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. .     

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.     

An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights

We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.     

可换环上矩阵代数的SZ-导子, PZ-导子 和 S-导子

Let R be a commutative ring with identity, Nn（R） the matrix algebra consisting of all n × n strictly upper triangular matrices over R with the usual product operation. An R-linear map φ ： Nn（R） → Nn（R） is said to be an SZ-derivation of Nn（R） if x2 = 0 implies that φ（x）x＋xφ（x） = 0. It is said to be an S-derivation of Nn（R） if φ（x2） = φ（x）x＋xφ（x） for any x ∈ Nn（R）. It is said to be a PZ-derivation of Nn（R） if xy = 0 implies that φ（x）y＋xφ（y） = 0. In this paper, by constructing several types of standard SZ-derivations of Nn（R）, we first characterize all SZ-derivations of Nn（R）. Then, as its application, we determine all S-derivations and PZ- derivations of Nn（R）, respectively.     

Series Representation of Daubechies' Wavelets

1,Iotroduction.InthispaPerwe8tudytherepresentationofDaubechies'wavelets.DaubechiesI1]constructedaf4milyofcompartlysupportedregularscallngfUnctionsrk.(x)andtheassoci4tedregularwpeletsop.(x)(N32):where4.eL'(R)definedbythep0lyn0mia:withZq.(k)=1'q.(k)ER,k=0,1,')N-1.Itisknownthat[1]f0reachN32,k=Osuppgh.=[0,2N-l],suppop.=[-(N-1),N]andthewaveletop.generatesbyitsdilatiOnsandtranslati0nsan0rth0rn0rmalbasis{m.(2ix-k)}i,k6Z0fL'(R).Thefunctionsrk.andop.havebeenprovedtobeveryusefulinnumericalanal…     

THE GLOBAL SOLUTION OF INITIALBOUNDARY VALUE PROBLEM OF HIGHER-ORDER MULTIDIMENSIONAL EQUATION OF CHANGING TYPE

In practical problems there appears the higher-order equations of changing type. But,there is only a few of papers, which studied the problems for this kind of equations. In this paper a kind of the higher-order m     

UNBOUNDED SOLUTIONS OF CONSERVATIVE OSCILLATORS UNDER ROUGHLY PERIODIC PERTURBATIONS

This note is concerned with the equation $$\[\frac{{{d^2}x}}{{d{t^2}}} + g(x) = p(t)\begin{array}{*{20}{c}} {}&{(1)} \end{array}\]$$ where g(x) is a continuously differentiable function of a $\[x \in R\]$, $\[xg(x) > 0\]$ whenever $\[x \ne 0\]$, and $\[g(x)/x\]$ tends to $\[\infty \]$ as \[\left| x \right| \to \infty \]. Let p(t) be a bounded function of $\[t \in R\]$. Define its norm by $\[\left\| p \right\| = {\sup _{t \in R}}\left| {p(t)} \right|\]$ The study of this note leads to the following conclusion which improves a result due to J. E. Littlewood, For any given small constants $\[\alpha > 0,s > 0\]$, there is a continuous and roughly periodic(with respect to $\[\Omega (\alpha )\]$) function p(t) with $\[\left\| p \right\| < s\]$ such that the corresponding equation (1) has at least one unbounded solution.     

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problem {p′(s)=f′(s)-λp^β(s)for s∈(0,1);β∈(0,1),p(0)=p(1)=0,p(s)&gt;0 if s∈(0,1),has a solution(λ^-，p^-（s）),where |λ^-| is the smallest parameter,under the minimal stringent restrictions on f(s), by applying the shooting and regularization methods. In a classic paper, Kohmogorov et.al.studied in 1937 a problem which can be converted into a special case of the above problem. The author also use the solution(λ^-,p^-(s)) to construct a weak travelling wave front solution u(x,t)=y(ξ),ξ=x-Ct,C=λ^-N/(N+1),of the generalized diffusion equation with reaction δ/δx（k（u）|δu/δx|^n-1 δu/δx)-δu/δt=g（u），where N&gt;0,k(s)&gt;0 a.e.on(0,1),and f(a):=n+1/N∫0ag(t)k^1/N(t)dt is absolutely continuous ou[0,1],while y(ξ) is increasing and absolutely continuous on (-∞,+∞) and (k(y(ξ))|y′(ξ)|^N)′=g(y(ξ))-Cy′(ξ)a.e.on(-∞，+∞）,y(-∞)=0,y(+∞)=1.     

