Nonautonomous second order Hamiltonian systems

Abstract

We study the existence of periodic solutions for a second order nonautonomous dynamical system. We make no assumptions on the gradient other than continuity. This allows both sublinear and superlinear problems. We also study the existence of nonconstant solutions.

Existence and multiplicity of periodic solutions for some second order Hamiltonian systems

“…Combining Morse theory, topological linking and bifurcation arguments, they established the existence of at least two or three nontrivial 2π-solutions of (HS) λ . Schechter [13] dealt with the existence of periodic solutions for the nonautonomous systems…”

Non-autonomous second order Hamiltonian systems

“…Schechter [113] gives conditions for existence of solutions when the potential is sufficiently bounded in a neighborhood of the origin, with weaker conditions if the potential also grows quadratically as |x| → ∞. Willem [142], Wu [146], Tang and Wu [129] and Feng and Han [61] have considered the existence of periodic solutions for a potential function which is periodic in at least some of the spatial dimensions.…”

“…These conditions are generalized by Bahri and Berestycki [11], Li [79], Ekeland and Ghoussoub [50] and Faraci [54]. Second order systems which satisfy a superquadratic growth condition in (5) [28] and Schechter [113].…”

“…Schechter [106,107,108] showed existence of a solution under both a lower and upper quadratic bound on the potential without convexity or symmetry assumptions. Schechter [113] showed existence of a solution under the condition that the average value of the potential on the interval I increases to ∞ as |x| → ∞, and the upper quadratic bound is of the form,…”

Existence of solutions for nonlinear operator equations

“…In recent years, linking methods have been still focused on and employed(see e.g. [1,4,5,10,11,12,13]). In this paper, we consider to generalize some theorems in [6,7] to operator equations.…”