Difference and Mean Value Type Functional Equations

Abstract

In these notes we consider problems of the following forms: let x, y, xⁱ, yⁱ, etc. denote elements of ℝ_n (we sometimes replace ℝ_n by a linear space χ), let t, r, tⁱ., rⁱ., etc. denote elements of ℝ Let f : ℝ_n →ℝ (occasionally, f : χ→y for anOtner linear space y ). Then our problems are of the

$${\text{Type}}\,{\text{I}}:\left\{ {\begin{array}{*{20}c} {{\text{f(x)}}\,{\text{ = }}\,{\text{F}}\left\{ {{\text{f}}\left( {{\text{x}}\,{\text{ + }}\,{\text{t}}_1 {\text{y}}} \right),\,...,\,{\text{f}}\left( {{\text{x}}\,{\text{ + }}\,{\text{t}}_{\text{k}} {\text{y}}} \right),\,{\text{x, y}}} \right\}} \\ {{\text{for}}\,{\text{all}}\,{\text{x,}}\,{\text{y}},\, \in \mathbb{R}^{\text{n}} {\text{(or}}\,{\text{some}}\,{\text{given}}\,{\text{subsets}}\,{\text{of}}\,\mathbb{R}^{\text{n}} {\text{)}}\,{\text{and}}\,{\text{for}}} \\ {{\text{fixed}}\,{\text{t}}_{\text{i}} \in \mathbb{R}.} \\ \end{array} } \right.$$

$${\text{Type}}\,{\text{II}}:\left\{ {\begin{array}{*{20}c} {{\text{f(x)}}\,{\text{ = }}\,{\text{F}}\left\{ {{\text{f}}\left( {{\text{x}}\,{\text{ + }}\,{\text{ty}}_1 } \right),\,...,\,{\text{f}}\left( {{\text{x}}\,{\text{ + }}\,{\text{ty}}_{\text{k}} } \right),\,{\text{x, t}}} \right\}} \\ {{\text{for}}\,{\text{all}}\,{\text{x}}\, \in \mathbb{R}^{\text{n}} {\text{, t}} \in \mathbb{R}\,\,\,{\text{(or}}\,\,{\text{subsets}}\,{\text{of}}\,\mathbb{R}^{\text{n}} {\text{ and }}\mathbb{R}{\text{)}}\,\,{\text{and}}\,{\text{for}}} \\ {{\text{fixed}}\,{\text{y}}_{\text{i}} \in \mathbb{R}^{\text{n}}.} \\ \end{array} } \right.$$

We do not solve all questions for these types of problems of course; on the other hand we occasionally consider generalizations of these problems.