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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 $g,h \in L^1((0,2\pi); {\Bbb R}^+)$ 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 $I_j: {\Bbb R} \to {\Bbb R}$ 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 of ODE 34B37 Boundary value problems for ODE with impulses 47J30 Variational methods (nonlinear operator equations)
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##### References:
 [1] Mawhin, J.; Wilem, M.: Critical point theory and Hamiltonian systems. (1989) [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, No. 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) [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) · 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) · Zbl 1171.35353