Abstract
This paper is concerned with nonautonomous second-order Hamiltonian systems with nondifferentiable potentials. By using the nonsmooth critical point theory for locally Lipschitz functionals, we obtain some new existence results for the periodic solutions.
Highlights
Introduction and main resultsIn this paper we consider the following second-order differential inclusions systems: u(t) + A(t)u(t) ∈ ∂F(t, u(t)) a.e. t ∈ [, T], ( )u( ) – u(T) = u ( ) – u (T) =, where T >, A(t) is a continuous symmetric matrix of order N and F : [, T] × RN → R is locally Lipschitz continuous in x and ∂F(t, x) denotes the Clarke subdifferential of F for x
F(t, x) ≤ a |x| b(t), ∇F(t, x) ≤ a |x| b(t) for all x ∈ RN and a.e. t ∈ [, T], where R+ is the set of all nonnegative real number
The main results of this paper are as follows
Summary
The following assumption is necessary: (A) F(t, x) is measurable in t for every x ∈ RN and continuously differentiable in x for a.e. t ∈ [ , T], and there exist a ∈ C(R+, R+), b ∈ L ( , T; R+) such that Throughout this paper, we always suppose that F : [ , T] × RN → R satisfies the following assumption: (A ) F(t, x) is integrable in t over [ , T] for each x ∈ RN and locally Lipschitz continuous in x for each t ∈ [ , T].
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