非线性边界条件下二阶奇异差分方程的正解

本文我们考虑如下二阶奇异差分边值问题\begin{equation*}\begin{cases}-\Delta^{2} u(t-1)=\lambda g(t)f(u) ,\ t\in [1,T]_\mathbb{Z},\\u(0)=0,\\ \Delta u(T)+c(u(T+1))u(T+1)=0,\end{cases}\end{equation*}正解的存在性. 其中, $\lambda>0$, $f:(0,\infty)\rightarrow \mathbb{R}$ 是连续的,且允许在~$0$ 处奇异.通过引入一个新的全连续算子, 我们建立正解的存在性.     

一类新四阶边值问题的唯一性结果

In this paper, we investigate the existence and uniqueness of solutions for a new fourth-order differential equation boundary value problem:{u(4)(t) = f(t, u(t))-b, 0 t 1,u(0) = u′(0) = u′(1) = u(3)(1) = 0,where f ∈ C([0,1] ×(-∞,+∞),(-∞, +∞)),b ≥ 0 is a constant. The novelty of this paper is that the boundary value problem is a new type and the method is a new fixed point theorem ofφ-(h,e)-concave operators.     

一类非线性悬臂梁方程正解的存在性与多解性

研究了非线性四阶常微分方程u(4)(t)=f(t,u(t),u'(t)),t ∈[0,1]\E在边界条件u(0)=u'(0)=u"(1)=u"'(1)=0下的正解,其中E(∩)[0,1]是一个零测度的闭集,而非线性项,(t,u,u)可以在t∈E时奇异.通过构造适当的积分方程并利用锥上的不动点定理证明了这个方程在满足与n有关的条件下存在n个正解,其中n是某个自然数.     

带参数的一阶泛函差分方程的定号周期解

In this paper,the authors obtain the existence of one-signed periodic solutions of the first-order functional difference equation ?u(n) = a(n)u(n)-λb(n)f(u(n-τ(n))),n ∈ Z by using global bifurcation techniques,where a,b:Z → [0,∞) are T-periodic functions with ∑T n=1 a(n) 0,∑T n=1 b(n) 0;τ:Z → Z is T-periodic function,λ 0 is a parameter;f ∈ C(R,R) and there exist two constants s_2 0 s_1 such that f(s_2) = f(0) = f(s_1) = 0,f(s) 0 for s ∈(0,s_1) ∪(s_1,∞),and f(s) 0 for s ∈(-∞,s_2) ∪(s_2,0).     

奇异非线性Sturm-Liouville边值问题正解的全局结构

该文利用拓扑方法讨论一类非线性Sturm-Liouville边值问题\[\left\{\begin{array}{lcl}-u'=\lambda f(x, u),\\\alpha_0 u(0)+\beta_0 u'(0)=0,\ \ \alpha_1 u(1)+\beta_1 u'(1)=0;\end{array}\right.\]作者在非线性项不奇异和奇异两种情况下研究了上述问题正解解集的全局结构,在非线性项$f$不满足条件f(x,u)≥0(u≥0)时 获得了正解的存在性.     

无限区间上二阶离散问题的正解

在本文中,通过运用离散的Arzel\''{a}-Ascoli引理和锥上的不动点定理,我们讨论了无限区间上二阶离散Sturm-Liouville边值问题$$\left\{\begin{array}{l}\Delta^{2}u(x-1)=f(x,u(x),\Delta u(x-1)),~~x\in\mathbb{N},\\ u(0)-a\Delta u(0)=B,~~\Delta u(\infty)=C\end{array}\right.$$ 正解的存在性,其中$\Delta u(x)=u(x+1)-u(x)$是前向差分算子,$\mathbb{N}=\{1,2,\ldots,\infty\}$且$f:\mathbb{N}\times\mathbb{R_{+}}\times\mathbb{R_{+}}\to\mathbb{R_{+}}$连续,$a>0, B, C$ 为非负实数,$\mathbb{R_{+}}=[0,+\infty)$, $\Delta u(\infty)=\lim_{x\rightarrow\infty}\Delta u(x)$.     

一类n-阶m-点奇异边值问题的正解

研究n-阶m-点奇异边值问题其中h(t)允许在t=0,t=1处奇异,f(t,v_0,v_1,…,v_(n-2))允许在v_i=0(i=0,1,…,n-2)处奇异.利用锥拉伸与压缩不动点定理得到了上述奇异边值问题正解的存在性.     

