两类泛函方程在几类空间中的稳定性

【作者】 刘满芹；

【导师】 王利广；

【作者基本信息】 曲阜师范大学 ， 基础数学， 2012， 硕士

【摘要】 泛函方程理论中一个典型的问题是稳定性问题.泛函方程的稳定性问题源自Ulam在1940年提出的关于群同态的稳定性问题：给定一个群(G1,*)和一个度量群(G2,.,d),其中d((?))为一个度量.给定一个ε>0,存在一个δ>0使得如果h：G1→G2为一个映射且对所有的x,y∈G1均有d(h(x*y),h(x)·h(y))<δ是否存在一个同态H：G1→G2使得对所有的x∈G1, d(h(x),H(x))<ε?1941年,D.H.Hyers解决了Banach空间上可加映射的稳定性问题.在接下来的几十年里,许多数学家对各种不同的泛函方程的稳定性进行了系统的研究,例如指数方程,二次泛函方程,三次泛函方程以及广义可加的泛函方程等.1978年Th.M.Rassias解决了线性映射在Banach空间中的稳定性问题；1999年Y.Lee和K.Jun研究了广义Jensen方程的稳定性.这些稳定性的成果在随机分析,金融数学和精算数学等领域中均有广泛的应用.本文共分为两章.在第一章中,我们研究了一个源自Jensen可加泛函方程和二次泛函方程的混合二次可加泛函方程在β-巴拿赫空间和拟巴拿赫空间中的稳定性问题.首先,我们讨论了上述方程在β-巴拿赫空间中的稳定性,接下来我们又考虑了这个方程在拟巴拿赫空间中的稳定性.在第二章中,我们研究了一个源自四次泛函方程的n维四次方程在非阿基米德巴拿赫模和随机巴拿赫模中的稳定性问题.在这里V={I(?)M：1∈I},M={1,2,...,n},且M/I=Ic更多还原

【Abstract】 The stability problem is a classical question in the theory of functional equations.The stability problem concerning the stability of group homomorphisms was firstly raisedby Ulam in1940:Give a group (G1,) and a metric group (G2,·, d) with the metric d(·.·). Give>0,does there exists a δ>0such that if h: G1→G2satisfies d(h(x y), h(x)· h(y))<δ forall x, y∈G1, then is there a homomorphism H: G1→G2with d(h(x), H(x))<ε for allx∈G1?In1941, D. H. Hyers solved the stability problem of additive mapping on Banachspaces. In the following decades, many mathematicians have studied the stability ofdiferent kinds of functional equations such as exponential equation, quadratic functionalequation, cubic functional equation, generalized additive equation and so on. In1978, Th.M. Rassias solved the stability problem of linear mapping in Banach space. In1999, Y.Lee and K. Jun studied the stability of generalized Jense equation. These stability resultshave applications in some related fields such as random analysis, financial mathematicsand actuarial mathematics.This thesis consists of two chapters.In chapter1, we consider the Hyers-Ulam stability of a mixed additive-quadraticfunctional equation deriving from the Jensen additive functional equationin β-Banach space and quasi-Banach space. We firstly discuss the Hyers-Ulam stability ofabove mixed additive-quadratic functional equation in β-Banach space. Then we considerthe Hyers-Ulam stability of this functional equation in quasi-Banach spaces.In chapter2, we consider the following n-dimensional functional equation (here V={I M:1∈I}, M={1,2,..., n}, M/I=Ic)deriving from the quarticfunctional equationsf(2x+y)+f(2x y)=4f(x+y)+4f(x y)+24f(x)6f(y),and discussed its Hyers-Ulam stability in non-Archimedean Banach Modules and randomBanach Modules.更多还原