OSCILLATORY BEHAVIOR OF SOLUTIONS OF CERTAIN THIRD ORDER MIXED NEUTRAL DIFFERENCE EQUATIONS

The objective of this paper is to study the oscillatory and asymptotic properties of the mixed type third order neutral difference equation of the form △(an△2(xn+bnxn-τ1+cnxn+τ2))+qnxβn+1-σ1+pnxβn+1+σ2=0,where{an},{bn},{cn},{qn}and{pn} are positive real sequences,β is a ratio of odd positive integers,τ1,τ2,σ1 and σ2 are positive integers.We establish some sufficient conditions which ensure that all solutions are either oscillatory or converges to zero.Some examples are presented to illustrate the main results.     

Brake Orbits of a Reversible Even Hamiltonian System Near an Equilibrium

In this paper,we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium.Let the Hamiltonian system(H S)x=J H(x)satisfies H(0)=0,H(0)=0,reversible and even conditions H(Nx)=H(x)and H(-x)=H(x)for all x∈R²ⁿ.Suppose the quadratic form Q(x)=1/2 is non-degenerate.Fixτ₀>0 and assume that R²ⁿ=E⊕F decomposes into linear subspaces E and F which are invariant under the flow associated to the linear system x=J H'(0)x and such that each solution of the above linear system in E isτ₀-periodic whereas no solution in F{0}isτ₀-periodic.Writeσ(τ₀)=σ_Q(τ₀)for the signature of Q|E.Ifσ(τ₀)≠=0,we prove that either there exists a sequence of brake orbits x_k→0 withτk-periodic on the hypersurface H^-1(0)whereτ_k→τ₀;or for eachλclose to 0 withλ_σ(τ₀)>0 the hypersurface H-1(λ)contains at least 1/2|σ(τ₀)|distinct brake orbits of the Hamiltonian system(HS)near 0 with periods nearτ₀.Such result for periodic solutions was proved by Bartsch in 1997.     

Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems

This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity but subcritical.Instead of the reduction approach used in Ding et al.(2014),we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al.(2014).We can find anε0>0 which is determined by terms of N,V,Q and F,and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for allε∈(0,ε0].     

Stability criteria for linear second order delay differential system

Consider second order delay differential system where r is a positive constant and all coefficients are real constants.Our main results are as follows:(1) The maximal length of the delay for which the stability of system (*) is maintained is given in the case where the zero solution of system (*) is asymptotically stable in the absence of delay.(2) The necessary and sufficient criteria for judging that asymptetical stability of system (*) is preserved for an arbitrary large delay are obtained.     

PERMANENCE AND PERIODIC SOLUTION FOR NONAUTONOMOUS COMPETITION-PREDATOR SYSTEM WITH TYPE II FUNCTIONAL RESPONSE

A nonautonomous multispecies competition-predator system with type Ⅱfunctional response is studied in this paper. It is proved that the system is per-manent under some appropriate conditions. Further, if the system is a periodic one, a set of easily verifiable suflficient conditions which guarantee the existence, uniqueness and global attractivity of a strictly positive periodic solution are ob-tained.     

Infinitely Many Solutions for Schr?dinger–Choquard–Kirchhoff Equations Involving the Fractional p-Laplacian

In this article,we show the existence of infinitely many solutions for the fractional pLaplacian equations of Schr?dinger-Kirchhoff type equation ■ ,where(-△)_p~s is the fractional p-Laplacian operator,[u]_(s,p) is the Gagliardo p-seminorm,0 s 1 q p N/s,α∈(0,N),M and V are continuous and positive functions,and k(x) is a non-negative function in an appropriate Lebesgue space.Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma,we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.     

A variational formula for controlled backward stochastic partial differential equations and some application

An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.     

The behavior of attractors for damped Schrdinger-Boussinesq equation

IntroductionIn the laser and plasma physics, the Schrsdinger-Boussinesq system has been raised. InRely. [1,2], the authors studied the echtence of the global solution of initial boundary conditionfor the system. Here we consider the behavior of attractors for this type of equationswhere a, 7 and A are positive constants, the eXternal forces g and h are given, and j(n) is asufficiently smooth real function with j(0) = 0. We first prove that the dissipative SchrsdingerBoussinesq system posses…     

Nonexistence of positive supersolutions to a class of semilinear elliptic equations and systems in an exterior domain

In this paper,we consider the following semilinear elliptic equation:■where Ω is an exterior domain in R~N with N≥3,h:Ω×R~+→R is a measurable function,and derive optimal nonexistence results of positive supersolutions.Our argument is based on a nonexistence result of positive supersolutions of a linear elliptic problem with Hardy potential.We also establish sharp nonexistence results of positive supersolutions to an elliptic system.     

Oscillatory behavior of solutions of general third order mixed neutral difference equations

The objective of this paper is to study the oscillatory and asymptotic properties of the general mixed type third order neutral difference equation of the form△(aαn△2(xn+ bnxn-τ1+ cnxn+τ2)) + qnx n+1-σ+ p1nxβn+1+= 0,σ2where {an}, {bn}, {cn}, {qn} and {pn} are positive real sequences, both α and β are ratios of odd positive integers, τ1, τ2, σ1 and σ2 are positive integers. We establish some sufficient conditions which ensure all solutions are either oscillatory or converge to zero.     

