On new stability results for composite functional equations in quasi-β-normed spaces

In this article, we prove the generalized Hyers-Ulam-Rassias stability for the following composite functional equation: f ( f ( x ) − f ( y ) ) = f ( x + y ) + f ( x − y ) − f ( x ) − f ( y ) , f(f\left(x)-f(y))=f\left(x+y)+f\left(x-y)-f\left(x)-f(y), where f f maps from a ( β , p ) \left(\beta ,p) -Banach space into itself, by using the fixed point method and the direct method. Also, the generalized Hyers-Ulam-Rassias stability for the composite s s -functional inequality is discussed via our results.