# Quasi-Additive Functions and Related Topics

• Donald H. Hyers
• George Isac
• Themistocles M. Rassias
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)

## 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.

## Keywords

Additive Function Related Topic Normed Vector Space Real Normed Space Baire Property
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Donald H. Hyers
• George Isac
• 1
• Themistocles M. Rassias
• 2