S-polar sets of super-brownian motions and solutions of nonlinear differential equations

This paper gives probabilistic expressions of the minimal and maximal positive solutions of the partial differential equation -1/2△v(x) γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its complement Dc is compact, γ(x) is a positive bounded integrable function in D, and 1 < α≤ 2. As an application, some necessary and sufficient conditions for a compact set to be S-polar are presented.     

Exceptional Cycles in the Bounded Derived Categories of Quivers

An exceptional n-cycle in a Hom-finite triangulated category with Serre functor has been recently introduced by Broomhead,Pauksztello and Ploog.When n=1,it is a spherical object.We explicitly determine all the exceptional cycles in the bounded derived category D^b(kQ)of a finite quiver Q without oriented cycles.In particular,if Q is an Euclidean quiver,then the length type of exceptional cycles in D^b(kQ)is exactly the tubular type of Q;if Q is a Dynkin quiver of type E_m(m=6,7,8),or Q is a wild quiver,then there are no exceptional cycles in D^b(kQ);and if Q is a Dynkin quiver of type An or D_n,then the length of an exceptional cycle in D^b(kQ)is either h or h/2,where h is the Coxeter number of Q.     

The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains

In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u0 in Ω,Np2,where Ω is a domain in RN,possibly unbounded,with empty or smooth boundary,εis a small positive parameter,f∈C1(R+,R)is of subcritical and V:RN→R is a locally Hlder continuous function which is bounded from below,away from zero,such that infΛVmin ■ΛV for some open bounded subset Λ of Ω.We prove that there is anε00 such that for anyε∈(0,ε0],the above mentioned problem possesses a weak solution uεwith exponential decay.Moreover,uεconcentrates around a minimum point of the potential V inΛ.Our result generalizes a similar result by del Pino and Felmer(1996)for semilinear elliptic equations to the p-Laplacian type problem.     

Differential Equations and Singular Vectors in Verma Modules over sl(n,C)

Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weightλover sl(n,C).He gave a differential-operator representation of the symmetric group S_n on the corresponding space of truncated power series and proved that the solution space of the system is spanned by{σ(1)|σ∈S_n}.It is known that S_n is also the Weyl group of sl(n,C)and generated by all reflections s_αwith positive rootsα.We present an explicit formula of the solution s_α(1)for every positive rootαand show directly that s_α(1)is a polynomial if and only if〈λ+ρ,α〉is a nonnegative integer.From this,we can recover a formula of singular vectors given by Malikov et al..     

Inequalities for Eigenvalues of a System of Equations of Elliptic Operator in Weighted Divergence Form on Metric Measure Space

Let A be a symmetric and positive definite(1, 1) tensor on a bounded domain Ω in an ndimensional metric measure space■. In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form■,where ■, α is a nonnegative constant and u is a vector-valued function.Some universal inequalities for eigenvalues of this problem are established. Moreover, as applications of these results, we give some estimates for the upper bound of ?k+1 and the gap of ?k+1-?k in terms of the first k eigenvalues. Our results contain some results for the Lam′e system and a system of equations of the drifting Laplacian.     

Fixed Points of Order Convex Maps in Ordered Banach Spaces

In this paper we discuss problems of the existence of fixed points of order convex maps. Lemma 1 Let (E,P) be an OBS,Suppose f: S_1→S_1 is a completely continuous map. Then f has fixed points in S_1. Lemma 2 Let (E,P) be an OBS whose positive cone is normal and has nonempty interior. Suppose f: P→P is a continuous, order increasing convex map, f(0)=0; such that. Then there exist a convex subset and a positive number r such that and.     

EMDEN-FOWLER TYPE SYSTEM：NOETHER SYMMETRIES AND FIRST INTEGRALS

We classify a generalized coupled singular Emden-Fowler type system +a(t)vn=0,v+b(t)um=0 with respect to the standard first-order Lagrangian according to the Noether point symmetries which it admits.First integrals of the various cases which admit Noether point symmetries are then obtained.This system was discussed in the literature from the view-point of existence and uniqueness of positive solutions.     

Strong Solutions for Nonhomogeneous Incompressible Viscous Heat-Conductive Fluids with Non-Newtonian Potential

We consider the Navier-Stokes system with non-Newtonian potential for heat-conducting incompressible fluids in a domain Ω R3. The viscosity, heat conduc- tion coefficients and specific heat at constant volume are allowed to depend smoothly on the density and temperature. We prove the existence of unique local strong solu- tions for all initial data satisfying a natural compatibility condition. The difficult of this type model is mainly that the equations are coupled with elliptic, parabolic and hyper- bolic, and the vacuum of density cause also much trouble, that is, the initial density need not be positive and may vanish in an open set.     

ON THE ALMOST PERIODIC KOLMOGOROV COMPETITIVE SYSTEMS WITH FEEDBACK CONTROLS

In this paper, we consider an almost periodic multi-species Kolmogorov type com-petitive system with feedback controls. Applying Schauder's fixed point theorem, a criterion on the existence of the positive almost periodic solution to the system is obtained. Our results improve and generalize some existing conclusions.     

Strong Laws of Large Numbers for Double Sums of Banach Space Valued Random Elements

For a double array {V_(m,n), m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p(1 ≤ p ≤ 2) Banach space and an increasing double array {b_(m,n), m ≥1, n ≥ 1} of positive constants, the limit law ■ and in L_p as m∨n→∞ is shown to hold if ■ This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space.