Periodic solution and wave front solution for delay equation

First, a general theorem on the existence of periodic solutions for equations with small discrete delay is obtained by employing a technique which is based on a result of the existence of inertial manifold for small discrete delay equation, meanwhile, this general theorem is applied to show the existence of periodic solution for a predator–prey system with small discrete delay. Second, this technique is also used to obtain the existence of travelling wave solution for a host–vector disease model with small discrete delay.     

Square of countably compact groups without non-trivial convergent sequences

We show in ZFC that the existence of a countably compact Abelian group without non-trivial convergent sequences implies the existence of a countably compact group whose square is not countably compact.This improves a result obtained by van Douwen in 1980: the existence of a countably compact Boolean group without non-trivial convergent sequences implies the existence of two countably compact groups whose product is not countably compact in ZFC.Hart and van Mill showed in 1991 the existence of a countably compact group whose square is not countably compact under Martin's Axiom for countable posets. We show that the existence of such an example does not depend on some form of Martin's Axiom.     

线性增长条件下一类非线性梁方程解的存在性

考察了一类非线性四阶两点边值问题的解的存在性.通过构造适当的Banach空间并且利用积分方程建立了一个存在定理.主要工具是Leray-Schauder非线性抉择.本文表明,如果非线性项在其定义域的某个有界子集上的增长速度是线性的,该问题可能存在解.     

一类混合型微分差分方程的周期解

利用Fenchel变换,我们推出一类微分差分方程存在周期解等价于某泛函具有临界点,并求出方程具有周期解的充分条件.     

On the role of supercompact and extendible cardinals in logic

It is proved that the existence of supercompact cardinal is equivalent to a certain Skolem-Löwenheim Theorem for second order logic, whereas the existence of extendible cardinal is equivalent to a certain compactness theorem for that logic. It is also proved that a certain axiom schema related to model theory implies the existence of many extendible cardinals.     

Measures of noncompactness and fixed points of random operators

The existence of random solutions is established for a class of random Volterra integral inclusions in which the orientor field has a stochastic domain. The proof is based on a stochastic analog of the Tietze extension theorem and on a deterministic existence result which are established as well in the paper. In particular the deterministic existence theorem is stated and proved for nonconvex orientor fields     

Uniform exponential attractors for second order non-autonomous lattice dynamical systems

In this paper, the existence of a uniform exponential attractor for a second order non-autonomous lattice dynamical system with quasiperiodic symbols acting on a closed bounded set is considered. Firstly, the existence and uniqueness of solutions for the considered systems which generate a family of continuous processes is shown, and the existence of a uniform bounded absorbing sets for the processes is proved. Secondly, a semigroup defined on a extended space is introduced, and the Lipschitz continuity, α-contraction and squeezing property of this semigroup are proved. Finally, the existence of a uniform exponential attractor for the family of processes associated with the studied lattice dynamical systems is obtained.     

On the theorem of T. Yamada and S. Watanabe

The well-known theorem of T. Yamada and S. Watanabe asserts that (weak) existence and pathwise uniqueness of the solution of a stochastic equation implies the existence of a strong solution. This is the most powerful tool for proving that a stochastic equation possesses a strong solution. However, pathwise uniqueness is far from being a necessary condition for this. Even if the solution is not unique in law it is also of interest to ask for strong solutions. In the present note, we will discuss in more detail the connection between pathwise uniqueness and the existence of a strong solution. We will state a condition which is not only sufficient but also necessary for the existence of a strong solution.     

Two-player stochastic games I: A reduction

This paper is the first step in the proof of existence of equilibrium payoffs for two-player stochastic games with finite state and action sets. It reduces the existence problem to the class of so-called positive absorbing recursive games. The existence problem for this class is solved in a subsequent paper.     

