算子方程组的正解及在半正微分边值系统的应用

证明了半正算子方程组{x=λK1F1(x,y),y=λk2f2(x,y)正解的存在性结果,其中λ>0为参数,P为实Banach空间E中一个完全锥,K1,K2:P→P为线性全连续算子,F1,F2:P→E为连续有界算子.作为应用,给出了一类半正微分边值系统正解存在性的结果.     

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 $\[t \to + \infty \]$.     

ON LINEAR AND NONLINEAR RIEMANN-HILBERT PROBLEMS FOR REGULAR FUNCTION WITH VALUES IN A CLIFFORD ALGEBRA

This paper deals with the boundary value problems for regular function with valuesin a Clifford algebra: ()W=O, x∈R~n\Г, w~+(x)=G(x)W~-(x)+λf(x, W~+(x), W~-(x)), x∈Г; W~-(∞)=0,where Г is a Liapunov surface in R~n the differential operator ()=()/()x_1+()/()x_2+…+()/()x_ne_n, W(x) =∑_A, ()_AW_A(x) are unknown functions with values in a Clifford algebra ()_n Undersome hypotheses, it is proved that the linear baundary value problem (where λf(x, W~+(x),W~-(x)) =g(x)) has a unique solution and the nonlinear boundary value problem has atleast one solution.     

On a semilinear double fractional heat equation driven by fractional Brownian sheet

In this paper, we consider the stochastic heat equation of the form $$\frac{\partial u}{\partial t}=(\Delta_\alpha+\Delta_\beta)u+\frac{\partial f}{\partial x}(t,x,u)+\frac{\partial^2W}{\partial t\partial x},$$ where $1<\beta<\alpha< 2$, $W(t,x)$ is a fractional Brownian sheet, $\Delta_\theta:=-(-\Delta)^{\theta/2}$ denotes the fractional Lapalacian operator and $f:[0,T]\times \mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear measurable function. We introduce the existence, uniqueness and H\"older regularity of the solution. As a related question, we consider also a large deviation principle associated with the above equation with a small perturbation via an equivalence relationship between Laplace principle and large deviation principle.     

Convergence results for a class of nonlinear fractional heat equations

In this article we study various convergence results for a class of nonlinear fractional heat equations of the form $\left\{ \begin{gathered} u_t (t,x) - \mathcal{I}[u(t, \cdot )](x) = f(t,x),(t,x) \in (0,T) \times \mathbb{R}^n , \hfill \\ u(0,x) = u_0 (x),x \in \mathbb{R}^n , \hfill \\ \end{gathered} \right.$ where I is a nonlocal nonlinear operator of Isaacs type. Our aim is to study the convergence of solutions when the order of the operator changes in various ways. In particular, we consider zero order operators approaching fractional operators through scaling and fractional operators of decreasing order approaching zero order operators. We further give rate of convergence in cases when the solution of the limiting equation has appropriate regularity assumptions.     

Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent

In this work, we give an existence result of entropy solutions for nonlinear anisotropic elliptic equation of the type $$- \mbox{div} \big( a(x,u,\nabla u)\big)+ g(x,u,\nabla u) + |u|^{p_{0}(x)-2}u = f-\mbox{div} \phi(u),\quad \mbox{ in } \Omega,$$ where $-\mbox{div}\big(a(x,u,\nabla u)\big)$ is a Leray-Lions operator, $\phi \in C^{0}(I\!\!R,I\!\!R^{N})$. The function $g(x,u,\nabla u)$ is a nonlinear lower order term with natural growth with respect to $|\nabla u|$, satisfying the sign condition and the datum $f$ belongs to $L^1(\Omega)$.     

Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator $D$ in the upper half-plane $R^{2}_+=R\times(0,\infty)$, where $$(Df)(x)=f'(x)+(\lambda/x)[f(x)-f(-x)]$$ for given $\lambda\ge0$. A $C^2$ function $u$ in $R^{2}_+$ is said to be $\lambda$-harmonic if $(D_x^2+\partial_{y}^2)u=0$. For a $\lambda$-harmonic function $u$ in $R^{2}_+$ and for a subset $E$ of $\partial R^{2}_+=R$ symmetric about $y$-axis, we prove that the following three assertions are equivalent: (i) $u$ has a finite non-tangential limit at $(x,0)$ for a.e. $x\in E$; (ii) $u$ is non-tangentially bounded for a.e. $x\in E$; (iii) $(Su)(x)<\infty$ for a.e. $x\in E$, where $S$ is a Lusin-type area integral associated with the Dunkl operator $D$.     