The Theory of Filled Function Method for Finding Global Minimizers of Nonlinearly Constrained Minimization Problems

This paper is an extension of [1]. In this paper the descent and ascent segments are introduced to replace respectively the descent and ascent directions in [1] and are used to extend the concepts of S-basin and basin of a minimizer of a function. Lemmas and theorems similar to those in [1] are proved for the filled function $$P(x,r,p)= \frac{1}{r+F(x)}exp(-|x-x^*_1|^2/\rho^2),$$ which is the same as that in [1], where $x^*_1$ is a constrained local minimizer of the problem (0.3) below and $$F(x)=f(x)+\sum^{m'}_{i=1}\mu_i|c_i(x)|+ \sum^m_{i=m'+1}\mu_i max(0, -c_i(x))$$ is the exact penalty function for the constrained minimization problem$\mathop{\rm min}\limits_x f(x)$,subject to $$c_i(x) = 0 , i = 1, 2, \cdots, m',$$ $$c_i(x) \ge 0 , i = m'+1, \cdots, m,$$ where $μ_i＞0 \ (i=1, 2, \cdots, m)$ are sufficiently large. When $x^*_1$ has been located, a saddle point or a minimizer $\hat{x}$ of $P(x,r,\rho)$ can be located by using the nonsmooth minimization method with some special termination principles. The $\hat{x}$ is proved to be in a basin of a lower minimizer $x^*_2$ of $F(x)$, provided that the ratio $\rho^2/[r+F(x^*_1)]$ is appropriately small. Thus, starting with $\hat{x}$ to minimize $F(x)$, one can locate $x^*_2$. In this way a constrained global minimizer of (0.3) can finally be found and termination will happen.     

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).     

A Completely Exponentially Fitted Difference Scheme for a Singular Perturbation Problem

A completely exponentially fitted difference scheme is considered for the singular perturbation problem: $\epsilon U^{''}+a(x) U^{'}-b(x) U=f(x) \ {\rm for} \ 0 \lt x \lt 1$, with U(0), and U(1) given, $\epsilon \in (0,1]$ and $a(x) \gt α \gt 0, b(x)\geq 0$. It is proven that the scheme is uniformly second-order accurate.     

INVARIANT SETS AND THE HUKUHARA-KNESER PROPERTY FOR PARABOLIC SYSTEMS

In this paper we are concerned with the nonlinear boundary value problem forparabolic system(Lu=f(x,t,u,▽u),x∈Ω,0相似文献   

ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION FOR PARAMETER OF ONE-DIMENSIONAL DISCRETE EXPONENTIAL FAMILIES

Consider the discrete exponential family written in the form P_θ(X=x)=h(x)β(θ)θ~x,x=0,1,2,…,where h(x)>0,x=0,1,2,…,The prior distribution of θ belongs to thefa     

Projektive Skalen,Tschebyscheff-Polynome und Fibonacci-Zahlen

Summary. Let $\widehat{\widehat T}_n$ and $\overline U_n$ denote the modified Chebyshev polynomials defined by $\widehat{\widehat T}_n (x) = {T_{2n + 1} \left(\sqrt{x + 3 \over 4} \right) \over \sqrt{x + 3 \over 4}}, \quad \overline U_{n}(x) = U_{n} \left({x + 1 \over 2}\right) \qquad (n \in \mathbb{N}_{0},\ x \in \mathbb{R}).$ For all $n \in \mathbb{N}_{0}$ define $\widehat{\widehat T}_{-(n + 1)} = \widehat{\widehat T}_n$ and $\overline U_{-(n + 2)} = - \overline U_n$, furthermore $\overline U_{-1} = 0$. In this paper, summation formulae for sums of type $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k}(\nu; x)$ are given, where $\bigl(\mathbf a_{\mathbf k}(\nu; x)\bigr)^{-1} = (-1)^k \cdot \Bigl( x \cdot \widehat{\widehat T}_{\left[k + 1 \over 2\right] - 1} (\nu) +\widehat{\widehat T}_{\left[k + 1 \over 2\right]}(\nu)\Bigr) \cdot \Bigl(x \cdot \overline U_{\left[k \over 2\right] - 1} (\nu) + \overline U_{\left[k \over 2\right]} (\nu)\Bigr)$ with real constants $ x, \nu $. The above sums will turn out to be telescope sums. They appear in connection with projective geometry. The directed euclidean measures of the line segments of a projective scale form a sequence of type $(\mathbf a_{\mathbf k} (\nu;x))_{k \in \mathbb{Z}}$ where $ \nu $ is the cross-ratio of the scale, and x is the ratio of two consecutive line segments once chosen. In case of hyperbolic $(\nu \in \mathbb{R} \setminus] - 3,1[)$ and parabolic $\nu = -3$ scales, the formula $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k} (\nu; x) = {\frac{1}{x - q_{{+}\atop(-)}}} - {\frac{1}{x - q_{{-}\atop(+)}}} \eqno (1)$ holds for $\nu > 1$ (resp. $\nu \leq - 3$), unless the scale is geometric, that is unless $x = q_+$ or $x = q_-$. By $q_{\pm} = {-(\nu + 1) \pm \sqrt{(\nu - 1)(\nu + 3)} \over 2}$ we denote the quotient of the associated geometric sequence.

