Study On The Stability Of The Functional Equation

The stability problem was posed by S.Ulam in 1940. Namely, whether a function whichapproximately satisfies a functional equation must be close to the exact solution of this equa-tion. Since its spectacular applications in harmonic analysis, relative theory, information theory,operator theory, Banach space geometry, etc., the study of the stability problem of functionalequations attracts much attention from many researchers worldwide. In this paper, we investi-gate the stability problem for several kinds of functional equations, including Jensen equation,Cauchy equation, quadratic equation, multi-quadratic equation, mixed type of cubic and quar-tic equation and trigonometric equation. Also, we find out the solution of the multi-quadraticequation and the mixed type of cubic and quartic equation.In chapter 1, we study the stability for the Cauchy equation, Jensen equation and theirPexider versions in the fuzzy normed linear space setting, namely, their fuzzy stability. In thischapter, firstly, we investigate the stability of the Jensen equation, then by applying the result,we obtain the stability of the Cauchy equation. Finally, we study the stability of the quadraticequation. In this part, we first examine the stability for odd and even functions and then applyour results to general function.In chapter 2, we introduce the notion of multi-quadratic functional equation and investigateits stability. Firstly, we find out the general solution of the equation. Secondly, a sufficient andnecessary condition used to justify whether a functional equation is a multi-quadratic equation isgiven. Finally, we study its stability, and obtain two important stability results of this equation.In chapter 3, we introduce a mixed type of cubic and quartic functional equation, for whichthe function f(x) = x3 + x4 is the solution of the equation. Also, for that reason, we call that itis of mixed type. In this chapter, firstly, we study the solution of the equation. We will see that ifan odd function satisfies the equation, then it is cubic; if an even function satisfies the equation,then it is quartic. Due to this two results, we obtain the general solution of the equation. Then,combine with the fixed point theory, we will investigate its stability by using the alternative offixed point. In this process, we also study the stability for odd and even functions firstly, thenby applying the results we obtain the stability of the general function.In chapter 4, we study the superstability of the generalized sine equation and the general-ized d'Alembet(cosine) equation, namely,whereλis a nonnegative real constant, f,g,h and k are non-zerofunctions from an uniquely 2-divisible abelian group (G, +) to the field of complex numbers C,σis an endomorphism of the group withσ(σ(x)) = x for all x∈G. Also, we will apply theresults to the Banach algebra and obtain some new results.