Necessary and sufficient conditions for the solvability of a problem of Hartman and Wintner

The equation (1) is regarded as a perturbation of (2) , where the latter is nonoscillatory at infinity. The functions are assumed to be continuous real-valued, , whereas is continuous complex-valued. A problem of Hartman and Wintner regarding the asymptotic integration of (1) for large by means of solutions of (2) is studied. A new statement of this problem is proposed, which is equivalent to the original one if is real-valued. In the general case of being complex-valued a criterion for the solvability of the Hartman-Wintner problem in the new formulation is obtained. The result improves upon the related theorems of Hartman and Wintner, Trench, Simsa and some results of Chen.

     

On Global Asymptotic Stability of Second Order Nonlinear Differential Systems

This paper investigates the global asymptotic stability of the autonomous planar systems $ \dot {x} = p_2(y)q_2(x)y $ , $ \dot {y} = p_3(y)q_3(x)x + p_3(y)q_4(x)y $ and $ \dot {x} = f_1(x) + h_2(x)y $ , $ \dot {y} = f_3(x) + h_4(x)y $ , under the assumption that all functions involved in the equations are continuous and that the origin is a unique equilibrium. We present necessary and sufficient conditions for the origin to be globally asymptotically stable.     

Limit Boundary Value Problems of a class of Nonlinear Differential Equations of the Second order

In this paper, the existence and uniqueness of solution of the limit boundary value problem $\[\ddot x = f(t,x)g(\dot x)\]$(F) $\[a\dot x(0) + bx(0) = c\]$(A) $\[x( + \infty ) = 0\]$(B) is considered, where $\[f(t,x),g(\dot x)\]$ are continuous functions on $\[\{ t \ge 0, - \infty < x,\dot x < + \infty \} \]$ such that the uniqueness of solution together with thier continuous dependence on initial value are ensured, and assume: 1)$\[f(t,0) \equiv 0,f(t,x)/x > 0(x \ne 0);\]$; 2) f(t,x)/x is nondecreasing in x>0 for fixed t and non-increasing in x<0 for fixed t, 3)$\[g(\dot x) > 0\]$, In theorem 1, farther assume: 4) $\[\int\limits_0^{ \pm \infty } {dy/g(y) = \pm \infty } \]$ Condition (A) may be discussed in the following three cases $x(0)=p(p \neq 0)$(A_1) $\[x(0) = q(q \ne 0)\]$(A_2) $\[x(0) = kx(0) + r{\rm{ }}(k > 0,r \ne 0)\]$(A_3) The notation $\[f(t,x) \in {I_\infty }\]$ will refer to the function f(t,x) satisfying $\[\int_0^{ + \infty } {\alpha tf(t,\alpha )dt = + \infty } \]$ for each $\alpha \neq 0$, Theorem. 1. For each $p \neq 0$, the boundary value problem (F), (A_1), (B) has a solution if and only if $f(t,x) \in I_{\infty}$ Theorem 2. For each$q \neq 0$, the boundary value problem (F), (A_2), (B) has a solution if and only if $f(t, x) \in I_{\infty}$. Theorem 3. For each k>0 and $r \neq 0$, the boundary value problem (F), (A_3), (B) has a solution if and only if f(t, x) \in I_{\infty}, Theorem 4. The boundary value problem (F), (A_j), (B) has at most one solution for j=l, 2, 3. .     

The modified KdV equation on a finite interval

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann–Hilbert problem in the complex k-plane. This Riemann–Hilbert problem has explicit (x,t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of the initial condition q(x,0)=q₀(x), while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic “global relation” that characterize the boundary values in spectral terms. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

σ-相关同伦元素的非平凡性

本文中,通过几何方法证明了σ相关同伦元素在球面稳定同伦群π_mS中是非平凡的,其中m=p~(n+1)q+2p~nq+(s+3)p~2q+(s+3)pq+(s+3)q-8,p≥7是奇素数,n3,0≤sp-3,且q=2(p-1).该σ相关同伦元素在Adams谱序列的E_2-项中由■_s+3■_ng0表示.     

一维$P$-Laplace二阶差分系统奇异边值问题正解的存在性

应用锥压缩锥拉伸不动点定理和Leray-Schauder 抉择定理研究了一类具有P-Laplace算子的奇异离散边值问题$$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~~i\in \{1,2,...,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.$$的单一和多重正解的存在性,其中$\phi(s) = |s|^{p-2}s, ~p>1$,非线性项$f_{k}(i,x,y)(k=1,2)$在$(x,y)=(0,0)$具有奇性.     

