TITLE MARCH 2012 - Pakistan Academy of Sciences

40 Rabha W. Ibrahim
(z )
is removed by requiring ( z
)
be real when ( z ) > 0.
log to
1.2. Definition: The fractional integral of order
> 0 is defined, for a function f (z),
by
I
z
1 z
f ( z) := f ( )( z
)
(
)
0
1
d
; > 0,
where the function f (z)
is analytic in simplyconnected
region of the complex z-plane (C)
containing the origin and the multiplicity of
1
( z
) is removed by requiring log ( z
)
to be real when ( z ) > 0.
1.1. Remark:
D z
and
I z
z
z
(
1)
= z , > 1
(
1)
(
1)
= z , > 1.
(
1)
In [17], it was shown the relation
I
z
D
z
f ( z)
= D I f ( z)
= f ( z).
z
z
Let U := { z C:|
z |< 1} be the open unit
disk in the complex plane C and H denote the
space of all analytic functions on U . Here we
suppose that H as a topological vector space
endowed with the topology of uniform
convergence over compact subsets of U . Also
for a C
and m N, let H [ a,
m]
be the
subspace of H consisting of functions of the
form
f ( z)
= a a z
m
m
a
m1
1z
, z U.
m
Definition 1.3. Let p be a real number. We say
that
a z
n
n
= f ( z)
(1)
n=0
has the generalized Hyers-Ulam stability if there
exists a constant K > 0 with the following
property:
for every > 0, wU
= U U,
if
p
| an
|
| | ( ),
2
pn ( 1)
n
aw
n
n=0 n=0
p (0,1)
then there exists some
equation (1) such that
i i
| z w | K,
( z , w U,
i N).
z U that satisfies
In the present paper, we study the generalized
Hyers-Ulam stability for holomorphic solutions
of the fractional differential equation in complex
Banach spaces X and Y
D z
f ( z)
= G(
f ( z),
zf (
z);
z),
1 < 2, (2)
2
where G : X U
Y
and f : U X are
holomorphic functions such that f (0) = (
is the zero vector in X ).
Recently, the authors studied the ulam
stability for different types of fractional
differential equations [20-22].
2. RESULTS
In this section we present extensions of the
generalized Hyers-Ulam stability to holomorphic
vector-valued functions. Let X , Y represent
complex Banach space. The class of admissible
functions G ( X , Y),
consists of those functions
2
g : X U
Y
that satisfy the admissibility
conditions:
g( r, ks; z) 1,
when r = 1, s = 1,
( z U,
k 1).
We need the following results:
(3)
2.1. Lemma: [23] Let g G( X,
Y).
If
f : U X is the holomorphic vector-valued
functions defined in the unit disk U with
f (0) = ,
then