## 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:

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