Periodic solutions for nonautonomous second order Hamiltonian systems with sublinear nonlinearity

Abstract

Some existence and multiplicity of periodic solutions are obtained for nonautonomous second order Hamiltonian systems with sublinear nonlinearity by using the least action principle and minimax methods in critical point theory. Mathematics Subject Classification (2000): 34C25, 37J45, 58E50.

Non-autonomous second order Hamiltonian systems

“…Similar conditions on the potential with some generalizations appear in the work of Tang [118,119,121,122], Tang and Wu [124,125,127] [41] gave conditions for solutions in the case when as |x| → ∞ the sign of the potential is negative and the magnitude is bounded above and below by a multiple of |x| 2 .…”

“…These results were generalized by Fei, Kim and Wong [58]. The method of Mawhin and Willem was adapted by Tang [121], Zhang and Zhou [156,157], Meng and Tang [93] and Tang and Wu [130] to find conditions for existence of solutions to a system with a subquadratic growth condition, including subadditive and subconvex potentials. The method was adapted by Ma and Tang [89], Wu and Zhao [148,149], Jiang and Tang [74], Tang and Ye [132], and Tang and Wu [130] to find conditions for existence of solutions to a system with a quadratically bounded potential.…”

Nonautonomous second order Hamiltonian systems

“…t ∈ I . Subsequently, Willem [1981], Mawhin [1987], Mawhin and Willem [1989], Tang [1995;1998], Tang and Wu [1999;2001; and others (see the references therein) proved existence under various conditions. The periodic problem (1-1) was studied by Mawhin and Willem [1986;1989], Long [1995], Tang and Wu [2003] and others.…”

Homoclinic solutions of nonlinear second-order Hamiltonian systems

“…We proved the existence of solutions of (1) under the condition that V (t, x) → ∞ as |x| → ∞ uniformly for a.e. t. Subsequently, Willem [55], Mawhin [25], Mawhin-Willem [27], Tang [48,49], Tang-Wu [52,53], Wu-Tang [56] and others proved existence under various conditions (cf. the references given in these publications).…”