Determining modes and almost periodic integrals for cocycles

Cocycles of general form on an arbitrary metric space are considered. The notion of determining modes for cocycles on a Hilbert space is introduced, and a theorem on the existence of finitely many determining modes for such cocycles is proved. The existence of a B-pullback attractor in the problem of microwave heating of a material is proved. The notion of almost periodic integral for a cocycle is introduced, and the existence of such an integral for a certain class of cocycles is proved.     

An existence theorem for a generalized self-dual Chern–Simons equation and its application

In this paper, we establish an existence theorem for a generalized self-dual Chern–Simons equation over a doubly periodic domain and use the existence theorem to prove the existence of doubly periodic self-dual vortices in a Maxwell–Chern–Simons model with non-minimal coupling. We find a necessary and sufficient condition for the existence of solutions of the generalized Chern–Simons equation. We prove the existence result by using two methods, a super- and sub-solution method and a constrained minimization method. Our main contribution is that we find a general inequality-type constraint by using the second method and it maybe applied to some related problems with the similar structures.     

A reaction-diffusion system with cross-diffusion modeling the spread of an epidemic disease

We provide existence results for nonnegative solutions to a class of reaction-diffusion systems with cross-diffusion modeling the spread of an epidemic disease. The existence of weak solutions is proved by means of an approximation process, the Faedo-Galerkin method, and a compactness argument. Under additional assumptions a global existence result for classical solutions is derived upon using interpolation results between Banach spaces.     

Equilibrium problems and variational inequalities in topological vector spaces

An existence result for the equilibrium problem is proved in a general topological vector space. As applications, existence results are derived for variational inequality problems, vector equilibrium problems and vector variational inequality problems. Our results extend and unify a number of existence theorems in non-compact cases     

Short time existence of the heat flow for Dirac-harmonic maps on closed manifolds

The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence, and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.     

一类多滞量周期扰动非线性系统的周期解

研究一类具有多个滞量的周期扰动非线性系统的T周期解.利用拓扑度的方法得到了系统存在T周期解的充分条件.作为应用,证明了具有滞后的单种群对数模型在一定条件下存在正周期解.     

两类分段对称系统周期解的存在性

本文研究了两类分段对称系统周期解的存在性,得到了系统相应的周期解存在的充分条件．作为应用,证明了一类具有分段对称性的时滞微分方程的周期解的存在性,推广了现有的相关结果．     

Multiple periodic solutions for Hamiltonian systems with not coercive potential

Under an appropriate oscillating behavior of the nonlinear term, the existence of infinitely many periodic solutions for a class of second order Hamiltonian systems is established. Moreover, the existence of two non-trivial periodic solutions for Hamiltonian systems with not coercive potential is obtained, and the existence of three periodic solutions for Hamiltonian systems with coercive potential is pointed out. The approach is based on critical point theorems.     

Asymptotic phase revisited

We discuss the classical problem on asymptotic phase of periodic orbits of planar systems. The existence of asymptotic phase for non-hyperbolic periodic orbits is completely determined with hypotheses on the derivatives of a Poincaré map and a return-time map. Smoothness of the vector field turns out to be crucial for existence of asymptotic phase. For hyperbolic periodic orbits, a new proof for the existence of invariant foliations is provided.     

The existence of fixed points of left-continuous monotone operators in spaces with a regular cone

In theorems on the existence of a fixed point of an operator the latter is usually assumed to be continuous. In this paper we prove a theorem with sufficient conditions for the existence of a fixed point of an operator which is not necessarily continuous (possibly it is left-continuous). The obtained theorem with the use of regular cones is applied for proving the existence of a fixed point of a nonlinear integral operator. We give an example illustrating the theorem.     

Resonance elliptic variational inequalities with discontinuous nonlinearities of linear growth

We consider resonance elliptic variational inequalities with second-order differential operators and discontinuous nonlinearities of linear growth. The theorem on existence of a strong solution is proved. The initial-value problem is reduced to the problem of existence of a fixed point for a compact multivalued mapping and then the existence of this point is established by the Leray–Schauder method.