Solutions For Second-order Hamiltonian Systems Research Articles

Existence of nontrivial rotating periodic solutions for second-order Hamiltonian systems

In this paper, we study existence of rotating periodic solutions for second-order Hamiltonian systems. We first define an index and give some properties of the index, and then build an index theory. By employing the index and the Leray–Schauder degree theory, the existence of nontrivial rotating periodic solutions is obtained.

Infinitely Many Solutions for a Generalized Periodic Boundary Value Problem without the Evenness Assumption

In this paper, we investigate infinitely many solutions for the generalized periodic boundary value problem −x″−B0tx+B1tx=λ∇xVt,xa.e.t∈0,1,x1=Mx0,x′1=Nx′0 under the potential function Vt,x without the evenness assumption and obtain two new existence results by the multiple critical point theorem. Meanwhile, we give two corollaries for the periodic solutions of second-order Hamiltonian systems and an example that illustrates our results.

Existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance

We investigate the existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance by means of the index theory, the critical point theory without compactness assumptions, the least action principle, the saddle point reduction theorem, and the minimax method. Applying the results to second-order HS satisfying periodic boundary value conditions, we obtain some new results.

Existence of solutions for nonlinear operator equations

We investigate solutions for nonlinear operator equations and obtain some abstract existence results by linking methods. Some well-known theorems about periodic solutions for second-order Hamiltonian systems by M. Schechter are special cases of these results.

Homoclinic solutions for second-order Hamiltonian systems with periodic potential

In this paper, we study the second-order Hamiltonian systems \t\t\tu¨−L(t)u+∇W(t,u)=0,t∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\ddot{u}-L(t)u+\\nabla W(t,u)=0,\\quad t\\in \\mathbb{R}, $$\\end{document} where Lin C(mathbb{R},mathbb{R}^{Ntimes N}) is a T-periodic and positive definite matrix for all tin mathbb{R} and W is superquadratic but does not satisfy the usual Ambrosetti–Rabinowitz condition at infinity. One ground homoclinic solution is obtained by applying the monotonicity trick of Jeanjean and the concentration–compactness principle. The main result improves the recent result of Liu–Guo–Zhang (Nonlinear Anal., Real World Appl. 36:116–138, 2017).

Existence of periodic solutions for second-order Hamiltonian systems with asymptotically linear conditions

We consider a class of asymptotically linear nonautonomous second-order Hamiltonian systems. Using the Saddle Point Theorem, we obtain the existence result, which extends some previously known results.

Existence of Solutions for Second-Order Hamiltonian Systems with Resonance

In 2014, J. Pipan and M. Schechter discussed periodic solutions for second-order Hamiltonian systems with resonance. In this paper, we will generalize their results. To this end, we will first establish an index theory for second-order linear Hamiltonian systems with coefficient matrix in $$L^1$$. Then we propose generalized assumptions. We also investigate multiple solutions for symmetric second-order Hamiltonian systems.

Infinitely Many Rotating Periodic Solutions for Second-Order Hamiltonian Systems

In this paper, we consider a class of second-order Hamiltonian system in $\mathbb {R}^{N}$ with combined nonlinearities. We will study the multiplicity of rotating periodic solutions, i.e., $x(t+T)=Qx(t)$ with $T>0$ and Q is an $N\times N$ orthogonal matrix. In the case $Q^{k}\neq I_{N}$ for any positive integer k, such a rotating periodic solution is just a quasi-periodic solution; In the case $Q^{k}=I_{N}$ for some positive integer k, such a rotating periodic solution is just a subharmonic solution. We will use the Fountain Theorem and its dual form to obtain two sequences of rotating periodic solutions with the corresponding energy tending to infinity and zero respectively.

Infinitely many periodic solutions for a class of second-order Hamiltonian systems

In this paper we study the existence of infinitely many periodic solutions for second-order Hamiltonian systems $\left\{ {\begin{array}{*{20}c} {\ddot u(t) + A(t)u(t) + \nabla F(t,u(t)) = 0,} \\ {u(0) - u(T) = \dot u(0) - \dot u(T) = 0,} \\ \end{array} } \right. $ , where F(t, u) is even in u, and ∇F(t, u) is of sublinear growth at infinity and satisfies the Ahmad-Lazer-Paul condition.

Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities

In this paper, we study the existence of homoclinic solutions to the following second-order Hamiltonian systems \begin{eqnarray} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\quad \forall t\in R, \end{eqnarray} where $L(t)$ is a symmetric and positive definite matrix for all $t\in R$. The nonlinear potential $W$ is a combination of superlinear and sublinear terms. By different conditions on the superlinear and sublinear terms, we obtain existence and nonuniqueness of nontrivial homoclinic solutions to above systems.

Homoclinic solutions for a second-order Hamiltonian system with a positive semi-definite matrix

In this paper, we study homoclinic solutions for second-order Hamiltonian systems u¨-L(t)u+Wu(t,u)=0, where L(t) is allowed to be a positive semi-definite symmetric matrix for all t∈R, and W∈C1(R×RN,R) is an indefinite potential satisfying asymptotically quadratic condition at infinity on u. We obtain some new results on the existence and multiplicity of homoclinic solutions for second-order systems. The proof is based on variational methods.

Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems

In this paper, we present some new results of homoclinic solutions for second-order Hamiltonian systems ü−λL(t)u+Wu(t,u)=0; here λ>0 is a parameter, L∈C(R,RN×N) and W∈C1(R×RN,R). Unlike most other papers on this problem, we require that L(t) is a positive semi-definite symmetric matrix for all t∈R, that is, L(t)≡0 is allowed to occur in some finite interval T of R. Under some suitable assumptions on W, we prove the existence of two different homoclinic solutions uλ(1), uλ(2), which both vanish on R∖T as λ→∞, and converge to u0(1), u0(2) in H1(R), respectively; here u0(1)≠u0(2)∈H01(T) are two nontrivial solutions of the Dirichlet BVP for Hamiltonian systems on the finite interval T.

Existence of subharmonic solutions for non-quadratic second-order Hamiltonian systems

In this paper, some existence theorems are obtained for subharmonic solutions of second-order Hamiltonian systems with linear part under non-quadratic conditions. The approach is the minimax principle. We consider some new cases and obtain some new existence results.MSC:34C25, 58E50, 70H05.

Multiplicity of solutions for second-order Hamiltonian systems with impulses

In this paper, we study the existence of infinitely many solutions for second-order Hamiltonian systems with impulses. By using an infinitely many critical points theorem and Fountain theorem, we obtain some new criteria for guaranteeing that the impulsive Hamiltonian systems have infinitely many solutions. No symmetric condition on the nonlinear term is assumed. Some examples are also given in this paper to illustrate our main results.

The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials

In this paper, we study the existence of homoclinic solutions for the second-order Hamiltonian system u ̈ − L ( t ) u + W u ( t , u ) = 0 , where L ( t ) and W ( t , u ) are supposed to be periodic in t . Under certain assumptions on L and W , we obtain two new existence results by using the variant mountain pass theorem and generalized linking theorem.

Infinitely many solutions for second-order Hamiltonian system with impulsive effects

In this paper, we study the existence of infinitely many solutions for a class of second-order impulsive Hamiltonian systems. By using the variational methods, we give some new criteria to guarantee that the impulsive Hamiltonian systems have infinitely many solutions under the assumptions that the nonlinear term satisfies superquadratics, asymptotically quadratic and subquadratics, respectively. Finally, some examples are presented to illustrate our main results.

Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities

In this paper we study the existence of homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities ü−L(t)u+Wu(t,u)=0, where L(t) and W(t,u) are not assumed to be periodic in t. We get, under certain assumptions on L and W, infinitely many homoclinic solutions for both subquadratic and superquadratic cases by using the fountain theorems in critical point theory.

Periodic solutions for second-order Hamiltonian systems with a p-Laplacian

In this paper, by using the least action principle, Sobolev’s inequality and Wirtinger’s inequality, some existence theorems are obtained for periodic solutions of second-order Hamiltonian systems with a p-Laplacian under subconvex condition, sublinear growth condition and linear growth condition. Our results generalize and improve those in the literature.

Periodic Solution of Second-Order Hamiltonian Systems with a Change Sign Potential on Time Scales

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.

Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems

In this paper, we consider the existence and multiplicity of solutions of second-order Hamiltonian systems. We propose a generalized asymptotically linear condition on the gradient of Hamiltonian function, classify the linear Hamiltonian systems, prove the monotonicity of the index function, and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear Hamiltonian systems by global analysis methods such as the Leray–Schauder degree theory, the Morse theory, the Ljusternik–Schnirelman theory, etc.