Study On The Stability Of The Function Equations In Multi-β-Normed Spaces

The stability problem was first posed by S.Ulam in1940. It mainly studies whether afunction which approximately satisfies a functional equation must be close to the exact solu-tion of this equation. The notion of multi-normed space was introduced by H.G.Dales andM.E.Polyakov. This concept is somewhat similar to operator sequence space and has someconnections with operator spaces.Because of its applications in and outside of mathematics,thestudy on the stability of functions various functional equations has become one of the mostimportant research subjects in the field of functional equations,and attracts much attention frommany researchers worldwide.In this paper,we first introduce the notion of multi--normed s-pace which is a generalization of multi-normed space,then, we investigate the stability problemfor several kinds of functional equations on multi--normed spaces, including alternative ad-ditive equation of the first form,alternative additive equation of the second form, quadraticequation, quartic equation and mixed type of cubic and quartic equation.In chapter1, we introduce the notion of multi--normed space and investigate the stabilityof alternative additive equation of the first form and alternative additive equation of the secondform on multi--normed spaces. firstly, we investigate the stability of the alternative additiveequation of the first form on unrestricted domains. Secondly, we study the stability of thealternative additive equation of the first form on restricted domains. finally, we investigate thestability of the alternative additive equation of the second form on unrestricted and restricteddomains.In chapter2, we introduce the notion of the Pexiderized orthogonally quadratic equa-tion、orthogonally quartic equation and investigate the stability of them on multi--normedspaces. and obtain some important results on the orthogonal stability of these two equations.In chapter3, By using fixed point method, we study the stability of cubic and quartic func-tional equation of mixed type on multi-Banach spaces. we first study the stability for odd andeven functions, then, by applying the results we obtain the stability of the general function onmulti-Banach spaces, and obtain two important stability results of this equation.