Abstract
Let X be a normed space,
{U\subset X\setminus\{0\}}
a non-empty subset, and
{(G,+)}
a commutative group equipped with a complete ultrametric d that is invariant (i.e.,
{d(x+z,y+z)=d(x,y}
) for
{x,y,z\in G}
).
Under some weak natural assumptions on U and on the function
{\gamma\colon U^{3}\to[0,\infty)}
, we study the new generalized hyperstability results when
{f\colon U\to G}
satisfies the inequality
d\biggl{(}\alpha f\biggl{(}\frac{x+y}{\alpha}+z\biggr{)},\alpha f(z)+f(y)+f(x)%
\biggr{)}\leq\gamma(x,y,z)
for all
{x,y,z\in U}
, where
{\frac{x+y}{\alpha}+z\in U}
and
{\alpha\geq 2}
is a fixed positive integer. The method is based on a quite recent fixed point theorem
(Theorem 1 in [J. Brzdȩk and K. Ciepliński,
A fixed point approach to the stability of functional equations in non-Archimedean metric spaces,
Nonlinear Anal. 74 2011, 18, 6861–6867])
(cf. [8, Theorem 1])
in some functions spaces.
On the probabilistic stability of some functional equations
