Abstract
In this paper, we study the existence of homoclinic solutions for the fractional Hamiltonian systems with left and right Liouville–Weyl derivatives. We establish some new results concerning the existence and multiplicity of homoclinic solutions for the given system by using Clark’s theorem from critical point theory and fountain theorem.
Highlights
IntroductionΑ are left and right Liouville–Weyl fractional derivatives of order α ∈ ( 1 , 1) on the where −∞ Dtα and t D∞
In this paper, we consider the following fractional Hamiltonian system (α α t D∞ (−∞ Dt u ( t )) +u ∈ H α (R), L(t)u(t) = ∇W (t, u(t)), t ∈ R, (1)α are left and right Liouville–Weyl fractional derivatives of order α ∈ ( 1, 1) on the where −∞ Dtα and t D∞whole axis R respectively, u ∈ Rn, W (t, u) is of indefinite sign and subquadratic as |u| → +∞ and L(t) is positive definite symmetric matrix for all t ∈ R.As usual, we say that a solution u(t) of (1) is homoclinic if u(t) → 0 as t → ±∞
The existence of homoclinic solutions for Hamiltonian systems and their importance in the study of behavior of dynamical systems can be recognized from Poincaré [1]
Summary
Α are left and right Liouville–Weyl fractional derivatives of order α ∈ ( 1 , 1) on the where −∞ Dtα and t D∞. The existence of homoclinic solutions for Hamiltonian systems and their importance in the study of behavior of dynamical systems can be recognized from Poincaré [1]. Entropy 2017, 19, 50 class of solutions, called multibump solutions The existence of such a class of solutions implies that the dynamics of the system is chaotic (in particular that its topological entropy is positive). Such a result has been obtained under a nondegeneracy condition which is verified when the set of homoclinic solutions is countable.
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