Abstract
This paper is concerned with the second-order Hamiltonian system on time scales𝕋of the formuΔΔ(ρ(t))+μb(t)|u(t)|μ−2u(t)+∇¯H(t,u(t))=0, Δ-a.e.t∈[0,T]𝕋 ,u(0)−u(T)=uΔ(ρ(0))−uΔ(ρ(T))=0,where0,T∈𝕋. By using the minimax methods in critical theory, an existence theorem of periodic solution for the above system is established. As an application, an example is given to illustrate the result. This is probably the first time the existence of periodic solutions for second-order Hamiltonian system on time scales has been studied by critical theory.
Highlights
The theory of calculus on time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988 1
To the best of our knowledge, there is no work on the existence of periodic solutions for second-order Hamiltonian systems on time scales
It is natural to use critical theory to consider the existence of periodic solutions for second-order Hamiltonian systems on time scales
Summary
The theory of calculus on time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988 1. It is natural to use critical theory to consider the existence of periodic solutions for second-order Hamiltonian systems on time scales. If f : T → R is a function and t ∈ Tκ, the delta derivative 4 of f at the point t is defined to be the number fΔ t provided it exists with the property that, for any > 0, there is a neighborhood U ⊂ T of t such that f σ t − f s − fΔtσt − s ≤ |σ t − s|, ∀ s ∈ U. We use the notation tsfΔ to denote the Lebesgue integral of a function f between s, t ∈ T when it is defined.
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