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Periodic solutions for sublinear systems via variational approach. (English) Zbl 1209.34046
This paper deals with a scalar second order equation of the form
\[ x''+f(t,x)=0, \] where \(f\) satisfies a variant of the Carathéodory conditions; moreover, a sublinear condition of the form \(|f(t,x)| \leq g(t) |x|^{\alpha} +h(t)\) is assumed, where \(g,h \in L^1((0,2\pi); {\mathbb R}^+)\) and \(\alpha \in [0,1)\). The first result deals with the periodic boundary value problem associated to the given equation. Assuming (setting \(F(t,x)=\int_0^x f(t,s)\,ds\)) that \(\liminf _{|x| \to +\infty}|x|^{-2\alpha}F(t,x) > (1/2) \|g\|^2_1\), the existence of at least one \(2\pi\)-periodic solution is proved.
In the second result, the periodic boundary condition is replaced by the impulsive condition \(x'(t_j^+)-x'(t_j^-)=I_j(x(t_j))\), \(j=1, \dots, p,\) where \(0=t_0<t_1< \dots < t_{p+1}=2\pi\) and the impulse functions \(I_j: {\mathbb R} \to {\mathbb R}\) are continuous for all \(j\). Besides the same assumptions on \(f\) considered in the first result, it is assumed that, for some \(a,b \geq 0\) and \(\gamma \in [0,\alpha)\), one has \(|I_j(x)| \leq a|x|^{\gamma}+b\) and \(|\int_0^x I_j(s) ds| \leq a|x|^{2\gamma}+b\) for all \(x,j\). Under these hypotheses, the existence of at least one \(2\pi\)-periodic solution is proved. The proofs are performed by applying the saddle point theorem.

MSC:
34C25 Periodic solutions to ordinary differential equations
34B37 Boundary value problems with impulses for ordinary differential equations
47J30 Variational methods involving nonlinear operators
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[1] Mawhin, J.; Wilem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag Berlin-Heidelberg-New York
[2] Tang, C., Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. amer. math. soc., 126, 3263-3270, (1998) · Zbl 0902.34036
[3] Tang, C.; Wu, X., Notes on periodic solutions of subquadratic second-order systems, J. math. anal. appl., 285, 1, 8-18, (2003) · Zbl 1054.34075
[4] Rabinowitz, P., On subharmonic solutions of Hamiltonian systems, Comm. pure. appl. math., 33, 609-633, (1980) · Zbl 0425.34024
[5] Jiang, Q.; Tang, C., Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems, J. math. anal. appl., 328, 380-389, (2007) · Zbl 1118.34038
[6] Tang, C.; Wu, X., Subharmonic solutions for nonautonomous sublinear second order Hamiltonian systems, J. math. anal. appl., 304, 383-393, (2005) · Zbl 1076.34049
[7] Fonda, A.; Lazer, A., Subharmonic solutions of conservative systems with nonconvex potentials, Proc. amer. math. soc., 115, 819-834, (1992) · Zbl 0752.34027
[8] Nieto, J.J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear anal. real world appl., 10, 680-690, (2009) · Zbl 1167.34318
[9] H. Zhang, Z. Li, Variational approach to impulsive differential equations with periodic boundary conditions, Nonlinear Anal. Real World Appl. doi:10.1016/j.nonrwa.2008.10.016
[10] Carter, T.E., Optimal impulsive space trajectories based on linear equations, J. optim. theory appl., 70, 277-297, (1991) · Zbl 0732.49025
[11] Carter, T.E., Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion, Dynam. control, 10, 219-227, (2000) · Zbl 0980.93058
[12] Zhou, J.; Li, Yo., Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects, Nonlinear anal. TMA, 71, 2856-2865, (2009) · Zbl 1175.34035
[13] Zhang, H.; Li, Z., Periodic solutions of second-order nonautonomous impulsive differential equations, Int. J. qualitative th. differential equations appl., 2, 112-124, (2008) · Zbl 1263.34020
[14] Jung, T.; Choi, Q., Critical point theory applied to a class of the systems of the superquadratic wave equations, Bound. value probl., (2008), Article ID 742030, 11 pages · Zbl 1177.35198
[15] Agarwal, R.P., Constant sign and nodal solutions for problems with the \(p\)-Laplacian and a nonsmooth potential using variational techniques, Boundary value problems, (2009), Article ID 820237, 32 pages · Zbl 1171.35353
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