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.
A NOTE ON STABILITY OF THE LINEAR FUNCTIONAL EQUATIONS OF HIGHER ORDER AND FIXED POINTS OF AN OPERATOR
We prove two general theorems, which appear to be very useful in investigation of the Hyers-Ulam stability of a higher order linear functional equation in single variable, with constant coecients. We give several examples of their applications. In particular we show that we obtain in this way several fixed point results for a particular operator. The main tool in the proofs is a complexification of a real normed (or Banach) space X, which can be described as the tensor product X R 2 endowed with the Taylor norm.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.