Class Of Second-order Hamiltonian Systems Research Articles

Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects

In this paper, a class of second-order Hamiltonian systems with impulsive effects are considered. We establish some sufficient conditions for the existence of solutions for the second-order Hamiltonian systems with impulsive effects { u ̈ ( t ) = ∇ F ( t , u ( t ) ) , a.e. t ∈ [ 0 , T ] ; u ( 0 ) − u ( T ) = u ̇ ( 0 ) − u ̇ ( T ) = 0 , Δ u ̇ i ( t j ) = u ̇ i ( t j + ) − u ̇ i ( t j − ) = I i j ( u i ( t j ) ) , i = 1 , 2 , … , N , j = 1 , 2 , … , p , where t 0 = 0 < t 1 < t 2 < ⋯ < t p < t p + 1 = T , u ( t ) = ( u 1 ( t ) , u 2 ( t ) , … , u N ( t ) ) ∈ R N , I i j : R → R ( i = 1 , 2 , … , N , j = 1 , 2 , … , p ) are continuous and F : [ 0 , T ] × R N → R . The solutions are sought by means of some critical point theorems. Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.

Existence of homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of boundary-value problems which are obtained by the minimax methods.

Homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems u ¨ ( t ) − L ( t ) u ( t ) + ∇ [ W 1 ( t , u ( t ) ) − W 2 ( t , u ( t ) ) ] = f ( t ) , where t ∈ R , u ∈ R n and W 1 , W 2 ∈ C 1 ( R × R n , R ) and f ∈ C ( R , R n ) are not necessary periodic in t. This result generalizes and improves some existing results in the literature.

Homoclinic solutions for a class of second-order Hamiltonian systems

Some new results for the existence of homoclinic orbits are obtained for the second-order Hamiltonian systems u ̈ ( t ) + ∇ V ( t , u ( t ) ) = f ( t ) , where t ∈ R , u ∈ R n and V ∈ C 1 ( R × R n , R ) , V ( t , x ) = − K ( t , x ) + W ( t , x ) is T -periodic with respect to t , T > 0 and f : R → R n is a continuous and bounded function. Our assumptions on K and f are rather loose.

Solutions of minimal period for a Hamiltonian system with a changing sign potential

We consider a class of second-order Hamiltonian systems with a potential indefinite in sign. Applying the fibering approach we prove some existence and multiplicity results of periodic solutions with minimal period. We also give an answer to the problem of the existence of solutions with prescribed period $T$ which is greater than the first eigenvalue $\frac{2\pi}{\omega_n}$ of the corresponding linear problem.