Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity.(English)Zbl 0944.34034

The author studies the existence of periodic solutions to the asymptotically linear Hamiltonian systems ${\mathbb-J\dot{z}}=H^{'}(t,z),\quad z \in {\mathbb{R}^{2N}},$ where $$J$$ is the standard symplectic matrix of $$2N \times 2N$$ and $$H: \mathbb{R}^1 \times \mathbb{R}^{2N}\rightarrow \mathbb{R}^{1}$$ is a $$C^{1}$$-function and is 1-periodic in $$t$$ satisfying the conditions $|H'(t,z) - B_{\infty}(t)z|/|z|\rightarrow 0, \quad \text{as }|z|\rightarrow \infty, \quad \forall t\in[0,1], \tag{H$$_{1}$$}$
$|H'(t,z) - B_{0}(t)z|/|z|\rightarrow 0, \quad \text{as }|z|\rightarrow 0, \quad \forall t\in[0,1], \tag{H$$_{2}$$}$ where $${\mathbb{B}}_{\infty} (t), {\mathbb{B}}_{0}(t)$$ are symmetric matrices in $${\mathbb{R} ^{2N}}$$, continuous and 1-periodic in $$t$$. Using Morse theory, Galerkin approximation method and Maslov-index theory, the authors consider the resonant case in which $${\mathbb{B}}_{\infty}(t)$$ is finitely degenerate and time dependent. Denoting by $$(i_{\infty},n_{\infty})$$ and $$(i_0,n_0)$$ the Maslov-type indices of $${\mathbb{B}}_{\infty}(t)$$ and $${\mathbb{B}}_0(t),n_\infty \neq 0$$ means that problem (1) is resonant at infinity and $$n_0 \neq 0$$ means that the origin $$z=0$$ is a degenerate solution or problem (1) is resonant at the origin.
Problem (1) was first studied considering constant matrices $${\mathbb{B}}_0$$ and $${\mathbb{B}}_{\infty}$$ such that $${\mathbb{B}}_{\infty}$$ was nondegenerate and $$i_\infty \notin [i_0,i_0+n_0]$$ and later the case $${\mathbb{B}}_0(t), {\mathbb{B}}_{\infty}(t)$$ nondegenerate and $$i_0 \neq i_{\infty}$$. Other authors followed the study for the case where $${\mathbb{B}}_{\infty}$$ is nondegenerate. All they required that $$H$$ be $$C^{2}$$. Sh. Li and J. Q. Liu [J. Differ. Equations 78, No. 1, 53-73 (1989; Zbl 0672.34037)] studied the case where $${\mathbb{B}}_{\infty}$$ is nondegenerate and $${\mathbb{B}}_0$$ and $$\mathbb{B}_\infty$$ are constant matrices but $$H$$ need not be $$C^{2}$$. They also consider the degenerate trivial solution of local linking type where $$i_\infty \neq i_0$$ or $$i_\infty\neq i_0 + n_0$$. G. Fei [J. Differ. Equations 121, No. 1, 121-133 (1995; Zbl 0831.34046)] introduced the concept of finitely degenerate and proved the case (i) of Theorem 1.2, where $$H$$ was required to be $$C^{2}$$ and to satisfy $$(H_{2})$$ and $(H_{3}') \quad H(t,z) = \tfrac{1}{2} ({\mathbb{B}}_{\infty} (t) z,z) + g_{\infty}(t,z), (g_{\infty}'(t,z),z) \geq c_{1} |z|^{s+1} -a, \;|g'_{\infty} (t,z) |\leq c_{2}|z|^{s} + b,$ with $$a,b \in {\mathbb{R}}^{1}$$, $$c_1,c_2>0$$, $$0<s<1$$. The author relaxes condition $$(H'_{3})$$ considering $$a = 0 = b$$ and the estimation valid for $$t\in[0,1]$$ a.e. and $$|z|$$ large . For example, he proves that this relaxed condition and $$B_{0}(t), B_{\infty}(t)$$ finitely degenerate imply the existence of one nontrivial 1-periodic solution to (1) if either $$n_{0}=0$$ and $$i_\infty + n_{\infty}\neq i_{\infty},$$ or, if $$n_{\infty}\neq 0$$, $$n_{0} \neq 0$$ and $$i_{\infty} + n_{\infty} \notin [i_{0},i_{0}+n_{0}]$$. Several criteria of this type are presented.

MSC:

 34C25 Periodic solutions to ordinary differential equations 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 70K30 Nonlinear resonances for nonlinear problems in mechanics 37N05 Dynamical systems in classical and celestial mechanics

Citations:

Zbl 0672.34037; Zbl 0831.34046
Full Text:

References:

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