Quasi-Additive Functions and Related Topics

Abstract

J. Tabor (1988) introduced the class of functions f: R → R which satisfy the inequality

$$\left| {f\left( {x + y} \right) - f\left( x \right) - f\left( y \right)} \right| \leqslant \varepsilon \min \left\{ {\left| {f\left( {x + y} \right)} \right|,\left| {f\left( x \right) + f\left( y \right)} \right|} \right\}$$

for all real x and y,where εis a fixed number satisfying 0 ≤ ε < 1. Later the same author (see J. Tabor (1990)) generalized the concept by considering functions f: X — Y, where XandYare real normed spaces. He called the class of functions satisfying the inequality

$$\left\| {f\left( {x + y} \right) - f\left( x \right) - f\left( y \right)} \right\| \leqslant \varepsilon \min \left\{ {\left\| {f\left( {x + y} \right)} \right\|,\left\| {f\left( x \right) + f\left( y \right)} \right\|} \right\}$$

(13.1)

for x,y in X, and for some ε∈ [0,1), quasi-additive.