# Article

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Keywords:
second order nonlinear ordinary differential equation; periodic problem; lower and upper functions
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.
References:
[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
[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. MR 1259048 | Zbl 0792.34040
[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
[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
[5] P. Habets, L. Sanchez: Periodic solutions of some Liénard equations with singularities. Proc. Amer. Math. Soc. 109 (1990), 1035–1044. MR 1009991
[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
[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. MR 1226930 | Zbl 0798.34025
[8] D. S. Mitrinović, J. E. Pečarić, A. M. Fink: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer, Dordrecht, 1991. MR 1190927
[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. MR 1347985 | Zbl 0831.34048
[10] I. Rachůnková: On the existence of more positive solutions of periodic BVP with singularity. Applicable Anal. 79 (2001), 257–275. MR 1880535
[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
[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
[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
[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.
[15] L. Scheeffer: Über die Bedeutung der Begriffe “Maximum und Minimum” in der Variationsrechnung. Math. Ann. 26 (1885), 197–208. MR 1510341
[16] P. Yan, M. Zhang: Higher order nonresonance for differential equations with singularities. Preprint.
[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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