Abstract
In this article, we study the existence of periodic solutions to second order Hamiltonian systems. Our goal is twofold. When the nonlinear term satisfies a strictly monotone condition, we show that, for any T>0, there exists a T-periodic solution with minimal period T. When the nonlinear term satisfies a non-decreasing condition, using a perturbation technique, we prove a similar result. In the latter case, the periodic solution corresponds to a critical point which minimizes the variational functional on the Nehari manifold which is not homeomorphic to the unit sphere.
Highlights
Denote by N, Z, R∗, R the sets of all positive integers, integers, nonnegative real numbers and real numbers, respectively.In the past 40 years, many authors have studied the existence of periodic solutions to classical Hamiltonian systems, x + V (x) = 0, x ∈ RN, (1)where N ∈ N and RN is the set of N -tuples of real numbers
Where N ∈ N and RN is the set of N -tuples of real numbers
In 1985, Ekeland and Hofer proved that, for any T > 0, there exists a T-periodic solution to system (1) with prescribed minimal period T when V is strictly convex with some additional conditions
Summary
Denote by N, Z, R∗, R the sets of all positive integers, integers, nonnegative real numbers and real numbers, respectively. In the past 40 years, many authors have studied the existence of periodic solutions to classical Hamiltonian systems, x + V (x) = 0, x ∈ RN ,. Where N ∈ N and RN is the set of N -tuples of real numbers. In 1978, Rabinowitz (cf [16]) has proved that, for any T > 0, system (1) admits a Tperiodic solution under the assumptions (V 1)–(V 3). He conjectured that such a solution has T as its minimal period. We describe some important contributions related to this conjecture
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