     

LIFE SPAN OF CLASSICAL SOLUTIONS TO $\[B \simeq Ma{t_m}(kD)\]$ IN TWO SPACE DIMENSIONS

The author studies the life span of classical solutions to the following Cauchy problem $\[B \simeq Ma{t_m}(kD)\]$, $t=0:u=\epsilon\phi(x),u_t=\epsilon\psi(x),x\in R^2$ where $\phi,\psi\in C_0^\infinity(R^2)$ and not both identically zero,$\[\square = \partial _t^2 - \partial _1^2 - \partial _2^2,p \geqslant 2\]$ is a real number and $\epsilon > 0$ is a small parameter, and obtains respectively upper and lower bounds of the same order of magnitude for the life span for $2\leq p \leq p_0$, where $p_0$ is the positive root of the quadratic $X^2-3X-2=0$.     

Characterizations of Bochner–Krall Orthogonal Polynomials of Jacobi Type

Let $\{P_n(x) \}_{n=0}^\infty$ be an orthogonal polynomial system relative to a compactly supported measure. We find characterizations for $\{P_n(x) \}_{n=0}^\infty$ to be a Bochner--Krall orthogonal polynomial system, that is, $\{P_n(x) \}_{n=0}^\infty$ are polynomial eigenfunctions of a linear differential operator of finite order. In particular, we show that $\{P_n(x) \}_{n=0}^\infty$ must be generalized Jacobi polynomials which are orthogonal relative to a Jacobi weight plus two point masses.     

具位势项的快扩散方程的第二临界指标

This paper considers a fast diffusion equation with potential ut= um V (x)um+upin Rn×(0,T), where 1 2αm+n< m ≤ 1, p > 1, n ≥ 2, V (x) ～ω|x|2with ω≥ 0 as |x| →∞,and α is the positive root of αm(αm + n 2) ω = 0. The critical Fujita exponent was determined as pc= m +2αm+nin a previous paper of the authors. In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p > pcvia the critical decay rates of the initial data.With u0(x) ～ |x| aas |x| →∞, it is shown that the second critical exponent a =2p m,independent of the potential parameter ω, is quite different from the situation for the critical exponent pc.     

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 $\[t \to + \infty \]$.     

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本文考虑离散时间风险模型$U_n=(U_{n-1}+Y_n)(1+r_n)-X_n$,$n=1,2,\cdots$, 其中$U_0=x>0$为保险公司的初始准备金,$r_n$为在第$n$个时刻的利率, $Y_n$为到时刻$n$为止的总保费收入,$X_n$为到时刻$n$为止的所支付的全部索赔,$U_n$表示保险公司在时刻$n$的盈余. 当$Y_n$和$r_n$满足某些温和条件时,我们得到了在\, $x\to\infty$时,有限时间破产概率$\psi(x,N)=\pr\big(\min\limits_{0\leq n\leqN}U_n<0|U_0=x\big)$关于$N\geq1$的一致渐近的关系式\,$\psi(x,N)\sim\tsm_{k=1}^{N}\ol{F}_X((1+r_1)\cdots(1+r_n)x)$,其中$\ol{F}_X(x)$是$X_1$的尾分布.