Solutions to Second-order Three-point Problems on Time Scales

In the first part of the paper, we establish the existence of multiple positive solutions to the nonlinear second-order three-point boundary value problem on time scales, u ^?? (t)+f(t,u(t))=0, u(0)=0, 𝛂u(𝛈)=u(T) for t∈[0,T]?╥, where ╥ is a time scale, 𝛂>0, η∈(0,p(T)?╥, and 𝛂η<T. We employ the Leggett-Williams fixed-point theorem in an appropriate cone to guarantee the existence of at least three positive solutions to this nonlinear problem. In the second part, we establish the existence of at least one positive solution to the related problem u ^??(t)+a(t)f(u(t))=0, u(0)=0, 𝛂u(η)=u(T), again using a fixed-point theorem for operators.     

Directional <Emphasis Type="Italic">∇</Emphasis>-derivative and Curves on <Emphasis Type="Italic">n</Emphasis>-dimensional Time Scales

The general ideal in this paper is to study a differential calculus for multivariable functions, directional ∇-derivative and curves of parametric equations on n-dimensional time scales.      

The Distribution of Generalized Zeros of Solutions of Delay Differential Equations on Time Scales1

In this paper, we consider the following equation where t is on a time scale T. We obtain the distribution of zeros of oscillatory solutions of the above equation.     

On First Order Impulsive Dynamic Equations on Time Scales

In this paper, the nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions for first order impulsive dynamic equations on time scales.     

Unification of calculus on Time Scales with mathematica

In 1990 Hilger defined the Time Scale Calculus which is the unification of discrete and continuous analysis in his PhD. In 2005 Yantir and Ufuktepe showed delta derivative with Mathematica[5]. In this study we give many computations of Time Scale Calculus with Mathematica such as the numerical and symbolic computation of forward jump operator and delta derivative for a particular time scale, graphs of functions, and definite integral on a time scale. We also improve and extent the Time Scale package for symbolics computations.     

Basic Partial Dynamic Equations on Time Scales

Work with partial differential equations on time scales is just beginning. In this paper we explore a basic partial differential equation, search for solutions, and find conditions which generate solutions of a given type.     

On Conformable Delta Fractional Derivative on Time Scales

In this paper, we introduce and investigate the concept of conformable delta fractional derivative on time scales. By using the theory of time scales, we obtain some basic properties of the conformable delta fractional derivative. Our results extend and improve both the results in [9] and the usual delta derivative.     

Eventual Right Disfocality on Time Scales

Eventual right disfocality of a dynamic equation y j j ( t )+ p 1 ( t ) y j ( t )+ p 2 ( t ) y ( t )=0, t ] T is established where T is a time scale. The results are obtained by imposing integrability conditions on p 1 and p 2 .     

时标动力方程的稳定性分析

近来，一种称之为时标动力系统的新理论得到了蓬勃的发展. 该文给出并建立了时标动力系统的各种稳定性概念和Lyapunov稳定性准则(稳定, 一致稳定, 渐近稳定, 指数渐近稳定等等).     

Oscillation Results for Second-order Linear Equations on a Time Scale

We are interested in extensions of certain averaging techniques for the second-order scalar differential equation ( r ( t ) x j ( t )) j + q ( t ) x σ ( t )=0, on a time scale (measure chain) T . These results include some earlier criteria for the difference equations case.     

时标上一类非自治捕食系统的周期解(英文)

本文在时标上运用迭合度理论中的Gaines和Mawhin连续性定理研究了一类非自治捕食系统的周期解的存在性,得到了此模型周期解存在的充分条件,该方法可将证明连续和离散微分方程的周期解的存在性统一起来.     

时间尺度上三阶Emden-Fowler动力方程的振动准则

运用Riccati 变换技术, 研究了时间尺度T上三阶Emden-Fowler时滞动力方程 (a(t)(r(t)x^? (t))^? ) ^?+p(t)x^γ(τ(t)) = 0 的振动性, 这里γ>0是正奇数的比, a, r, p是定义在T上的正的实值rd -连续函数. 得到一些新的振动结果, 推广和丰富了已有文献中的结论. 另外, 还给出了几个例子说明主要结果的合理性.     

Three Point Boundary Value Problems on Time Scales

This work formulates existence theorems for solutions to three-point boundary value problems on time scales. The ideas are based on a relationship between the three point boundary conditions, lower and upper solutions and topological degree theory.     

时间标度上的Taylor公式及链式法则

致力于研究时间标度上二元函数的链式法则,以期其在最优控制上有广泛的应用.同时,对一元函数的泰勒公式给出一种新的较为简单的证明方法.     

Positivity of Quadratic Functionals on Time Scales: Necessity

In this work we establish that disconjugacy of a linear Hamiltonian system on time scales is a necessary condition for the positivity of the corresponding quadratic functional. We employ a certain minimal normality (controllability) assumption. Hence, the open problems stated by the author in [17], [18] are solved with the result that positivity of the quadratic functional is equivalent to disconjugacy of the Hamiltonian system on the interval under consideration. The general approach on time scales 𝕋 involves, as special cases, the well–known continuous case for 𝕋 = ℝ and recently developed discrete one for 𝕋 = ℤ so that they are unified. As applications, Sturmian type separation and comparison theorems on time scales are also provided.     

On some averages at the zeros of the derivatives of the Riemann zeta-function

In this article we study two problems raised by a work of Conrey and Ghosh from 1989. Let ζ(k)(s) be the k-th derivative of the Riemann zeta-function, and χ(s) be factor in the functional equation of the Riemann zeta-function. We calculate the average values of ζ(j) and χ at the nontrivial zeros of ζ(k).     

测度链上p-Laplacian边值问题的三个正对称解

该文研究了p-Laplacian动力边值问题（g（u^△（t）））△＋a（t）f（t，u（t））=0，t∈[0，T]T,u（0）=u（T）=w，u△（0）=-u^△（T）正解的存在性．其中W是非负实数，g（v）=｜v｜p-2v1 P＞1．根据对称技巧和五泛函不动点定理，证明了边值问题至少有三个正的对称解，同时，给出了一个例子验证了我们的结果。     

一类时标上具状态依赖时滞的中立型泛函微分方程的周期正解

该文利用一个严格集压缩不动点定理,得到了如下形式的一类时标上具状态依赖时滞的中立型泛函微分方程周期正解存在性的充分条件x~Δ(t)=x(t)[r(t)-a(t)x(t)-sum from j=1 to n a_j(t)x(t-Υ_j(t,x(t)))-sum from j=1 to n c_j(t)x~Δ(t-σ_j(t,x(t)))],其中r,a,a_j,c_j∈C(T,R~+)(j=1,2,…,n)是ω-周期函数,Υ_j,σ_j∈C(T×R,T)(j=1,2,…,n)分别是其第一变元的ω-周期函数.