Nodal Solutions of the Brezis-Nirenberg Problem in Dimension 6

We show that the classical Brezis-Nirenberg problem $$-\Delta u=u|u|+\lambda u \ \ \ \ \ \ \ in \ \ \ \Omega, \\ u=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ on \ \ \ \partial\Omega,$$ when $\Omega$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $\Omega$ as $\lambda$ approaches a suitable value $\lambda_0>0.$     

Infinitely many bound state solutions of Schrodinger-Poisson equations in $\mathbb{R}^3$

In this paper, we study a system of Schr\"odinger-Poisson equation \[ \left\{ \begin{array}{c} -\Delta u+a(x)u+K(x)\phi u=|u|^{p-2}u,\quad \quad \quad \ \ \ \ \ \ x\in \mathbb{R}^3, \-\Delta \phi=K(x)u^2,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ x\in \mathbb{R}^3, \end{array} \right. \] where $p\in (4,6)$ and $ K\geq (\not\equiv) 0$. Under some suitable decay assumptions but without any symmetry property on $a$ and $K$, we obtain infinitely many solutions of this system.     

On the numerical computation of blowing‐up solutions for semilinear parabolic equations

Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end , with a finite unknown ‘blow‐up’ time T_b have been studied in a previous work. Specifically, for ε a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ‘mass control’ property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve‐\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon‐u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end , \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow‐up time T_b and the blow‐up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end , with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end . Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end , we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end , defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end . In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end , the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end , verifies, $\|(u‐u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1‐\alpha}$\nopagenumbers\end , \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end . The two special cases of a single‐point blow‐up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p‐1})$\nopagenumbers\end . The estimate $|T_\eta‐T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd.     

On the asymptotic behavior of a class of third order nonlinear neutral differential equations

The objective of this paper is to study asymptotic properties of the third-order neutral differential equation

On approximation properties of non-convolution type nonlinear integral operators

In the present paper we state some approximation theorems concerning pointwise convergence and its rate for a class of non-convolution type nonlinear integral operators of the form:Tλ (f;x) = B A Kλ (t,x, f (t))dt , x ∈< a,b >, λ∈Λ. In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 of f as (x,λ ) → (x0,λ0) in L1 < A,B >, where < a,b > and < A,B > are is an arbitrary intervals in R, Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology. The results of the present paper generalize several ones obtained previously in the papers [19]-[23].     

The Growth of Composite Meromorphic Functions

Let f be a transcendental meromorphic function of order $ \rho _f $ , g be a transcendental entire function of lower order $\lambda _g (\lambda _g \lt + \infty ) $ with $ \sum _{a\not = \infty }\delta (a,g)= 1 $ , then $$\overline {\mathop {{\rm lim}}\limits_{r \to \infty } } \log {{\left( {T\left( {r,f\left( g \right)} \right)} \right)} \mathord{\left/{\vphantom {{\left( {T\left( {r,f\left( g \right)} \right)} \right)} {T\left( {r,g} \right)}}} \right. \kern-\nulldelimiterspace} {T\left( {r,g} \right)}} = \pi \rho f.$$     

Double Biproduct Hom-Bialgebra and Related Quasitriangular Structures

Let(H, β) be a Hom-bialgebra such that β~2= id_H.(A, α_A) is a Hom-bialgebra in the left-left Hom-Yetter-Drinfeld category (_H~H)YD and(B, α_B) is a Hom-bialgebra in the right-right Hom-Yetter-Drinfeld category YD_H~H. The authors define the two-sided smash product Hom-algebra(A■H■B, α_A ? β ? α_B) and the two-sided smash coproduct Homcoalgebra(A◇H◇B, α_A ? β ? α_B). Then the necessary and sufficient conditions for(A■H■B, α_A ? β ? α_B) and(A◇H◇B, α_A ? β ? α_B) to be a Hom-bialgebra(called the double biproduct Hom-bialgebra and denoted by(A_◇~■H_◇~■B, α_A ? β ? α_B)) are derived. On the other hand, the necessary and sufficient conditions for the smash coproduct Hom-Hopf algebra(A◇H, α_A ? β) to be quasitriangular are given.     

Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity

We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space

逼近Banach空间中渐近非扩张映象的不动点

设E是一致凸Banach空间,C是E的非空闭凸子集, T:C→C是具有不动点的渐近非扩张映象. 该文证明了, 在某些适当的条件下, 由下列修改了的Ishikawa迭代程序所定义的序列{x\-n},\$\$x\-\{n+1\}=t\-nT\+n(s\-nT\+nx\-n+(1-s\-n)x\-n)+(1-t\-n)x\-n,\$\$弱收敛到T的不动点, 其中{t\-n},{s\-n}是区间\[0,1\]中满足某些限制的实数列.     

