## Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem.(English)Zbl 1190.34049

Summary: We study the existence of nontrivial periodic solutions to the following nonlinear differential equation
$\begin{cases} u''(t)+a(t)u(t)=f(t,u(t)),\quad t\in\mathbb R,\\ u(0)=u(\omega),\quad u'(0)=u'(\omega),\end{cases}$
where $$a:\mathbb R\to\mathbb R^+$$ is an $$\omega$$-periodic continuous function with $$a(t)\not\equiv 0$$, $$f:\mathbb R\times \mathbb R\to\mathbb R$$ is continuous, may take negative values and can be sign-changing. Without making any nonnegative assumption on nonlinearity, by using the first eigenvalue corresponding to the relevant linear operator and the topological degree, the existence of nontrivial periodic solutions to the above periodic boundary value problem is established. Finally, three examples are given to demonstrate the validity of our main results.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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### References:

 [1] Jiang, D.; Chu, J.; O’Regan, D.; Agarwal, R.P., Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, J. math. anal. appl., 286, 563-576, (2003) · Zbl 1042.34047 [2] Jiang, D.; Chu, J.; Zhang, M., Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. differential equations, 211, 282-302, (2005) · Zbl 1074.34048 [3] Li, X.; Zhang, Z., Periodic solutions for second-order differential equations with a singular nonlinearity, Nonlinear anal., 69, 3866-3876, (2008) · Zbl 1162.34316 [4] Mehri, B.; Niksirat, M.A., On the existence of periodic solutions for certain differential equations, J. comput. appl. math., 174, 239-249, (2005) · Zbl 1069.34064 [5] Nkashama, M.N.; Santannilla, J., Existence of multiple solutions for some nonlinear boundary value problems, J. differential equations, 84, 148-164, (1990) · Zbl 0693.34011 [6] Rachu̇nková, I.; Stane˘k, S.; Tvrdý, M., Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations, Handbook of differential equations: ordinary differential equations, 3, 607-723, (2006) [7] Rachu̇nková, I.; Tvrdý, M.; Vrkoc, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008 [8] Torres, P.J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. differential equations, 190, 643-662, (2003) · Zbl 1032.34040 [9] Zhang, Z.; Wang, J., On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations, J. math. anal. appl., 281, 99-107, (2003) · Zbl 1030.34024 [10] Chu, J.; Torres, J.P., Applications of Schauder fixed point theorem to singular differential equations, Bull. London math. soc., 39, 653-660, (2007) · Zbl 1128.34027 [11] Yao, Q., Positive solutions of nonlinear second-order periodic boundary value problems, Appl. math. lett., 20, 583-590, (2007) · Zbl 1131.34303 [12] Li, Y., Positive periodic solutions of nonlinear second order ordinary differential equations, Acta. math. sinica., 45, 481-488, (2002), (in Chinese) · Zbl 1018.34046 [13] Li, F.; Liang, Z., Existence of positive periodic solution to nonlinear second order differential equations, Appl. math. lett., 18, 1256-1264, (2005) · Zbl 1088.34038 [14] Yang, Z., Existence of nontrivial solutions for a nonlinear sturm – liouville problem with integral boundary conditions, Nonlinear anal., 68, 216-225, (2008) · Zbl 1132.34022 [15] Krasnoselskii, M.A.; Zabreiko, B.P., Geometrical methods of nonlinear analysis, (1984), Springer [16] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin, Heidelberg · Zbl 0559.47040
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