Periodic Solutions of a Nonlinear Evolution Problem

In this paper we prove existence of periodic solutions to a nonlinear evolution system of second order partial differential equations involving the pseudo-Laplacian operator. To show the existence of periodic solutions we use Faedo-Galerkin method with a Schauder fixed point argument.     

Recurrent solutions of the derivative Ginzburg–Landau equation with boundary forces

In this paper, we will establish the existence of the bounded solutions, periodic solutions, quasi-periodic solutions and almost periodic solutions for the derivative Ginzburg–Landau equation with time-dependent boundary external forces. A smoothing effect for the semigroup associated with linear Ginzburg–Landau operator is the crucial tool in establishing the main results.     

一类积分方程的向量值遥远概周期解

讨论了向量值遥远概周期函数空间上一类积分算子的不变性,并应用此结果对一类积分方程的向量值遥远概周期解的存在唯一性进行了研究,得到了其遥远概周期解存在唯一的一个充分性条件.     

Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response

A class of reaction-diffusion equations with time delay and nonlocal response is considered. Assuming that the corresponding reaction equations have heteroclinic orbits connecting an equilibrium point and a periodic solution, we show the existence of traveling wave solutions of large wave speed joining an equilibrium point and a periodic solution for reaction-diffusion equations. Our approach is based on a transformation of the differential equations to integral equations in a Banach space and the rigorous analysis of the property for a corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by the application of Liapunov-Schmidt method and the Implicit Function Theorem.     

Periodic problem for the nonlinear damped wave equation with convective nonlinearity

We study the nonlinear damped wave equation with a linear pumping and a convective nonlinearity. We consider the solutions, which satisfy the periodic boundary conditions. Our aim is to prove global existence of solutions to the periodic problem for the nonlinear damped wave equation by applying the energy-type estimates and estimates for the Green operator. Moreover, we study the asymptotic profile of global solutions.     

First kind symmetric periodic solutions and their stability for the Kepler problem and anisotropic Kepler problem plus generalized anisotropic perturbation

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As an application of this general situation, we discuss the existence of symmetric periodic solutions for the anisotropic Kepler problem plus a generalized anisotropic perturbation, (shortly, p-AKPQ problem) and for the Kepler problem plus a generalized anisotropic perturbation (shortly, p-KPQ problem), as continuation of circular orbits of the two-dimensional Kepler problem. To get this objective, we consider different types of perturbations and then we apply our main result.     

带有外部输入项的时间周期SIR传染病模型的周期行波解北大核心CSCD

研究了一类带有外部输入项的时间周期SIR传染病模型周期行波解的存在性和不存在性.首先,通过构造辅助系统适当的上下解并定义闭凸锥,将周期行波解的存在性转化为定义在这个闭凸锥上的非单调算子的不动点问题,利用Schauder不动点定理建立辅助系统周期解的存在性,并利用Arzela-Ascoli定理证明了原模型周期行波解的存在性.其次,借助分析技术得到了周期行波解的不存在性.     

Odd periodic waves and stability results for the defocusing mass-critical Korteweg-de Vries equation

In this paper, we present results of existence and stability of odd periodic traveling wave solutions for the defocusing mass-critical Korteweg-de Vries equation. The existence of periodic wave trains is obtained by solving a constrained minimization problem. Concerning the stability, we use the Floquet theory to determine the behavior of the first three eigenvalues of the linearized operator around the wave, as well as the positiveness of the associated Hessian matrix.     

Banach空间中一类周期非线性受控系统最优控制的存在性

研究一类强非线性发展方程的周期解及相应的最优控制问题的存在性，首先，证明了Banach空间中一类包含非线性单调算子和非线性非单调扰动的强非线性发展方程周期解的存在性；其次，给出了保证相应的Lagrange最优控制的充分条件；最后，举例说明理论结果在拟线笥抛物方程周期问题及相应的最优控制问题中的应用。     

Multiple periodic solutions for a second-order problem on periodic time scales

Green's function for a second-order delta dynamic equation is found, then used as the kernel in an integral operator to guarantee the existence of positive periodic solutions to a second-order periodic boundary value problem with periodic coefficient functions.     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay

In this work, we give a variation of constants formula for partial functional differential equations with infinite delay. We assume that the linear part is not necessarily densely defined and its resolvent operator satisfies the Hille-Yosida condition. We establish a reduction of the problem to a finite-dimensional space which allows us to prove the existence of almost periodic solutions.     

一类二阶迭代泛函微分方程周期解的存在性

利用k-集压缩算子抽象连续性定理和一些分析技巧,研究了一类二阶迭代泛函微分方程周期解的存在性,得到了一些保证该类方程存在周期解的新结果.     

Periodic solutions of a class of degenerate parabolic system with delays

This paper is concerned with a class of periodic degenerate parabolic system with time delays in a bounded domain under mixed boundary condition. Under locally Lipschitz condition on reaction functions, we apply Schauder fixed point theorem to obtain the existence of periodic solutions of the periodic problem. With quasi-monotonicity in addition, we also show that the periodic problem has a maximal and a minimal periodic solutions. Applications of the obtained results are also given to some nonlinear diffusion models arising from ecology.     

Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential

In this article, we give necessary and sufficient conditions for the existence of a weak solution of a Kolmogorov equation perturbed by an inverse-square potential. More precisely, using a weighted Hardy's inequality with respect to an invariant measure μ, we show the existence of the semigroup solution of the parabolic problem corresponding to a generalized Ornstein–Uhlenbeck operator perturbed by an inverse-square potential in L ²(? ^N ,?μ). In the case of the classical Ornstein–Uhlenbeck operator we obtain nonexistence of positive exponentially bounded solutions of the parabolic problem if the coefficient of the inverse-square function is too large.     

Spreading in space–time periodic media governed by a monostable equation with free boundaries,Part 2: Spreading speed

This is Part 2 of our work aimed at classifying the long-time behavior of the solution to a free boundary problem with monostable reaction term in space–time periodic media. In Part 1 (see [2]) we have established a theory on the existence and uniqueness of solutions to this free boundary problem with continuous initial functions, as well as a spreading-vanishing dichotomy. We are now able to develop the methods of Weinberger [15], [16] and others [6], [7], [8], [9], [10] to prove the existence of asymptotic spreading speed when spreading happens, without knowing a priori the existence of the corresponding semi-wave solutions of the free boundary problem. This is a completely different approach from earlier works on the free boundary model, where the spreading speed is determined by firstly showing the existence of a corresponding semi-wave. Such a semi-wave appears difficult to obtain by the earlier approaches in the case of space–time periodic media considered in our work here.     

Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation

In the paper, we first show the existence of global periodic conservative solutions to the Cauchy problem for a periodic modified two-component Camassa-Holm equation. Then we prove that these solutions, which depend continuously on the initial data, construct a semigroup.     

Nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions

In this paper, existence of nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions is considered by using the monotone operator principle, Mountain pass lemma, Linking theorem and some results in Morse theory. The obtained results are also valid and new for the corresponding difference periodic boundary value problem.     

Periodic boundary-value problem for a fourth-order differential equation

In the present paper we obtain sufficient conditions for solvability of a periodic boundary-value problem for a fourth-order ordinary differential equation. The research technique is based on a solvability theorem for a quasi-linear operator equation in the resonance case. We formulate sufficient conditions for existence of periodic solutions in terms of the initial equation. The main result of the paper clarifies the existence theorem established by B. Mehry and D. Shadman in Sci. Iran. 15 (2), 182–185 (2008).