H?lder-type inequalities involving unitarily invariant norms

General H?lder-type inequalities involving unitarily invariant norms for sums and products of Hilbert space operators are given. Among other inequalities, it is shown that if A, B and X are operators on a complex Hilbert space, then $$\left\vert \left\vert \left\vert {} \left\vert A^{\ast }XB\right\vert^{r} \right\vert \right\vert \right\vert ^{2}\leq \left\vert \left\vert \left\vert \left( A^{\ast }\left\vert X^{\ast} \right\vert A\right) ^{\frac{ pr}{2}} \right\vert \right\vert \right\vert ^{\frac{1}{p}} \left\vert \left\vert \left\vert \left( B^{\ast }\left\vert X\right\vert B\right) ^{ \frac{qr}{2}} \right\vert \right\vert \right\vert ^{\frac{1}{q}}$$ for all positive real numbers r, p and q such that p ^?1?+?q ^?1?=?1 and for every unitarily invariant norm. The results in this article generalize some known H?lder inequalities for operators.     

On the oscillatory behaviour of a class of nonlinear delay difference equations of second order

In this paper, sufficient conditions have been obtained for oscillation of all solutions of a class of nonlinear neutral delay difference equations of the form

具有脉冲的二阶三点边值问题存在性定理

In this paper, two existence theorems are given concerning the following 3-point boundary value problem of second order differential systems with impulses     

THE FIRST BOUNDARY VALUE PROBLEM FOR SOLUTIONS OF DEGENERATE QUASUJNEAR PARABOLIC EQUATIONS

In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation $$\[\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}} = \sum {\frac{\partial }{{\partial {x_i}}}\left( {v(u){A_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}} \right) + \sum {{B_i}(x,t,u)} \frac{{\partial u}}{{\partial {x_i}}}} + C(x,t,u)u\begin{array}{*{20}{c}} {}&{(x,t) \in [0,T]} \end{array},}\{u{|_{t = 0}} = {u_0}(x),x \in \Omega ,}\{u{|_{x \in \partial \Omega }} = \psi (s,t),0 \le t \le T} \end{array}} \right.\]$$ $$\[\left( {\frac{1}{\Lambda }{{\left| \alpha \right|}^2} \le \sum {{A_{ij}}{\alpha _i}{\alpha _j}} \le \Lambda {{\left| \alpha \right|}^2},\forall a \in {R^n},0 < \Lambda < \infty ,v(u) > 0\begin{array}{*{20}{c}} {and}&{v(u) \to 0\begin{array}{*{20}{c}} {as}&{u \to 0} \end{array}} \end{array}} \right)\]$$ under some very weak restrictions, i.e. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$     

On some problems involving Hardy’s function

Some problems involving the classical Hardy function

Asymptotics of solutions to the periodic problem for a Burgers type equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear Burgers type equation

Some integral inequalities for the polar derivative of a polynomial

If P(z) is a polynomial of degree n which does not vanish in |z| 1,then it is recently proved by Rather [Jour.Ineq.Pure and Appl.Math.,9 (2008),Issue 4,Art.103] that for every γ 0 and every real or complex number α with |α|≥ 1,{∫02π |D α P(e iθ)| γ dθ}1/γ≤ n(|α| + 1)C γ{∫02π|P(eiθ)| γ dθ}1/γ,C γ ={1/2π∫0 2π|1+eiβ|γdβ}-1/γ,where D α P(z) denotes the polar derivative of P(z) with respect to α.In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J.Approx.Theory,54 (1988),306-313] as a special case.     

Unitarily invariant norm and $Q$-norm estimations for the Moore--Penrose inverse of multiplicative perturbations of matrices

Let $B$ be a multiplicative perturbation of $A\in\mathbb{C}^{m\times n}$ given by $B=D_1^* A D_2$, where $D_1\in\mathbb{C}^{m\times m}$ and $D_2\in\mathbb{C}^{n\times n}$ are both nonsingular. New upper bounds for $\Vert B^\dag-A^\dag\Vert_U$ and $\Vert B^\dag-A^\dag\Vert_Q$ are derived, where $A^\dag,B^\dag$ are the Moore-Penrose inverses of $A$ and $B$, and $\Vert \cdot\Vert_U,\Vert \cdot\Vert_Q$ are any unitarily invariant norm and $Q$-norm, respectively. Numerical examples are provided to illustrate the sharpness of the obtained upper bounds.     

Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle

For suitable classes of random Verblunsky coefficients, including independent, identically distributed, rotationally invariant ones, we prove that if $$ \bbE \biggl( \int\f{d\theta}{2\pi} \biggl|\biggl( \f{\calC + e^{i\theta}}{\calC-e^{i\theta}} \biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-\kappa_1 \abs{k-\ell}} $$ for some $\kappa_1 < 0$ and $p < 1$, then for suitable $C_2$ and $\kappa_2 >0$, $$ \bbE \Bigl( \sup_n \abs{(\calC^n)_{k\ell}}\Bigr) \leq C_2 e^{-\kappa_2 \abs{k-\ell}}. $$ Here $\calC$ is the CMV matrix.     

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

A Variational Approach to a Kirchhoff-type Problem Involving Two Parameters

In this paper, the authors establish the existence of at least three weak solutions for the Kirchhoff-type problem $$\left\{\begin{array}{ll}-K \left( \int_{\Omega}| \nabla u(x)|^{2}dx \right) \Delta u(x)= \lambda f(x,u)+\mu g(x,u),\quad {\rm in}\; \Omega,\\u=0, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad {\rm on}\; \partial \Omega, \end{array} \right.$$ under appropriate hypotheses. The proofs are based on variational methods.