Let
$(X,+)$
be an Abelian group and
$E$
be a Banach space. Suppose that
$f:X\rightarrow E$
is a surjective map satisfying the inequality
$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(x-y)\Vert \,|\leq {\it\varepsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(x-y)\Vert ^{p}\}\end{eqnarray}$$
for some
${\it\varepsilon}>0$
,
$p>1$
and for all
$x,y\in X$
. We prove that
$f$
is an additive map. However, this result does not hold for
$0<p\leq 1$
. As an application, we show that if
$f$
is a surjective map from a Banach space
$E$
onto a Banach space
$F$
so that for some
${\it\epsilon}>0$
and
$p>1$
$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(u)-f(v)\Vert \,|\leq {\it\epsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(u)-f(v)\Vert ^{p}\}\end{eqnarray}$$
whenever
$\Vert x-y\Vert =\Vert u-v\Vert$
, then
$f$
preserves equality of distance. Moreover, if
$\dim E\geq 2$
, there exists a constant
$K\neq 0$
such that
$Kf$
is an affine isometry. This improves a result of Vogt [‘Maps which preserve equality of distance’,
Studia Math.45 (1973) 43–48].