【作者】 徐冰；

【导师】 张伟年；

【作者基本信息】 四川大学 ， 应用数学， 2003， 博士

【摘要】 迭代是自然界乃至人类生活中的一种普遍现象。迭代方程就是以迭代为基本运算形式的方程。漫长的历史沉淀使迭代方程成为与微分方程、差分方程、积分方程及动力系统紧密相关的现代数学分支，在实验科学和工程科学研究中起着重要的作用。在本文的绪言中介绍了几类迭代函数方程，并对与之相关的一些基本结果作了一个简要介绍。 第二章研究了两类迭代方程连续解的存在性。本章首先讨论了与逐段常时滞泛函微分方程不变曲线有关的一类迭代函数方程的连续解的存在唯一性和连续依赖性，不但弱化了已有结果的C~1光滑性的条件，还讨论了连续解的对称性，并根据对称性将一些结果推进到高维。本章还讨论了线性型迭代方程的递减解与非单调解的存在唯一性及连续依赖性，并将相关结果推广到拟线性型迭代方程。 第三章研究了线性型迭代方程的拟凸解、拟凹解、凸解及凹解的存在性。凸性是函数的最重要的性质之一。曾经有人讨论了凸迭代根和凹迭代根的存在性，但对更一般的方程——线性型迭代方程的解的凹凸性尚无结果。本章在连续函数构成的紧凸集上构造一个连续自映射算子，利用均差理论和不动点理论证明了线性型迭代方程的凹凸解的存在唯一性及连续依赖性。 第四章研究了迭代方程的解析解。已有的许多关于迭代方程的解析解结果都是运用优级数法得到的，并要求一个作为未知函数在其不动点处的线性部分的特征值的常数α不在单位圆周上或者α在单位圆周上但满足Diophantine条件。本章同样使用优级数法讨论一类带时滞的迭代微分方程解析解的存在性。突破了已有工作的Diophantine条件限制，讨论了常数α在单位圆周上但又不满足Diophantine条件的情形。此外，、本章还使用Sehauder不动点定理，通过建立辅助方程，研究了变系数的线性型迭代方程解析解的存在性。 第五章研究了单变量的函数方程的Hyers一Ulam稳定性.本章在综述单变量的函数方程Hyers一Ulam稳定性的有关结果的基础上，简化了有关广义r一函数方程在三种意义下的Hyers一Ulam稳定性的条件并给出了一些不同于经典r一函数方程的例子.进而还研究了一类非线性型迭代方程的Hyers一Ulam稳定性，证明了在这类非线性型迭代方程的近似解附近存在唯一的真解.更多还原

【Abstract】 Iteration is an extensive phenomenon in nature and human life. Iterative equations are those equations which involve iteration as a basic operation. In recent years, iterative equations play an important role in experimental sciences and engineerings and closely link with differential equations, difference equations, integral equations and dynamical systems. Some kinds of iterative functional equations together with some basic results are introduced in Chapter 1.Chapter 2 is devoted to existence of continuous solutions of some iterative equations. Existence, uniqueness and continuous dependence of C0 solutions for an iterative functional equation related to invariant curves of functional differential equations with piecewise constant arguments are given under weaker conditions than that known results of C1 solutions. Symmetry is also considered so that some obtained results are generalized to RN. Moreover, existence, uniqueness and continuous dependence of decreasing solutions and non-monotonic solutions for a linear iterative functional equation are discussed. Some corresponding results are generalized to a quasi-linear iterative equation.In Chapter 3, existence of quasi-convex, quasi-concave, convex and concave solutions of a linear iterative functional equation are studied. Although some results on convex and concave iterative roots are known, there are no results about convexity for more general iterative equations. In this chapter, convexity of both increasing solutions and decreasing solutions is investigated by the divided difference theory and fixed point theory.In Chapter 4, existence of analytic solutions of some iterative equations are studied. As we know, many results on analytic solutions of iterative equations are obtained by constructing a majorant series. For technical reasons, in previous works an indeterminate constant a, as the eigenvalue of the linearization of unknown function at its fixed point, is required to be off the unit circle or lie on the circle with the Diophantine condition. In this chapter, existence of analytic solutions of an iterative differential equation with state-dependent delays is studied by using a similar method. We breakthrough the restriction of the Diophantine condition and study the case that the constant a is a unity root, which offends the Diophantine condition. Moreover, existence of analytic solutions for a linear iterative equation with variable coefficients is given by reducing to an auxiliary equation and using Schauder’s fixed point theorem.In Chapter 5, Hyers-Ulam stability of functional equations in single variable is studied. We summarize some known results on Hyers-Ulam stability of functional equations in single variable, simplify conditions in three senses for the generalized gamma functional equation. Furthermore, we discuss Hyers-Ulam stability of a nonlinear iterative equation and prove existence and uniqueness of solutions near its approximate solutions.更多还原