×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Amann, H.; Zehnder, E., Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuol. Norm. Sup. Pisa Cl. Sci (4), 8, 539-603 (1980) · Zbl 0452.47077
[2] Amann, H.; Zehnder, E., Periodic solutions of an asymptotical linear Hamiltonian system, Manuscr. Math., 32, 149-189 (1980) · Zbl 0443.70019
[3] Bartsch, T.; Li, S. J., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28, 419-441 (1997) · Zbl 0872.58018
[4] 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
[5] Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems (1993), Birkhäuser: Birkhäuser Boston
[6] Chang, K. C., Solutions of asymptotically linear operator equation via Morse theory, Comm. Pure Appl. Math., 34, 693-712 (1981) · Zbl 0444.58008
[7] Chang, K. C., Applications of homology theory to some problem in differential equations, Nonlinear Functional Analysis. Nonlinear Functional Analysis, Proceedings of Symposic in Pure Mathematics (1986), Amer. Math. Soc: Amer. Math. Soc Providence, p. 253-261
[8] Chang, K. C., On the homology method in the critical point theory, (Miranda, M., PDE and Related Subjects (1992), Pitman), 59-77 · Zbl 0798.58012
[9] Conley, C.; Zehnder, E., Morse type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37, 207-253 (1984) · Zbl 0559.58019
[10] Ding, Y. H.; Liu, J. Q., Periodic solutions of asymptotically linear Hamiltonian systems, J. System Sci. Math. Sci., 9, 30-39 (1990)
[11] Fei, G. H., Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity, J. Differential Equations, 121, 121-133 (1995) · Zbl 0831.34046
[12] Li, S. J.; Liu, J. Q., Morse theory and asymptotically Hamiltonian systems, J. Differential Equations, 78, 53-73 (1989) · Zbl 0672.34037
[13] Long, Y. M., Maslov index, degenerate critical point and asymptotically Hamiltonian systems, Sci. China Ser A, 33, 1409-1419 (1990) · Zbl 0736.58022
[14] Long, Y. M.; Zehnder, E., Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, Stochastic Processes in Physics and Geometry (1990), World Scientific: World Scientific Singapore, p. 528-563
[15] Gromoll, D.; Meyer, W., On differential functions with isolated critical points, Topology, 8, 361-369 (1969) · Zbl 0212.28903
[16] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0676.58017
[17] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Application to Differential Equations. Minimax Methods in Critical Point Theory with Application to Differential Equations, CBMS, 65 (1986), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0609.58002
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.