On Solutions to Riemann's Functional Equation

In 1921 Hamburger proved that Riemann's functional equation characterizes the Riemann zeta function in the space of functions representable by ordinary Dirichlet series satisfying certain regularity conditions. We consider solutions to a more general functional equation with real weight k. In the case of Hamburger's theorem, k = . We show that, under suitable conditions, the generalized functional equation admits no nontrivial solutions for k > 0 unless k = . Our proof generalizes an elegant proof of Hamburger's theorem given by Siegel, and employs a generalized integral transform.1997 Sunrise Way     

一类迭代函数方程的连续解

本文用Schauder和Banach不动点定理讨论了一类非多项式形式的迭代函数方程的连续解的存在性唯一性与稳定性,其结果推广了文[1]工作．     

一个函数方程及其它的实数解

运用求函数条件极值的方法研究了一类函数方程的可解性,并获得了它的所有实数解,从而推广了美籍罗马尼亚著名数论专家F.Smaranche教授的一个问题.     

Singular Solutions of the Generalized Dhombres Functional Equation

We consider singular solutions of the functional equation ${f(xf(x)) = \varphi (f(x))}$ where ${\varphi}$ is a given and f an unknown continuous map ${\mathbb R_{+} \rightarrow \mathbb R_{+}}$ . A solution f is regular if the sets ${R_f \cap (0, 1]}$ and ${R_f \cap [1, \infty)}$ , where R _f is the range of f, are ${\varphi}$ -invariant; otherwise f is singular. We show that for singular solutions the associated dynamical system ${({R_f}, \varphi|_{R_f})}$ can have strange properties unknown for the regular solutions. In particular, we show that ${\varphi |_{R_f}}$ can have a periodic point of period 3 and hence can be chaotic in a strong sense. We also provide an effective method of construction of singular solutions.     

Nonsingle-valley Continuous Solutions of p-order Feigenbaum''''s Functional Equation

The author considers the Feigenbaum's functional equation fp(λx) = λf(x) for each p > 2. The existence of nonsingle-valley continuous solutions to this equation is discussed and a feasible method to construct such solutions is given.     

非单谷的p阶Feigenbaum映射

The author considers the Feigenbaum's functional equation f^P(λx) = λf(x) for each p ≥ 2. The existence of nonsingle-valley continuous solutions to this equation is discussed and a feasible method to construct such solutions is given.     

Hirzebruch Functional Equations and Krichever Complex Genera

As is well known, the two-parameter Todd genus and the elliptic functions of level d define n-multiplicative Hirzebruch genera if d divides n + 1. Both cases are special cases of the Krichever genera defined by the Baker–Akhiezer function. In the present paper, the inverse problem is solved. Namely, it is proved that only these properties define n-multiplicative Hirzebruch genera among all Krichever genera for all n.

     

具有p-Laplace算子的中立型泛函微分方程周期解

利用重合度延拓定理和分析技巧,讨论了一类具有p-Laplace算子的分布时滞中立型泛函微分方程的周期解问题,得到了其存在周期解的充分性结论,并且该结果是新的.     

Approximations of Solutions to Abstract Neutral Functional Differential Equation

In the present work, we study the approximations of solutions to the abstract neutral functional differential equations with bounded delay. We consider an associated integral equation and a sequence of approximate integral equations. We establish the existence and uniqueness of the solutions to every approximate integral equation using the fixed point arguments. We then prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next, we consider the Faedo–Galerkin approximations of the solutions and prove some convergence results. Finally, we demonstrate the application of the results established.     

中立型泛函微分方程周期解问题

利用Fourier级数理论研究了一类k-阶线性中立型泛函微分方程周期解问题,给出了周期解存在唯一性的充要条件。利用此结果并结合Schauder不动点原理,进一步研究了一类k-阶非线性中立型泛函微分方程,得到了存在周期解的新的结果。这些结果改进和推广了近期文献中的已有结论。     

泛函积分方程的正解与应用

运用锥拉伸与压缩不动点定理,研究一类泛函积分方程的正解．把所得结果应用于研究常微分方程积分边值问题正解的存在性,所得结果推广和本质上改进了马如云等作者最近的一些结果．     

Dhombres型函数方程的连续解

本文用构造的方法给出了Dhombres型函数方程g(x/kg(x))=9(x)~r的连续解,其中k为满足k0(k≠1)的实参数,推广了已有文献关于r=2的结论.     

On Non-Negative Measurable Solutions of a Difference Functional Equation

Following M. Laczkovich [5] we consider the iterative functional equation

随机KdV方程的白色泛函解

利用一个辅助性方程和埃米尔特变换研究一类非线性随机偏微分方程的精确解.此外,还获得了方程的随机雅克比椭圆函数类波解,随机类孤子解和随机三角函数波解.     

一类多项式型迭代函数方程在共振点附近的解析解

In this paper existence of local analytic solutions of a polynomial-like iterative functional equation is studied. As well as in previous work, we reduce this problem with the SchrSder transformation to finding analytic solutions of a functional equation without iteration of the unknown function f. For technical reasons, in previous work the constant α given in the Schroder transformation, i.e., the eigenvalue of the linearized f at its fixed point O, is required to fulfill that α is off the unit circle S^1 or lies on the circle with the Diophantine condition. In this paper, we obtain results of analytic solutions in the case of α at resonance, i.e., at a root of the unity and the case of α near resonance under the Brjuno condition.     

Classification of Nonoscillatory Solutions of a Higher Order Neutral Difference Equation

In this paper, classification schemes are given for the nonoscillatory solutions of higher order neutral difference equations. Several necessary and/or sufficient conditions are obtained for the existence of each kind of solution.     

二阶迭代泛函微分方程解析解的存在性

在复数域中讨论二阶迭代泛函微分方程x″(z)=x(az+bx(z)),z∈c解析解的存在性.     

Monotonic Solutions of a Functional Equation Arising from Simultaneous Utility Representations

The functional equation ?(h(x)) ? ?(h(y)) + ?(y) = ?(h(x ? y) + y) was solved by Aczél, Luce and Marley on the assumption that the functions are different iable. They posed the question of its strictly monotonic continuous solutions. The problem is solved using a uniqueness theorem. The continuity assumption on the functions is removed and the equation is also solved on a restricted domain.