【摘要】：泛函方程理論是泛函分析的一個重要研究方向,其理論和方法在非線性方程、最優(yōu)化理論、數(shù)學模型等諸多領域有著廣泛的應用.該理論對量子力學、信息理論、模糊集理論、數(shù)理經(jīng)濟學、人工智能等相關學科也產(chǎn)生了重要影響。泛函不等式的Hyers-Ulam-Rassias穩(wěn)定性理論將映射的拓撲性質和線性性質相聯(lián)系,作為泛函分析領域的經(jīng)典問題,吸引了眾多研究者的深入探索和研究,出現(xiàn)了諸多有價值的結果.本文在前人工作的基礎上,研究了三類泛函不等式在兩類空間上的Hyers-Ulam-Rassias穩(wěn)定性,在泛函不等式的結構及空間類型等方面推廣了前人的結果。本文共分三個部分.在第一章中,主要闡述了泛函方程及不等式穩(wěn)定性問題的來源及發(fā)展概況,系統(tǒng)介紹了前人在泛函方程及不等式穩(wěn)定性問題上的主要工作,同時介紹了本論文的主要研究內容和研究方法.在第二章中,首先回顧了不動點理論的基本結果,給出了β-齊次F-空間的基本定義,進而在此空間中利用直接法及不動點方法對泛函不等式進行了討論,得到了如下結果：如果對任意的x,y,z∈X,映射f：X→Y且f(0)=0及ψ：X~3→[0,∞)滿足不等式且那么對任意的x∈X,存在唯一的可加映射A：X→Y使得以上結果說明上述泛函不等式可化為一個可加映射A與一個擾動函數(shù)(?)的和,即該泛函不等式在β-齊次F-空間中具備Hyers-Ulam-Rassias穩(wěn)定性。在第三章中,首先介紹了一般擬Banach空間的定義及C. Baak和C. Park的代表性工作,進而對泛函不等式在該空間上的Hyers-Ulan-Rassias穩(wěn)定性進行了討論,得到以下結果：如果對任意的x,y,z ∈X,函數(shù)f,9,h, p: X→ y 及φ:X~3→[0,∞)滿足不等式此時,g(0)=h(0)=p(0)=0,φ(0,0,0)=0且那么對任意的x∈X,存在唯一的可加映射A：X→Y使得對于泛函不等式得到以下結果：如果對任意的x,y,z ∈X,函數(shù)f,g,h,p:X → y,φ:X~3→[0,∞)滿足不等式此時,g(0)=h(0)=p(0)=0,φ(0,0,0)=0且那么對任意的x∈X,存在唯一的可加映射A：X→Y使得以上結果說明在一般擬Banach空間上,我們構造的兩類泛函不等式具備Hyers-Ulam-Rassias穩(wěn)定性,將C. park等給出的結果推廣到了更一般的情形。
[Abstract]:Functional equation theory is an important research direction in functional analysis. Its theory and methods have been widely used in many fields, such as nonlinear equations, optimization theories, mathematical models and so on. The theory also has an important impact on quantum mechanics, information theory, fuzzy set theory, mathematical economics, artificial intelligence and other related disciplines. The Hyers-Ulam-Rassias stability theory of functional inequalities relates the topological properties and linear properties of mappings as a classical problem in the field of functional analysis which attracts many researchers to explore and study deeply and presents many valuable results. In this paper, we study the Hyers-Ulam-Rassias stability of three functional inequalities on two kinds of spaces on the basis of previous work, and generalize the previous results in terms of the structure and space type of functional inequalities. This paper is divided into three parts. In the first chapter, the origin and development of the stability problems of functional equations and inequalities are introduced, and the main works of predecessors on the stability of functional equations and inequalities are systematically introduced. At the same time, the main research contents and methods of this paper are introduced. In the second chapter, the basic results of fixed point theory are reviewed, and the basic definition of 尾 -homogeneous F-spaces is given, and then the functional inequalities are discussed by using direct method and fixed point method in this space. The following results are obtained: if f (0) = 0 and 蠄 (0) = 0 and 蠄: X ~ 3 [0, 鈭，

本文編號：2266223