二阶中立型微分方程非振动解的存在性

考虑了一类变系数的具有强迫项的二阶中立型微分方程(x(t)+R(t)x(h(t)))″+P(t)x(g_1(t))-Q(t)x(g_2(t))=f(t)非振动解的存在性问题.通过Banach压缩映像原理,分别得到了方程存在满足■|x(t)|>0的非振动解x(t)的充分条件与必要条件,推广了一阶变系数方程的相应结果.     

边界条件含有特征参数的Sturm-Liouville算子的唯一性定理

建立了一类Sturm-Liouville问题的唯一性定理.对于固定的n∈Z,证明了该Sturm-Liouville问题的第n个特征值λ_n（q,a）关于a是严格单调的.对不同系数的a_k,如果能够测得第n个特征值的谱集合{λ_n（q,a_k）}_k=1^+∞,则谱集合{λ_n（q,a_k）}_k=1^+∞能够唯一确定[0,π]上的势函数q（x）.     

The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach

We consider the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u _t −u _txx +2u _x +3uu _x =2u _x u _xx +uu _xxx on the half-line x≥0. In this article, we aim to provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann–Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated to initial and boundary values of the solution. The construction requires more boundary data than those needed for a well-posed IBV problem. Their dependence is expressed in terms of an algebraic relation to be satisfied by the spectral functions. This RH formulation gives us the long-time asymptotics of a solution of the CH-equation. Dedicated to Gennadi Henkin in great admiration.     

GLOBAL EXISTENCE OF THE SOLUTIONS TO NONLINEAR HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.     

EXISTENCE AND NONEXISTENCE OF GLOBAL SOLUTION OF NONLINEAR PARABOLIC EQUATION WITH NONLINEAR BOUNDARY CONDITION

ＥＸＩＳＴＥＮＣＥＡＮＤＮＯＮＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＯＦＮＯＮＬＩＮＥＡＲＰＡＲＡＢＯＬＩＣＥＱＵＡＴＩＯＮＷＩＴＨＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＣＯＮＤＩＴＩＯＮ￥ＷＵＹＯＮＧＨＵＩ；ＷＡＮＧＭＩＮＧＸＩＮＡｂｓｔｒ...     

Oscillation Criteria for Functional Nonlinear Dynamic Equations with $${\gamma}$$ -Laplacian,Damping and Nonlinearities Given by Riemann–Stieltjes Integrals

In this paper, we present oscillation criteria for the second-order nonlinear dynamic equation \({[a(t)\phi_{\gamma} (x^{\Delta}(t))]^{\Delta} + p(t)\phi_{\gamma}(x^{\Delta^{\sigma}}(t)) + q_{0}(t) \phi_{\gamma}(x(g_{0}(t)))+\sum_{i=1}^{2}\int_{a_{i}}^{b_{i}}q_{i}(t,s)\phi_{\alpha_{i}(s)}(x(g_{i}(t,s))) \Delta \zeta_{i}(s)=0}\) on a time scale \({\mathbb{T}}\) which is unbounded above. Our results generalize and improve some known results for oscillation of second-order nonlinear dynamic equation. Some examples are given to illustrate the main results.     

稳定性理論中第一临界情形的微分方程与微分差分方程的等价性問題

<正> §1.問題与方法.在[1]中提出了等价性問題,并对于一般n的情形作了系統的研究.本文是处理在第一临界情形下的微分方程与微分差分方程的等价性問題. 問題是研究微分方程組     

On Stability and Convergence of the Finite Difference Methods for the Nonlinear Pseudo-Parabolic System

In this paper, we deal with the finite difference method for the initial boundary value problem of the nonlinear pseudo-parabolic system $(-1)^Mu_t+A(x,t,u,u_x,\cdots,u_x 2M-1)u_x2M_t=F(x,t,u,u_x,\cdots,u_x 2M)$,$u_xk(o,t)=\psi_{0k}(t), u_xk(L,t)=\psi_{1k}(t),k=0,1,\cdots,M-1,u(x,0)=\phi (x)$ in the rectangular domain $D=[0\leq X\leq L,0\leq t\leq T]$, where $u(x,t)=(u_1(x,t),u_2(x,t),\cdots,u_m(x,t)),\phi (x),\psi_{0k}(t),\psi_{1k}(t),F(x,t,u,u_x,\cdots,u_x 2M)$ are $m$-dimensional vector functions, and $A(x,t,u,u_x,\cdots,u_x2M-1)$ is an $m\times m$ positive definite matrix. The existence and uniqueness of solution for the finite difference system are proved by fixed-point theory. Stability, convergence and error estimates are derived.     