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

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].     

A Singular Parabolic Boundary Value Problem in Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

We consider the operator ${\cal A}$ formally defined by ${\cal A}u(x)=\alpha(x)\Delta u(x)$ for any $x$ in a sufficiently smooth bounded open set $\Om\subset\R^N$ ($N\ge 1$), where $\alpha\in C(\ov\Omega)$ is a continuous nonnegative function vanishing only on $\partial\Omega$, and such that $1/\alpha$ is integrable in $\Omega$. We prove that the realization $A_p$ of ${\cal A}$, equipped with suitable nonlinear boundary conditions is an m-dissipative operator in suitably weighted $L^p(\Omega)$-spaces in the case where either $(p,N)\in (1,+\infty)\times\{1\}$ or $(p,N)=\{2\}\times\N$. Moreover, we prove that $A_p$ is a densely defined closed operator. We consider nonlinear boundary conditions of the following type: in the one dimensional case we take $\Omega=(0,1)$ and we assume that $u(j)=(-1)^j\beta_j(u(j))$ ($j=0,1$), $\beta_0$ and $\beta_1$ being nondecreasing continuous functions in $\R$ such that $\beta_0(0)=\beta_1(0)=0$; in the $N$-dimensional setting we assume that $(D_{\nu}u)_{|\partial\Omega}=-\beta(u_{|\partial\Omega})$, $\beta$ being a nondecreasing Lipschitz continuous function in $\R$ such that $\beta(0)=0$. Here $\nu$ denotes the unit outward normal to $\partial\Om$.     

Anisotropic Elliptic Nonlinear Obstacle Problem with Weighted Variable Exponent

In this paper, we are concerned with a show the existence of a entropy solution to the obstacle problem associated with the equation of the type :$\begin{cases}Au+g(x,u,∇u) = f & {\rm in} & Ω \\ u=0 & {\rm on} & ∂Ω \end{cases}$where $\Omega$ is a bounded open subset of $\;\mathbb{R}^{N}$, $N\geq 2$, $A\,$ is an operator of Leray-Lions type acting from $\; W_{0}^{1,\overrightarrow{p}(.)} (\Omega,\ \overrightarrow{w}(.))\;$ into its dual $\; W_{0}^{-1,\overrightarrow{p}'(.)} (\Omega,\ \overrightarrow{w}^*(.))$ and $\,L^1\,-\,$deta. The nonlinear term $\;g\,$: $\Omega\times \mathbb{R}\times \mathbb{R}^{N}\longrightarrow \mathbb{R} $ satisfying only some growth condition.     

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.     

Positive Gegenbauer Polynomial Sums and Applications to Starlike Functions

Let $s_n(f,z):=\sum_{k=0}^{n}a_kz^k$ be the $n$th partial sum of $f(z)=\sum_{k=0}^{\infty{}}a_kz^k$. We show that $\RE s_n(f/z,z)>0$ holds for all $z\in\D,\ n\in\N$, and all starlike functions $f$ of order $\lambda$ iff $\lambda_0\leq\lambda<1$ where $\lambda_0=0.654222...$ is the unique solution $\lambda\in(\frac{1}{2},1)$ of the equation $\int_{0}^{3\pi/2}t^{1-2\lambda}\cos t \,dt=0$. Here $\D$ denotes the unit disk in the complex plane $\C$. This result is the best possible with respect to $\lambda_0$. In particular, it shows that for the Gegenbauer polynomials $C_{n}^{\mu}(x)$ we have $\sum_{k=0}^n C_{k}^{\mu}(x)\cos k \theta>0$ for all $n\in\N,\ x\in[-1,1]$, and $0<\mu\leq\mu_0:=1-\lambda_0=0.345778...$. This result complements an inequality of Brown, Wang, and Wilson (1993) and extends a result of Ruscheweyh and Salinas (2000).     

一类奇异三阶三点边值问题的正解

In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positive parameter and 0 ≤ η 1 2 .By using the classical Krasnosel’skii’s fixed point theorem in cone,we obtain various new results on the existence of positive solution,and the solution is strictly increasing.Finally we give an example.     

ASYMPTOTIC STABILITY FOR A CLASS OF NONAUTONOMOUS NEUTRAL DIFFERENTIAL EQUATIONS

ＡＳＹＭＰＴＯＴＩＣＳＴＡＢＩＬＩＴＹＦＯＲＡＣＬＡＳＳＯＦＮＯＮＡＵＴＯＮＯＭＯＵＳＮＥＵＴＲＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ＊＊ＹＵＪＩＡＮＳＨＥ＊ＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＪｕｌｙ４，１９９５．ＲｅｖｉｓｅｄＭａｒｃｈ２...     

