, M. Tvrdý, I. Vrkoč: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems.J. Differ. Equations 176 (2001), 4450–469. Reference: [15] L. Scheeffer: Über die Bedeutung der Begriffe “Maximum und Minimum” in der Variationsrechnung.Math. Ann. 26 (1885), 197–208. MR 1510341 Reference: [16] P. Yan, M. Zhang: Higher order nonresonance for differential equations with singularities.Preprint. Reference: [17] M. Zhang: A relationship between the periodic and the Dirichlet BVP’s of singular differential equations.Proc. Royal Soc. Edinburgh 128A (1998), 1099–1114. MR 1642144 .

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