Inverse scattering up to smooth functions for the Dirac-ZS-AKNS system

We consider the Dirac-ZS-AKNS system (1) where (the space of functions with n derivatives in L ¹), (2) We consider for (1) the transition matrix and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case the scattering matrix We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions, of the inverse scattering problem for (1). More preciseley, we show that, under condition (2) with , the following formulas are valid: (3) and, in addition, for the case of the Dirac system (4) where denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac system under condition (2) with III. As applications of the results mentioned above, we show that 1) for any real-valued initial data , the Cauchy problem for the sh-Gordon equation has a unique solution such that and for any t > 0, 2) in addition, for , for such a solution the following formula is valid: where denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results.     

GLOBAL EXISTENCE OF THE SOLUTIONS OF NONLINEAR PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear parabolic equation: $$\[\left\{ {\begin{array}{*{20}{c}} {{u_t} = \Delta u + F(u,{D_x}u,D_x^2u),(t,x) \in {B^ + } \times \Omega ,}\{u(0,x) = \varphi (x),x \in \Omega }\{u{|_{\partial \Omega }} = 0} \end{array}} \right.\]$$ where $\[\Omega \]$ is the exterior domain of a compact set in $\[{R^n}\]$ with smooth boundary and F satisfies $\[\left| {F(\lambda )} \right| = o({\left| \lambda \right|^2})\]$, near $\[\lambda = 0\]$. It is proved that when $\[n \ge 3\]$, under the suitable smoothness and compatibility conditions, the above problem has a unique global smooth solution for small initial data. Moreover, It is also proved that the solution has the decay property $\[{\left\| {u(t)} \right\|_{{L^\infty }(\Omega )}} = o({t^{ - \frac{n}{2}}})\]$, as $\[t \to + \infty \]$.     

Chemotaxis-fluild 系统的全局弱解

We investigate the existence of the global weak solution to the coupled Chemotaxisfluid system ■in a bounded smooth domain ??R~2. Here, r≥0 and μ 0 are given constants,?Φ∈L~∞(?) and g∈L~2((0, T); L_σ~2(?)) are prescribed functions. We obtain the local existence of the weak solution of the system by using the Schauder fixed point theorem. Furthermore, we study the regularity estimate of this system. Utilizing the regularity estimates, we obtain that the coupled Chemotaxis-fluid system with the initial-boundary value problem possesses a global weak solution.     

EXISTENCE AND NON-EXISTENCE OF GLOBAL SMOOTH SOLUTIONS FOR QUASILINEAR HYPERBOLIC SYSTEMS~*

Consider initial value probiom v_t-u_x=0, u_t+p(v)_x=0, (E), v(x, 0)=v_0(x), u(x, 0)=u_0(x), (I), where A≥0, p(v)=K~2v~(-γ), K>0, 0<γ<3. As 0<γ≤1, the authors give a sufficient condition for that (E), (I) to have a unique global smooth solution, As 1≤γ<3, a necessary condition is given for that.     

扩散方程系数的半逆问题

一般地,扩散方程的系数q(x)与p(x)是由两组谱或者一组谱及其标准常数唯一确定的.运用Hochstadt与Lieberman的方法证明了:(a)如果给定区间[π/2,π]上的p(x)及区间[0,π]上的q(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数p(x);(b)如果给定区间[π/2,π]上的q(x)及区间[0,π]上的p(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数q(x).     

An $L_1$ Minimization Problem by Generalized Rational Functions

Let $P,Q \subset L_1(X,\Sigma,\mu)$ and $q(x)>0$ a. e. in $X$ for all $q\in Q$. Define $R=\{p/q:p\in P,q\in Q\}$. In this paper we discuss an $L_1$ minimization problem of a nonnegative function $E(z,x)$, i.e. we wish to find a minimum of the functional $\phi(r)=\int _X qE(r,x)d\mu$ form $r=p/q\in R$. For such a problem we have established the complete characterizations of its minimum and of uniqueness of its minimum, when both $P,Q$ are arbitrary convex subsets.