Study On The Stability Of Functional Equations

In1940, S. D. Ulam raised the stability problem of functional equations:let G1be a group and let G2be a metric group with the metric d(·,·). Given ε＞0, does there exist a δ＞0such that if f: G1â†'G2satisfies d(f(x· y),f(x)· f(y))<δ for all x,y∈G1, then a homomorphism h:G1â†'G2exist with d(f(x),h(x))<ε for all x∈G1? Because the stability of functional equations is closely linked with many branches of mathematics, and has spectacular applications in harmonic analysis, relative theory, operator theory, Banach space geometry etc, the study of the stability problem of functional equations attracts more attention from many researchers. Mathematicians have solved a lot of problems by applying the stability of functional equations.So far, mathematicians have investigated the stability of different functional equations in non-Archimedean normed spaces, multi-normed spaces and fuzzy normed spaces etc. The dissertation separately investigate the stability problem for several kinds of functional equations in fuzzy β-normed spaces, matrix β-normed spaces and matrix fuzzy normed spaces. The main results are the following four aspects:1. The dissertation introduces the notion of fuzzy β-norm which is a generalization of fuzzy norm, and then defines fuzzy β-normed spaces. In fuzzy β-normed spaces, this paper investigates the stability of orthogonally quartic functional equation by using the fixed point method and the direct method.2. The dissertation introduces the notion of matrix β-normed spaces which is a general-ization of matrix normed spaces, and then investigates the stability of the Pexider equation in matrix β-normed spaces.3. The dissertation investigates the stability of an additive-quadratic functional equation in matrix β-normed spaces. The paper separately discusses the stability of an additive-quadratic functional equation under the condition that the function is odd or even.4. The dissertation studies the stability of a cubic-quartic functional equation in matrix fuzzy normed spaces. The paper discusses its stability under the condition that the functional equation is odd or even:if an odd function satisfies this equation, the paper gets the stability of cubic; if an even function satisfies this equation, the paper gets the stability of quartic.