Positive periodic solutions for a nonlinear differential system with two parameters

In this article, we investigate a nonlinear system of differential equations with two parameters $$\left\{ \begin{array}{l} x"(t)=a(t)x(t)-\lambda f(t, x(t), y(t)),\y"(t)=-b(t)y(t)+\mu g(t, x(t), y(t)),\end{array}\right.$$ where $a,b \in C(\textbf{R},\textbf{R}_+)$ are $\omega-$periodic for some period $\omega > 0$, $a,b \not\equiv 0$, $f,g \in C(\textbf{R} \times \textbf{R}_+ \times \textbf{R}_+ ,\textbf{R}_+)$ are $\omega-$periodic functions in $t$, $\lambda$ and $\mu$ are positive parameters. Based upon a new fixed point theorem, we establish sufficient conditions for the existence and uniqueness of positive periodic solutions to this system for any fixed $\lambda,\mu>0$. Finally, we give a simple example to illustrate our main result.     

Necessary and Sufficient Conditions for Associated Differential Operators in Quaternionic Analysis and Applications to Initial Value Problems

This paper deals with the initial value problem of type $$\begin{array}{ll} \qquad \frac{\partial u}{\partial t} = \mathcal{L} u := \sum \limits^3_{i=0} A^{(i)} (t, x) \frac{\partial u}{\partial x_{i}} + B(t, x)u + C(t, x)\\ u (0, x) = u_{0}(x)\end{array}$$ in the space of generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation $$\mathcal{D}_{\lambda}u := \mathcal{D} u + \lambda u = 0,$$ where ${t \in [0, T]}$ is the time variable, x runs in a bounded and simply connected domain in ${\mathbb{R}^{4}, \lambda}$ is a real number, and ${\mathcal{D}}$ is the Cauchy-Fueter operator. We prove necessary and sufficient conditions on the coefficients of the operator ${\mathcal{L}}$ under which ${\mathcal{L}}$ is associated with the operator ${\mathcal{D}_{\lambda}}$ , i.e. ${\mathcal{L}}$ transforms the set of all solutions of the differential equation ${\mathcal{D}_{\lambda}u = 0}$ into solutions of the same equation for fixedly chosen t. This criterion makes it possible to construct operators ${\mathcal{L}}$ for which the initial value problem is uniquely soluble for an arbitrary initial generalized regular function u ₀ by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also generalized regular for each t.     

On critical exponent for the existence and stability properties of positive weak solutions for some nonlinear elliptic systems involving the (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian and indefinite weight function

This paper deals with the existence and stability properties of positive weak solutions to classes of nonlinear systems involving the (p,q)-Laplacian of the form

Transplantation Theorems for Ultraspherical Polynomials in Re H 1 and BMO

We consider the uniformly bounded orthonormal system of functions $$ u_n^{(\l)}(x)= \varphi_n^{(\lambda)}(\cos x)(\sin x)^\lambda, \qquad x\in [0,\pi], $$ where $\{\varphi_n^{(\lambda)}\}_{n=0}^\infty \,\, (\lambda > 0)$ is the normalized system of ultraspherical polynomials. R. Askey and S. Wainger proved that the $L^p$-norm $(1 < p < \infty)$ of any linear combination of the first $N+1$ functions $u_n^{(\lambda)}(x)$ is equivalent to the $L^p$-norm of the even trigonometric polynomial of degree $N$ with the same coefficients. This theorem fails if $p=1 $ or $p=\infty.$ Studying these limiting cases, we prove (for $0 < \lambda < 1$) similar transplantation theorems in $\mbox{Re } H^1$ and $\mbox{BMO}.$     

Uniqueness of solutions for an integral boundary value problem with fractional $q$-differences

This paper deals with uniqueness of solutions for integral boundary value problem$\left\{\begin{array}{l}(D_q^{\alpha}u)(t)+f(t, u(t))=0,\ \ \ t\in(0,1),\ u(0)=D_qu(0)=0,\ \ u(1)=\lambda\int_0^1u(s){\mbox d}_qs, \end{array}\right.$ where $\alpha\in(2,3]$, $\lambda\in (0,[\alpha]_q)$, $D_q^{\alpha}$ denotes the $q$-fractional differential operator of order $\alpha$. By using the iterative method and one new fixed point theorem, we obtain that there exist a unique nontrivial solution and a unique positive solution.