Spline Finite Difference Methods and Their Extrapolation for Singular Two-Point Boundary Value Problems

In this paper we consider a class of singular two-point boundary value problem: $(x^αu')'=f(x,u),u(0)=A,u(1)=B,0＜α＜1$. The asymptotic expression of the spline finite difference solution for the problem is obtained. By using the asymptotic expression, Richardson's extrapolation can be done and the accuracy of numerical solution can be improved. Some numerical examples are given in illustration of this theory.     

Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form(P_(a,b)){D~αu(x) + f(x, u(x)) = 0, x ∈(0, 1),u(0) = u(1) = 0, D~(α-3)u(0) = a, u(1) =-b,where 3 α≤ 4, Dαis the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) =-p(x)t~σ, with σ∈(-1, 1)and p being a nonnegative continuous function that may be singular at x = 0 or x = 1and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch¨auder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem(P_(0,0)).Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b 0 and f(x, t) = tφ(x, t), with φ(x, t) being a nonnegative continuous function in(0, 1)×[0, ∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green's function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem(P_(a,b)), which behaves like the unique solution of the homogeneous problem corresponding to(P_(a,b)). Some examples are given to illustrate the existence results.     

Some remarks on Liouville type results for quasilinear elliptic equations

For a wide class of nonlinearities satisfying

0$\space in $(0,a)$\space and $f(u)<0$\space in $(a,\infty)$ ,}\end{displaymath}">

we show that any nonnegative solution of the quasilinear equation over the entire must be a constant. Our results improve or complement some recently obtained Liouville type theorems. In particular, we completely answer a question left open by Du and Guo.

     

单圈图的无符号狄利克雷谱半径

Let G be a simple connected graph with pendant vertex set ?V and nonpendant vertex set V_0. The signless Laplacian matrix of G is denoted by Q(G). The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ≠ 0 on V(G) such that Q(G)f(u) = λf(u) for u ∈ V_0 and f(u) = 0 for u ∈ ?V. The signless Dirichlet spectral radiusλ(G) is the largest signless Dirichlet eigenvalue. In this paper, the unicyclic graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with a given degree sequence are characterized.     

弱半正三阶三点边值问题的正解

The positive solutions are studied for the nonlinear third-order three-point boundary value problem u′″（t）=f（t,u（t））,a.e,t∈[0,1],u（0）=u′（η）=u″（1）=0, where the nonlinear term f（t, u） is a Caratheodory function and there exists a nonnegative function h ∈ L^1[0, 1] such that f（t, u） 〉 ≥-h（t）. The existence of n positive solutions is proved by considering the integrations of ＂height functions＂ and applying the Krasnosel＇skii fixed point theorem on cone.     

Eigenvalues of Fourth-order Singular Sturm-Liouville Boundary Value Problems

In this paper, by using Krasnoselskii''s fixed-point theorem, some sufficient conditions of existence of positive solutions for the following fourth-order nonlinear Sturm-Liouville eigenvalue problem:\begin{equation*}\left\{\begin{array}{lll} \frac{1}{p(t)}(p(t)u'')''(t)+ \lambda f(t,u)=0, t\in(0,1), \\ u(0)=u(1)=0, \\ \alpha u''(0)- \beta \lim_{t \rightarrow 0^{+}} p(t)u''(t)=0, \\ \gamma u''(1)+\delta\lim_{t \rightarrow 1^{-}} p(t)u''(t)=0, \end{array}\right.\end{equation*} are established, where $\alpha,\beta,\gamma,\delta \geq 0,$ and $~\beta\gamma+\alpha\gamma+\alpha\delta >0$. The function $p$ may be singular at $t=0$ or $1$, and $f$ satisfies Carath\''{e}odory condition.     

On Numerical Solution of Quasilinear Boundary Value Problems with Two Small Parameters

We consider the singular perturbation problem $$-\varepsilon^2u"+\mu b(x,u)u'+c(x,u)=0,u(0),u(1)$$ given with two small parameters $\varepsilon$ and $\mu$ , $\mu =\varepsilon^{1+p},p>0$. The problem is solved numerically by using finite difference schemes on the mesh which is dense in the boundary layers. The convergence uniform in $\varepsilon$ is proved in the discrete $L^1$ norm. Some convergence results are given in the maximum norm as well.     

