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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
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