## Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems.(English)Zbl 1043.34045

The authors obtain some existence theorems for periodic solutions of the class of unbounded nonautonomous nonconvex subquadratic second-order Hamiltonian systems $\ddot u(t)+A(t)u(t)+\nabla F(t,u(t))= 0, \quad\text{a.e. }t\in [0,T], \qquad u(0)- u(T)= \dot u(0)- \dot u(T)= 0,$ where $$T>0$$, $$A(t)$$ is a continuous symmetric matrix of order $$N$$. $$F:[0,T]\times \mathbb{R}^N\to \mathbb{R}$$ and satisfies the following condition: $$F(t,x)$$ is measurable in $$t$$ for every $$x\in \mathbb{R}^N$$ and continuously differentiable in $$x$$ for a.e. $$t\in [0,T]$$, and there exist $$a\in \mathbb{C}(\mathbb{R}^+, \mathbb{R}^+)$$, $$b\in L^1(0,T;\mathbb{R}^+)$$ such that $$| F(t,x)|\leq a(| x|)b(t)$$, $$|\nabla F(t,x)|\leq a(| x|) b(t)$$ for all $$x\in \mathbb{R}^N$$ and a.e. $$t\in [0,T]$$, where $$\mathbb{R}^+$$ is the set of all nonnegative real numbers. The results are obtained by the minimax methods in critical point theory.

### MSC:

 34C25 Periodic solutions to ordinary differential equations
Full Text:

### References:

 [1] Berger, M.S.; Schechter, M., On the solvability of semilinear gradient operator equations, Adv. math., 25, 97-132, (1977) · Zbl 0354.47025 [2] Long, Y.M., Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials, Nonlinear anal., 24, 1665-1671, (1995) · Zbl 0824.34042 [3] Rabinowitz, P.H., On subharmonic solutions of Hamiltonian systems, Comm. pure appl. math., 33, 609-633, (1980) · Zbl 0425.34024 [4] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag New York · Zbl 0676.58017 [5] Tang, C.L., Periodic solutions for nonautonomous second systems with sublinear nonlinearity, Proc. amer. math. soc., 126, 3263-3270, (1998) · Zbl 0902.34036 [6] Han, Z.Q., 2π-periodic solutions to ordinary differential systems at resonance, Acta math. sinica, 43, 639-644, (2000), in Chinese · Zbl 1027.34050 [7] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, CBMS regional conf. ser. in math., 65, (1986), American Mathematical Society Providence, RI [8] Tang, C.L.; Wu, X.P., Periodic solutions of second order systems with not uniformly coercive potential, J. math. anal. appl., 259, 386-397, (2001) · Zbl 0999.34039 [9] Tang, C.L., Existence and multiplicity of periodic solutions for nonautonomous second order systems, Nonlinear anal., 32, 299-304, (1998) · Zbl 0949.34032 [10] Wu, X.P.; Tang, C.L., Periodic solutions of a class of nonautonomous second order systems, J. math. anal. appl., 236, 227-235, (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.