Polynomial growth rates

We consider linear equations v^′=A(t)v with a polynomial asymptotic behavior, that can be stable, unstable and central. We show that this behavior is exhibited by a large class of differential equations, by giving necessary and sufficient conditions in terms of generalized “polynomial” Lyapunov exponents for the existence of polynomial behavior. In particular, any linear equation in block form in a finite-dimensional space, with three blocks having “polynomial” Lyapunov exponents respectively negative, positive, and zero, has a nonuniform version of polynomial trichotomy, which corresponds to the usual notion of trichotomy but now with polynomial growth rates. We also obtain sharp bounds for the constants in the notion of polynomial trichotomy. In addition, we establish the persistence under sufficiently small nonlinear perturbations of the stability of a nonuniform polynomial contraction.     

Remark on asymptotic behaviors of solutions to a quasi-linear elliptic Dirichlet problem with large diffusion

We study asymptotic behaviors of nontrivial solutions to the Dirichlet problem of a quasi-linear elliptic equation and obtain a lower bound for growth of L^∞-norm of the solutions, which implies the L^∞-norm of the solutions goes to infinity as the diffusion coefficient goes to infinity.     

Asymptotic stability of rarefaction waves for the generalized KdV–Burgers equation on the half-line

We investigate the asymptotic behavior of solutions of the initial boundary value problem for the generalized KdV–Burgers equation _ut+f_(u)x=_uxx−_uxxx$u_t+f{\left(u\right)}_x=u_{xx}-u_{xxx}$ on the half-line with the boundary condition u(0,t)=_u−$u(0,t)=u_{-}$. The corresponding Cauchy problems of the behaviors of weak and strong rarefaction waves have respectively been studied by Wang and Zhu [Z.A. Wang, C.J. Zhu, Stability of the rarefaction wave for the generalized KdV–Burgers equation, Acta Math. Sci. 22B (3) (2002) 309–328] and Duan and Zhao [R. Duan, H.J. Zhao, Global stability of strong rarefaction waves for the generalized KdV–Burgers equation, Nonlinear Anal. TMA 66 (2007) 1100–1117]. In the present problem, on the basis of the Dirichlet boundary conditions, the asymptotic states are divided into five cases dependent on the signs of the characteristic speeds f^′(_u±)$f^{\prime }\left(u_{\pm }\right)$. In the cases of 0≤f^′(_u−)<f^′(_u+)$0\le f^{\prime }\left(u_{-}\right)<f^{\prime }\left(u_{+}\right)$, we prove the global existence of solutions and asymptotic stability of the weak rarefaction waves when the initial disturbance is small. Also, we can get asymptotic stability of the strong rarefaction waves when f(u)$f\left(u\right)$ satisfies a certain growth condition.     

On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

On measure solutions of backward stochastic differential equations

We consider backward stochastic differential equations (BSDEs) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. For the case where the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. For the case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in the case of non-uniqueness, classical solutions that fail to be measure solutions can coexist with different measure solutions.     

弱向量变分不等式的解集及其连通性

建立Hausdorff拓扑向量空间的非空凸子集到其值域为连续线性映射空间L(X, Y) 内的C -单调映射的弱向量变分不等式和它的纯量型变分不等式问题解的存在性, 讨论该弱向量变分不等式与之相联系的纯量型变分不等式解集的关系, 利用映射的C -弱次连续和C -单调性及其集值映射的不动点定理,通过纯量型变分不等式解集所诱导的集值映射所具有的特性给出弱向量变分不等式解集的连通性.     

一类广义Liénard方程的概周期解

研究广义 Liénard方程: x'=φ(y)-F(x), y'=-g(x)+p(t), 利用 Amerio的结果证明方程的解部分变元的最终有界性意味着概周期解的存在性, 推广了Cieutat^[1]的主要结果.     

An empirical central limit theorem with applications to copulas under weak dependence

We state a multidimensional Functional Central Limit Theorem for weakly dependent random vectors. We apply this result to copulas. We get the weak convergence of the empirical copula process and of its smoothed version. The finite dimensional convergence of smoothed copula densities is also proved. A new definition and the theoretical analysis of conditional copulas and their empirical counterparts are provided.      

Extremes of Shepp statistics for Gaussian random walk

Let (ξ _i , i ≥ 1) be a sequence of independent standard normal random variables and let be the corresponding random walk. We study the renormalized Shepp statistic and determine asymptotic expressions for when u,N and T→ ∞ in a synchronized way. There are three types of relations between u and N that give different asymptotic behavior. For these three cases we establish the limiting Gumbel distribution of when T,N→ ∞ and present corresponding normalization sequences.      

Asymptotic expression of the linear discrete best -approximation

Let h_p, 1<p<∞, be the best ℓ_p-approximation of the element from a proper affine subspace K of , hK, and let denote the strict uniform approximation of h from K. We prove that there are a vector and a real number a, 0a1, such that

for all p>1, where with γ_p=o(a^p/p).