Additive functional equation and inequality are stable in Banach space and its applications

Abstract

In this paper, the authors established the solution of the additive functional equation and inequality$$f(x)+f(y+z)-f(x+y)=f(z)$$and$$\|f(x)+f(y+z)-f(x+y)\| \leq\|f(z)\| .$$We also prove that the above functional equation and inequality are stable in Banach space in the sense of Ulam, Hyers, Rassias. An application of this functional equation is also studied.