由卷积和从属关系刻画的单叶调和函数某些子类

Let SH be the class of functions f = h + ˉg that are harmonic univalent and sensepreserving in the open unit disk U = {z ∈ C : |z| 1} for which f(0) = f′(0)-1 = 0. In the present paper, we introduce some new subclasses of SH consisting of univalent and sensepreserving functions defined by convolution and subordination. Sufficient coefficient conditions,distortion bounds, extreme points and convolution properties for functions of these classes are obtained. Also, we discuss the radii of starlikeness and convexity.     

THE PHYSICAL ENTROPY OF SINGLE CONSERVATION LAWS

1.IntroductionAsthesimplestrepresentationofgeneralconservationlaws,thesingleconservationlawswithonespacevariablehavebeenthoroughlydiscussed.Inthecasesthatthefluxfunctionisconvexorconcave,P.D.LaxobtainedageneralexpressionofthesolutionsforCauchyproblems[2];inordertoguaranteetheuniquenessofthesolution,O.A.OlejnikpresentedherfamousE--conditionwhichcanbeappliedtogeneralconservationlawswithonespacevariablel4];TheLax'sconceptsofentropyfunctionsandrelatedinequalityalsocanbeusedinthisspecialcase[3]…     

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.     

一类复线性微分差分方程亚纯解的唯一性

本文主要研究一类复线性微分差分方程超越亚纯解的唯一性.特别地,假设$f(z)$为复线性微分差分方程: $W_{1}(z)f''(z+1)+W_{2}(z)f(z)=W_{3}(z)$的一个有穷级超越亚纯解,其中$W_{1}(z)$, $W_{2}(z)$, $W_{3}(z)$为增长级小于1的非零亚纯函数并且满足$W_{1}(z)+W_{2}(z)\not\equiv 0$.若$f(z)$与亚纯函数$g(z)$, $CM$分担0,1,$\infty$,则$f(z)\equiv g(z)$或$f(z)+g(z)\equiv f(z)g(z)$或$f^{2}(z)(g(z)-1)^2+g^{2}(z)(f(z)-1)^2=g(z)f(z)(g(z)f(z)-1)$或存在一个多项式$\varphi(z)=az+b_{0}$使得$f(z)=\frac{1-e^{\varphi(z)}}{e^{\varphi(z)}(e^{a_{0}-b_{0}}-1)}$与$g(z)=\frac{1-e^{\varphi(z)}}{1-e^{b_{0}-a_{0}}}$,其中$a(\neq 0)$, $a_{0}$ $b_{0}$均为常数且$a_{0}\neq b_{0}$.     

二阶三点边值问题的正解

该文讨论了二阶三点边值问题$-u'(t)=b(t)f(u(t))$满足$u'(0)=0$, $u(1)={\alpha}u({\eta})$ 正解的存在性与多重性, 其中常数$\alpha, \eta\in(0,1)$, $f\in C ([0,\infty),[0,\infty) )$, $b\in C ([0,1],[0,\infty) )$且存在$t_0\in[0,1]$使$b(t_0)>0$. 利用该问题相应的Green函数, 将其转化为Hammerstein型积分方程, 借助于锥上的不动点指数理论,给出了该问题单个正解和多个正解存在的与其相应线性问题的第一特征值有关的最佳充分性条件.     

An integral boundary value problem of conformable integro-differential equations with a parameter

In this article, we consider some properties of positive solutions for a new conformable integro-differential equation with integral boundary conditions and a parameter $$ \left\{ \begin{array}{l} T_{\alpha}u(t)+\lambda f(t,u(t),I_{\alpha}u(t))=0,t\in[0,1],\u(0)=0,u(1)=\beta\int_{0}^{1}u(t)dt ,\beta\in[\frac 32,2), \ \end{array}\right.\nonumber $$ where $\alpha\in(1,2]$, $\lambda$ is a positive parameter, $T_{\alpha}$ is the usual conformable derivative and $I_{\alpha}$ is the conformable integral, $f:[0,1]\times\mathbf{R^{+}}\times\mathbf{R^{+}}\rightarrow \mathbf{R^{+}} $ is a continuous function, where $\mathbf{R^{+}}=[0,+\infty)$. We use a recent fixed point theorem for monotone operators in ordered Banach spaces, and then establish the existence and uniqueness of positive solutions for the boundary value problem. Further, we give an iterative sequence to approximate the unique positive solution and some good properties of positive solution about the parameter $\lambda$. A concrete example is given to better demonstrate our main result.