The Stability Of Nonlinear Functional Inequalities

The theory of functional equations is an important research direction of functional analysis, which theory and methods are widely used in nonlinear equations, optimiza-tion theory, mathematical model and many other fields. Moreover, it also has a great influence on the quantum mechanics, information theory, fuzzy set theory, mathematical economics, artificial intelligence and other related disciplines. As a typical problem in the field of functional analysis, Hyers-Ulam-Rassias stability theory of functional inequalities associated topological properties with linear character of a mapping attracted many re-searchers to explore and study. What's more, many valuable results appeared. Based on the previous work, this paper studied the Hyers-Ulam-Rassias stability of three kinds of functional inequalities in different spaces, and generalized the previous results in terms of the structure and the space types of functional inequalities.This paper includes three chapters. The first chapter described the background of the sources, the development and main previous research of the stability of functional equations and inequalities. The research contents and methods were also introduced in this part.The second chapter reviews the basic results of the fixed point theory, Gave the basic definitions of ?-homogeneous F-spaces, studied the functional inequality which was discussed by the direct method and fixed point method in the space and got the following results:Let f:X ? Y be a mapping with f(0)= 0. A function ?:X~3 ? [0, ?) satisfying the above functional inequality such that for all x, y, z € X, then there exists a unique additive mapping A:X ? Y such that for all x ? X.The result shows that we can find an additive mapping A and a perturbation function ?, and the functional inequality can be transformed into a sum of an additive function and a perturbation function. That means that the functional inequality has Hyers-Ulam-Rassias stability.The third chapter introduced the basic definitions of generalized quasi-Banach space and the previous works by C. Baak and C. Park, and studied the Hyers-Ulam-Rassias stability of the functional inequality We get the following results:Assume that mappings f,g,h,p:X?Y with g(0)= h(0)= p(0)= 0 satisfy the inequality where ?:X~3? [0, ?) satisfies ?(0,0,0)= 0 and for all x, y, z € X.Then there exists a unique additive mapping A:X ?Y such that for all x € X.For another type functional inequalityWe get the following results in generalized quasi-Banach space about functional in-equality.Assume that mappings f,g,h,p:X ? Y with g(0)= h(0)= p(0)= 0 satisfy the inequality where ?:X~3 ? [0, ?) satisfies 0(0,0,0)= 0 and for all x,y, z ? X.Then there exists a unique additive mapping A:X ? Y such that for all x ?X.As a conclusion, we can see that the functional inequalities have Hyers-Ulam-Rassias stability in quasi-Banach space, improve and generalize the previous results of C. park.