Functional Equation fx pfx − 1 − qfx − 2 and Its Hyers-Ulam Stability
2.Moreover,asubstantialpartofproofofTheorem 2.1canbederivedfromtheoremspresentedinthebooks 19,20 .However,the theorems in 19, 20 deal with solutions of the linear equation under some regularityconditions, for example, the continuity, convexity, diﬀerentiability, analyticity and so on,while Theorem 2.1 deals with the general solution of 1.7 without regularity conditions.
Note on nonstability of the linear functional equation of higher order
We provide a complete solution of the problem of Hyers-Ulam stability for a large class of higher order linear functional equations in single variable, with constant coefficients. We obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characteristic equation is of module 1. Our results are related to the notions of shadowing (in dynamical systems and computer science) and controlled chaos. They also correspond to some earlier results on approximate solutions of functional equations in single variable.
On a Local Stability of the Jensen Functional Equation
The problem of s tab i l i t y of Cauohy's femotional equation on a r e s t r i c t domain D in R has a positive answer [ 5 ] . In this paper -we shal l show that CauOhy's functional equation and Jensen's funotional equation on a res t r io t domain D being some subset of ft" are stable . Moreover, we shal l use a suoh type result to give a positive answer to a problem of K. Nikodem ( [ 3 ] , [4]) . Let (X, ||*||) be a real Banach space and let D be a subset of X. We say that a function f i D — X i s ¿-addit ive (t3>0 i s f ixed) in D i f f