Existence of dichotomies and invariant splittings for linear differential systems,II

This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy or equivalently an invariant splitting. The conditions are more general than those given in Part I of this paper and include the case in which the coefficients lie in a base space which is chain-recurrent under the translation flow and also the case in which compatible splittings are known to exist over invariant subsets of the base space. When the compatibility fails, the flow in the base space is shown to exhibit a gradient-like structure with attractors and repellers. Sufficient conditions are given guaranteeing the existence of bounded solutions of a linear system. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. Sufficient conditions are given for a diffeomorphism to be an Anosov diffeomorphism.     

On the permutability of sylow subgroups with Schmidt subgroups

A Schmidt group is a finite nonnilpotent group in which every proper subgroup is nilpotent. Sufficient conditions are established for the p-solvability of a finite group in which a Sylow p-subgroup is permutable with some Schmidt subgroups. Sufficient conditions for the solvability of a finite group in which some Schmidt subgroups are permutable are also obtained.     

The emergence of eigenvalues of a PT-symmetric operator in a thin strip

The Schrödinger operator in a thin infinite strip with PT -symmetric boundary conditions and a localized potential is studied. The case of a virtual level on the threshold of the essential spectrum of an efficient one-dimensional operator is considered. Sufficient conditions for the transformation of this level into an isolated eigenvalue are obtained and the first terms of the asymptotic expansion are calculated for this eigenvalue. Sufficient conditions for the absence of such an eigenvalue are also obtained.     

Permanence of a nonlinear integro-differential prey-competition model with infinite delays

A nonlinear integro-differential prey-competition model with infinite delays is investigated. Sufficient conditions are obtained for the permanence of the system.     

Periodic Solution for High-order Differential System

Sufficient conditions are presented for theexistence and stability of periodic solutions for a high-order differential system.Besides, an example is given to illustrate the result.     

Martingale problems for switched processes

Sufficient and necessary conditions are given for existence and uniqueness for the martingale problem associated with weakly coupled operator. Some result of convergence of martingale solutions is also obtained.     

Convergence of multistep methods for retarded differential equations with parameters

A class of numerical methods for nonlinear boundary-value problems for retarded differential equations with a parameter is considered. Sufficient conditions for convergence and error estimates are given     

不育控制下的具有性别结构的单种群模型

建立了一个简单的不育控制下的具有性别结构的单种群模型,判断了平衡点存在和稳定的条件,得到了种群一致持续生存的充分条件.     

On the periodic boundary-value problem for systems of second-order nonlinear ordinary differential equations

The periodic boundary value problem for systems of secondorder ordinary nonlinear differential equations is considered. Sufficient conditions for the existence and uniqueness of a solution are established.     

Existence of dichotomies and invariant splittings for linear differential systems I

This paper is concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy for a class of equations which includes those with Bohr almost-periodic coefficients. The problem is treated in the context of linear skew-product flows, where it becomes clear how to generalize to the case of fiber-preserving flows on vector bundles. Both continuous and discrete flows are treated and the results apply to the linearized variational equation for a time-varying vector field on a manifold as well as the linearization of a diffeomorphism acting on a manifold. Sufficient conditions are given for a diffeomorphism on a manifold to be an Anosov diffeomorphism. For linear skew-product flows arising from ordinary differential equations our theory is a partial generalization of Floquet theory to the almost-periodic case.