## Notes on periodic solutions of subquadratic second order systems.(English)Zbl 1054.34075

The authors study the second order system $-\ddot{u}(t)=\nabla F(t,u(t))\quad\text{ a.e. } t\in[0,T],\;u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \tag{*}$ where $$T>0$$ and $$F:[0,T]\times \mathbb R^{N}\to \mathbb R$$ satisfies the following assumptions
(A) $$F(t,x)$$ is measurable in $$t$$ for each $$x\in \mathbb R^N$$ and continuously differentiable in $$x$$ for a.e. $$t\in[0,T]$$, and there exists $$a\in C(\mathbb R^+,\mathbb R^+)$$, $$b\in L^{1}(0,T,\mathbb R^+)$$ such that $$| F(t,x)| +| \nabla F(t,x)| \leq a(| x| )b(t)$$ for all $$x\in \mathbb R^N$$ and a.e. $$t\in[0,T]$$.
(3) $$F(t,x)$$ is subquadratic in Rabinowitz’s sense, that is, there exists $$0<\mu<2$$, $$M>0$$ such that $$(\nabla F(t,x),x)\leq\mu F(t,x)$$ for all $$| x| \geq M$$ and a.e. $$t\in[0,T]$$.
The main results are existence theorems on periodic solutions of (*) in $H_{T}^{1}:=\{u:[0,T]\to \mathbb R^N \mid u \text{ is absolutely continuous, } u(0)=u(T) \text{ and }\dot{u}\in L^2(0,T;\mathbb R^N)\},$ which extend some similar results of P. H. Rabinowitz [Commun. Pure Appl. Math. 33, 609–633 (1980; Zbl 0425.34024)] and C. L.Tang [J. Math. Anal. Appl. 189, 671–675 (1995; Zbl 0824.34043)]. A typical result of the paper is given in the next theorem: Suppose that $$F$$ satisfies assumptions (A) and (3). Assume that $$F(t,x)\to+\infty\quad\text{as }| x| \to\infty$$ uniformly for a.e. $$t\in[0,T]$$. Then problem (*) has at least one solution in $$H_{T}^{1}$$.

### MSC:

 34C25 Periodic solutions to ordinary differential equations

### Citations:

Zbl 0824.34043; Zbl 0425.34024
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] Willem, M., Oscillations forcées de systèmes hamiltoniens, () · Zbl 0482.70020 [3] Mawhin, J., Semi-coercive monotone variational problems, Acad. roy. belg. bull. cl. sci., 73, 118-130, (1987) · Zbl 0647.49007 [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 of nonautonomous second order systems with γ-quasisubadditive potential, J. math. anal. appl., 189, 671-675, (1995) · Zbl 0824.34043 [6] Tang, C.L., Periodic solutions of nonautonomous second order systems with sublinear nonlinearity, Proc. amer. math. soc., 126, 3263-3270, (1998) · Zbl 0902.34036 [7] Tang, C.L.; Wu, X.P., Periodic solutions for second order systems with not uniformly coercive potential, J. math. anal. appl., 259, 386-397, (2001) · Zbl 0999.34039 [8] Wu, X.P.; Tang, C.L., Periodic solutions of a class of nonautonomous second order systems, J. math. anal. appl., 236, 227-235, (1999) [9] Tang, C.L., Periodic solutions of nonautonomous second order systems, J. math. anal. appl., 202, 465-469, (1996) · Zbl 0857.34044 [10] Rabinowitz, P.H., On subharmonic solutions of Hamiltonian systems, Comm. pure appl. math., 33, 609-633, (1980) · Zbl 0425.34024 [11] Mawhin, J.; Willem, M., Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance, Ann. inst. H. Poincaré anal. non linéaire, 3, 431-453, (1986) · Zbl 0678.35091 [12] Long, Y.M., Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials, Nonlinear anal., 24, 1665-1671, (1995) · Zbl 0824.34042 [13] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear anal., 7, 981-1012, (1983) · Zbl 0522.58012 [14] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, () · Zbl 0152.10003
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.