Periodic solutions for a class of superquadratic Hamiltonian systems

We prove the existence of nontrivial critical points for a class of superquadratic nonautonomous second-order Hamiltonian systems by applying condition ^∗(C) to critical point theory, and some new solvability conditions of nontrivial periodic solutions are obtained.     

Periodic and subharmonic solutions for a class of non-autonomous Hamiltonian systems

Some existence theorems are obtained for periodic and subharmonic solutions of a class of non-autonomous Hamiltonian systems. Our technical approach is based on a version of the Local Linking Theorem, and the Generalized Mountain Pass Theorem.     

Fourier multipliers and periodic solutions of delay equations in Banach spaces

In this paper we characterize the existence and uniqueness of periodic solutions of inhomogeneous abstract delay equations and establish maximal regularity results for strong solutions. The conditions are obtained in terms of R-boundedness of linear operators determined by the equations and Lp- Fourier multipliers. Periodic mild solutions are also studied and characterized.     

Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials

In this paper we study the following nonperiodic second order Hamiltonian systems

     

Spectral properties and their applications of frequency response operators in linear continuous-time periodic systems

Spectral properties related to the frequency response operators of finite-dimensional linear continuous-time periodic (FDLCP) systems are examined rigorously and thoroughly in the paper. As applications of the spectral features, positive realness of FDLCP systems is scrutinized and then a harmonic Hamiltonian criterion is derived for the H_∞ norm of FDLCP systems. In particular, positive realness of FDLCP systems is interpreted in term of Toeplitz operators for the first time in this study, together with a testing algorithm. Deriving the harmonic Hamiltonian criterion with a frequency-domain approach bridges the spectra of the frequency response operators in FDLCP systems with their time-domain behaviors.     

Limit-point criteria for semi-degenerate singular Hamiltonian differential systems with perturbation terms

Some limit-point criteria are obtained for higher-dimensional semi-degenerate singular Hamiltonian differential systems with perturbation potential terms by using M(λ)-theory. Results in this paper cover many previous results of Hartman, Levinson, Titchmarsh and Read.     

Low-regularity solutions of the periodic general Degasperis-Procesi equation

This paper studies low-regularity solutions of the periodic general Degasperis-Procesi equation with an initial value. The existence and the uniqueness of solutions are proved. The results are illustrated by considering the periodic peakons of the periodic general Degasperis-Procesi equation.     

Multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Sturm-Liouville boundary conditions

This paper consider the multiple solutions for even Hamiltonian systems satisfying Sturm-Liouville boundary conditions. The gradient of Hamiltonian function is generalized asymptotically linear. The solutions obtained are shown to coincide with the critical points of a dual functional. Thanks to the index theory for linear Hamiltonian systems by Dong (2010) [1], we find critical points of this dual functional by verifying the assumptions of a lemma about multiple critical points given by Chang (1993) [2].     

Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and (2+1)-dimensional breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota-Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.     

Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959-1000] obtained the strong convergence of uniform bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein-Weierstrass theorem, and obtaining the strong convergence of uniform L^∞ or bounded viscosity solutions for scalar conservation law without convexity.     

Existence and multiplicity of semiclassical solutions for a Schrödinger equation

We establish the existence and multiplicity of solutions for the semiclassical nonlinear Schrödinger equation

     

Multiple solutions of two-point nonlinear boundary value problems

A quasi-linear boundary value problem has a solution with properties induced by oscillatory properties of the linear part of an equation. This result is proved for two dimensional systems. Consequences for Φ-Laplacian equations and problems with resonant linear parts are discussed.     

Entropy solutions for a doubly nonlinear elliptic problem with variable exponent

We study the boundary value problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN (N?3) and is a p(x)-Laplace type operator with p(.):Ω→[1,+∞) a measurable function and b a continuous and nondecreasing function from R→R. We prove the existence and uniqueness of an entropy solution for L¹-data f.     

Uniform stability of solutions of Boltzmann equation for soft potential with external force

The study of Cauchy problem of the Boltzmann equation is important in both theory and applications. Existence of global solutions to the equation and uniform stability of solutions in the absence of external force were introduced in the previous work on the Boltzmann equation. In this paper, we will investigate the uniform stability of solutions in L¹ for the Cauchy problem of the Boltzmann equation when there is an external force for the case of soft potentials.     

Multiple solutions for some nonlinear Schrödinger equations with indefinite linear part

We consider a class of nonlinear Schrödinger equation with indefinite linear part in RN. We prove that the problem has at least three nontrivial solutions by means of Linking Theorem and (∇)-Theorem.     

Multiplicity of solutions for a class of elliptic systems with critical Sobolev exponent

In this paper we consider systems of p-q-Laplacian elliptic equations with critical Sobolev exponent. The existence and multiplicity results of solutions are obtained by a limit index method.     

Chaos in Kolmogorov systems with proliferation - general criteria and applications

We provide general criteria for existence of chaotic dynamics in Kolmogorov systems with proliferation. As applications, we provide a new proof of chaoticity of a class of constant coefficient birth-and-death systems and also show that a finite difference discretization of a class of drift diffusion equations generates chaotic dynamics. We also present a general method of constructing chaotic Kolmogorov systems.     

Asymptotic stability for anisotropic Kirchhoff systems

We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, involving the p(x)-Laplacian operator, governed by time-dependent nonlinear damping forces and strongly nonlinear power-like variable potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The results are further extended to the more delicate problem involving higher order damping terms.     

Large time behavior of solutions to a nonlinear integro-differential system

Asymptotic behavior of solutions as t→∞ to the nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is studied. Initial-boundary value problems with two kinds of boundary data are considered. The first with homogeneous conditions on whole boundary and the second with non-homogeneous boundary data on one side of lateral boundary. The rates of convergence are given too.     

Parseval frames for ICC groups

We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of equal norm vectors of norm , the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by a Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left-regular representation of a free group has this property.