In this paper, we present a local Lipchitz condition for the local existence of solution to a class of stochastic differential equations with finite delay in a real separable Hilbert space which has the following form:
In this paper, we present a local Lipchitz condition for the local existence of solution to a class of stochastic differential equations with finite delay in a real separable Hilbert space which has the following form:
The purpose of this paper focuses on the local existence of mild solution to a class of the following stochastic differential equations with finite delay in a real separable Hilbert space 
where 
is a linear (possibly unbound) operator, and with a constant
we define
by
In which, 
is the space of all continuous functions from 
into 
equipped with the norm
.
(
and 
are continuous functions; 
is a 
Weiner process defined in Section 2).
In [
; 
satisfies Lipchitz and linear growth conditions then the mild solution of Equation (1) is exponentially stable in mean square.
In this paper, we prove the local existence of solution for Equation (1) with the weaker condition on
; and
.
In this section, we will recall some notions from Bezandry and Diagana [
Let H, K be real separable Hilbert spaces, 
be a filtered probability space; and 
is a sequence of real-valued standard Brownian motions mutually independent on this space. Furthermore,
.
where 
are nonnegative real numbers; and
is the complete orthonormal basis in
.
In addition, we suppose that 
is an operator defined by 
such that
Then, 
and for all 
the distribution of 
is
. The K-valued stochastic process 
is called a 
-Weiner process.
The subset 
is a Hilbert space equipped with the norm
and we define the space of all Hilbert-Schmidt operators from 
into 
by
Clearly, 
is a separable Hilbert space with norm
.
Let 
be all 
valued predictable processes 
such that
.
Then, for all 
the stochastic integral 
is well-defined by
.
where 
is the 
-Weiner process defined above. We have
In the following, we assume the stochastic integrals are well defined. For stochastic differential equation and stochastic calculus, we refer to [1-8].
For
, a stochastic process 
is said to be a strong solution of Equation (1) on 
if 1) 
is adapted to 
for all
;
2) 
is continuous in 
almost sure;
3) 
for any 
almost surely for any
, and
for all 
with probability one.
4) 
almost surely.
For
, a stochastic process 
is said to be a mild solution of Equation (1) on 
if 1) 
is adapted to 
for all
;
2) 
is continuous in 
almost sure;
3) 
is measureable with 
almost surely for any 
and
for all 
with probability one;
4) 
almost surely.
We assume that
(*) The operator 
generates a strongly semi-group
in
.
(**)
and 
satisfy local Lipchitz conditions respects to second argument that means for 
be a given real number, there exits 
such that with 
, and
, we have
If condition (*) holds, we will prove that if 
is a strong solution of Equation (1) then it also is a mild one. It is expressed by Theorem 3.1.
If (*) holds then (3) can be written in the form (4).
Proof: Applying Fubini theorem, we have
On the other hand
From (5) and (6), one has
or
By the definition of strong solution, we have
Since
We have
Substituting equation above for (8), we received
Hence,
Now, we turn our attention to the local existence of mild solution of Equation (1).
If the condition (*) and (**) are satisfied, then (1) has only mild solution.
Proof: Let 
be a fixed number in
, for each
, there exists 
, such that
where
For any
, we chose
. Let 
be a subspace of 
containing all functions X which adapt with 
such that 
and
is continuous. Then 
is a Banach space with norm
.
Let us consider a set Z which is defined by
It is easy to verify that 
is a closed subspace of
.
Let 
be the operator defined by
We now prove that
. Indeed,
Since
, 
with
, we have 
for any
.
Furthermore,
Hence
with
.
If we choose 
small enough, such that
Then, for any 
we have 
. In other words, we have 
.
For any
,
In addition, for any 
and
, we have:
Therefore,
Finally, if
, we have 
is contraction map in 
respects to the norm
Because this norm is equivalent to
, by applying fixed point principle we conclude that (1.1) has only mild solution on
.