Affiliation:
1. Department of Mathematics, Faculty of Sciences , Ibn Tofail University , B.P. 133, Kenitra , Morocco ; e-mail: izelfassi.math@gmail.com
Abstract
Abstract
Let G be an Abelian group with a metric d and E be a normed space. For any f : G → E we define the Drygas difference of the function f by the formula
Λ
f
(
x
,
y
)
:
=
2
f
(
x
)
+
f
(
y
)
+
f
(
-
y
)
-
f
(
x
+
y
)
-
f
(
x
-
y
)
$$\Lambda {\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right): = 2{\rm{f}}\left( {\rm{x}} \right) + {\rm{f}}\left( {\rm{y}} \right) + {\rm{f}}\left( {{\rm{ - y}}} \right) - {\rm{f}}\left( {{\rm{x + y}}} \right) - {\rm{f}}\left( {{\rm{x - y}}} \right)$$
for all x, y ∈ G. In this article, we prove that if ˄f is Lipschitz, then there exists a Drygas function D : G → E such that f − D is Lipschitz with the same constant. Moreover, some results concerning the approximation of the Drygas functional equation in the Lipschitz norms are presented.