Existence of positive periodic solutions to nonlinear second order differential equations. (English) Zbl 1088.34038

Summary: We discuss the existence of positive periodic solutions to the nonlinear differential equation \[ u''(t)+a(t)u(t)=f\bigl(t,u(t)\bigr),\;t\in \mathbb{R}, \] where \(a:\mathbb{R}\to[0,+\infty)\) is an \(\omega\)-periodic continuous function with \(a(t)\not \equiv 0\), \(f:\mathbb{R}\times[0,+\infty)\to[0,+\infty)\) is continuous and f\((\cdot,u):\mathbb{R} \to[0,+\infty)\) is also an \(\omega\)-periodic function for each \(u\in[0,+ \infty)\). Using the fixed-point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.


34C25 Periodic solutions to ordinary differential equations
47H10 Fixed-point theorems
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[1] Erbe, L.H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. amer. math. soc., 120, 743-748, (1994) · Zbl 0802.34018
[2] Erbe, L.H.; Hu, S.; Wang, H., Multiple positive solutions of some boundary value problems, J. math. anal. appl., 184, 640-648, (1994) · Zbl 0805.34021
[3] Liu, Z.; Li, F., Multiple positive solutions of nonlinear two-point boundary value problems, J. math. anal. appl., 203, 610-625, (1996) · Zbl 0878.34016
[4] Henderson, J.; Wang, H., Positive solutions for nonlinear eigenvalue problems, J. math. anal. appl., 208, 252-259, (1997) · Zbl 0876.34023
[5] Li, Y., Positive periodic solutions of nonlinear second order ordinary differential equations, Acta math. sinica, 45, 481-488, (2002), (in Chinese) · Zbl 1018.34046
[6] Zhang, Z.; Wang, J., On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equation, J. math. anal. appl., 281, 99-107, (2003) · Zbl 1030.34024
[7] Guo, D., Nonlinear functional analysis, (1985), Shangdong Science and Technique Publishing House Jinan, (in Chinese)
[8] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press, Inc London · Zbl 0661.47045
[9] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin Heidelberg · Zbl 0559.47040
[10] Krein, M.G.; Rutman, M.A., Linear operators leaving invariant a cone in a Banach space, Trans. amer. math. soc., 10, 199-325, (1962) · Zbl 0030.12902
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