Existence Results for Some Damped Second-Order Volterra Integro-Differential Equations

In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem.Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1402.563

Weighted pseudo almost periodic solutions of second-order neutral-delay differential equations with piecewise constant argument

AbstractBy introducing the method of decomposition of weighted pseudo almost periodic sequence, we present some existence theorems of weighted pseudo almost periodic solutions for second order neutral differential equations with piecewise constant argument of the form d2dt2(x(t)+px(t−1))=qx(2[t+12])+f(t), where |p|=1, [⋅] denotes the greatest integer function, q is a nonzero constant and f(t) is weighted pseudo almost periodic. Our results are new and can be regarded as a complement of some known results even in the special cases of almost periodicity and pseudo almost periodicity

On the existence of periodic solutions to second order Hamiltonian systems

In this paper, the existence of periodic solutions to the second order Hamiltonian systems is investigated. By introducing a new growth condition which generalizes the Ambrosetti–Rabinowitz condition, we prove a existence result of nontrivial T-periodic solution via the variational methods. Our result is new because it can deal with not only the superquadratic case, but also the anisotropic case which allows the potential to be superquadratic growth in only one direction and asymptotically quadratic growth in other directions

Infinitely many periodic solutions for second order Hamiltonian systems

In this paper, we study the existence of infinitely many periodic solutions for second order Hamiltonian systems $\ddot{u}+\nabla_u V(t,u)=0$, where $V(t, u)$ is either asymptotically quadratic or superquadratic as $|u|\to \infty$.Comment: to appear in JDE(doi:10.1016/j.jde.2011.05.021

Periodic solutions of second-order systems with subquadratic convex potential

In this paper, we investigate the existence of periodic solutions for the second order systems at resonance: \begin{equation} \begin{cases} \ddot u(t)+m^2\omega^2u(t)+\nabla F(t,u(t))=0\qquad \mbox{a.e. }t\in [0,T],\\ u(0)-u(T)=\dot u(0)-\dot u(T)=0, \end{cases} \end{equation} where $m>0$, the potential $F(t,x)$ is convex in $x$ and satisfies some general subquadratic conditions. The main results generalize and improve Theorem 3.7 in J. Mawhin and M. Willem [Critical point theory and Hamiltonian systems, Springer-Verlag, New York, 1989]

Subharmonic solutions for nonautonomous sublinear second order Hamiltonian systems

AbstractSome existence theorems are obtained for subharmonic solutions of nonautonomous second order Hamiltonian systems by the minimax methods in critical point theory

Bifurcation for Second-Order Hamiltonian Systems with Periodic Boundary Conditions

Through variational methods, we study nonautonomous systems of second-order ordinary differential equations with periodic boundary conditions. First, we deal with a nonlinear system, depending on a functionu, and prove that the set of bifurcation points for the solutions of the system is notσ-compact. Then, we deal with a linear system depending on a real parameterλ>0and on a functionu, and prove that there existsλ∗such that the set of the functionsu, such that the system admits nontrivial solutions, contains an accumulation point

Homoclinic orbits for periodic second order Hamiltonian systems with superlinear terms

We obtain the existence of nontrivial homoclinic orbits for nonautonomous second order Hamiltonian systems by using critical point theory under some different superlinear conditions from those previously used in Hamiltonian systems. In particular, an example is given to illustrate our result