非线性泛函不等式的稳定性研究

发布时间：2018-10-12 13:05

【摘要】：泛函方程理论是泛函分析的一个重要研究方向,其理论和方法在非线性方程、最优化理论、数学模型等诸多领域有着广泛的应用.该理论对量子力学、信息理论、模糊集理论、数理经济学、人工智能等相关学科也产生了重要影响。泛函不等式的Hyers-Ulam-Rassias稳定性理论将映射的拓扑性质和线性性质相联系,作为泛函分析领域的经典问题,吸引了众多研究者的深入探索和研究,出现了诸多有价值的结果.本文在前人工作的基础上,研究了三类泛函不等式在两类空间上的Hyers-Ulam-Rassias稳定性,在泛函不等式的结构及空间类型等方面推广了前人的结果。本文共分三个部分.在第一章中,主要阐述了泛函方程及不等式稳定性问题的来源及发展概况,系统介绍了前人在泛函方程及不等式稳定性问题上的主要工作,同时介绍了本论文的主要研究内容和研究方法.在第二章中,首先回顾了不动点理论的基本结果,给出了β-齐次F-空间的基本定义,进而在此空间中利用直接法及不动点方法对泛函不等式进行了讨论,得到了如下结果：如果对任意的x,y,z∈X,映射f：X→Y且f(0)=0及ψ：X~3→[0,∞)满足不等式且那么对任意的x∈X,存在唯一的可加映射A：X→Y使得以上结果说明上述泛函不等式可化为一个可加映射A与一个扰动函数(?)的和,即该泛函不等式在β-齐次F-空间中具备Hyers-Ulam-Rassias稳定性。在第三章中,首先介绍了一般拟Banach空间的定义及C. Baak和C. Park的代表性工作,进而对泛函不等式在该空间上的Hyers-Ulan-Rassias稳定性进行了讨论,得到以下结果：如果对任意的x,y,z ∈X,函数f,9,h, p: X→ y 及φ:X~3→[0,∞)满足不等式此时,g(0)=h(0)=p(0)=0,φ(0,0,0)=0且那么对任意的x∈X,存在唯一的可加映射A：X→Y使得对于泛函不等式得到以下结果：如果对任意的x,y,z ∈X,函数f,g,h,p:X → y,φ:X~3→[0,∞)满足不等式此时,g(0)=h(0)=p(0)=0,φ(0,0,0)=0且那么对任意的x∈X,存在唯一的可加映射A：X→Y使得以上结果说明在一般拟Banach空间上,我们构造的两类泛函不等式具备Hyers-Ulam-Rassias稳定性,将C. park等给出的结果推广到了更一般的情形。
[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，

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