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Abstract
In the current work, we deal with a class of stochastic time-fractional integral equations in Hilbert space by studying their well-posedness and regularity. Precisely, we use the celebrity fixed point theorem to prove the well-posedness of the problem by imposing the global Lipschitz and the linear growth conditions. Further, we prove the spatial and temporal regularity by imposing only a regularity condition on the initial value. An important example is considered in order to confirm and support the validity of our theoretical results.
1. Introduction
There are many phenomena in applied sciences as anomalous diffusion in some physical processes and dynamical systems with memory in medicine, which cannot be described adequately by the classical differential and integral equations. This fact gives rise of the theory of fractional differential and integral equations. Such kinds of equations became an effective tool to model such phenomena in the past four decades, although the roots of the fractional calculus that extends to the year 1695, see for a short list [Citation1–9].
However, the nondeterministic nature of the most of such phenomena obliges us to incorporate randomness into their mathematical descriptions and provide more realistic mathematical models of them, and this frequently results in stochastic fractional differential and/or integral equations. For example, Arab et al. [Citation10] studied the fractional stochastic Burgers-type equation and proved not only its well-posedness in the Hölder space, but its numerical approximations as well. The authors in Ref. [Citation11] developed the basic theory of fractional calculus and anomalous diffusion from probability's point of view. While Metzler et al. [Citation12] discussed extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.
Recently, a few contributions have been considered by some authors in the analysis of stochastic integro-differential equations driven by a fractional Brownian motion. For details, Caraballo et al. [Citation13] studied the asymptotic behaviour for neutral stochastic integro-differential equations driven by a fractional Brownian motion. Arthi et al. [Citation14] proved the existence and exponential stability of a same class of equations with impulses. In Ref. [Citation15], Sathiyara et al. dealt with the contrabillity of fractional higher order stochastic integro-differential systems with fractional Brownian motion. Moreover, a class of fractional stochastic Itö integral or Skorokhod integral equations has been studied by El-borai et al. [Citation16,Citation17], where the well-posedness of such classes has been established and in the recent literature various qualitative behaviors of solutions of Volterra integral equations, ordinary and stochastic differential delay equations of second order and integro-differential equations of first order have been investigated in [Citation18–22] and some new qualitative results have been obtained in these papers.
From these facts and regarding the importance of the stochastic time-fractional integral equations in the description of some anomalous diffusions, our contribution in the current paper will be the study of such class of equations, which is given in the following general form:
(1)
(1)
where, , for fixed T>0,
, the initial condition
is a H-valued
-measurable random variable, with H be a real Hilbert space,
is a linear closed operator generates a semigroup
,
and
are two operators, W is a H-valued cylindrical Wiener process.
In our work, we consider the fractional integrals in the Riemann–Liouville approach.
It is worth mentioning that in Ref. [Citation16], the authors studied an equation similar to (Equation1(1)
(1) ), perturbed by a multiplicative noise, where W is a Wiener process on the real separable Hilbert space K, with covariance operator Q. They proved only the well-posedness of the problem, where the mild solution
satisfies
is an
-valued random variable, for any
. In our case, the equation is perturbed by an additive noise, where we prove the well-posedness, with
is an
-valued random variable, for
. Further, to the best of the authors knowledge, there is no work has been dealt with the regularity of solutions for such class of equations. For this, our main contribution is not only to study the temporal regularity, but to study the spatial regularity as well.
This work is ordered by the following: Section 2 is devoted to prove the well-posedness of the problem. In Section 3 we prove the spatial regularity of the problem. The temporal regularity is postponted to Section 4. Finally, conclusion is presented in Section 6.
We close this introduction by giving the following notations. For an operator we mean by
its domain of definition, H is a real Hilbert space endowed with the norm
,
is the space of linear bounded operators defined on H into it self endowed with the norm
, HS is the space of Hilbert-Schmidt operators defined from the Hilbert space H into it self, and we indicate its norm by
. Let
be a filtered probability space, where
is a normal filtration and X be a Banach space,
, for
is the space of X-valued pth integrable random variables on Ω, its norm is denoted by
,
is a Banach space equipped with the norm
. We use the abbreviation RHS for right hand side.
2. Well-posedness of problem (1)
The current section is devoted to prove the existence and the uniqueness of a mild solution u defined as follows (see Refs. [Citation16,Citation23,Citation24]).
Definition 2.1
Let be an H-valued stochastic process. u is said to be a mild solution of problem (Equation1
(1)
(1) ) if
for all
,
is
-adapted,
u satisfies the following equality in H,
-a.s.,
(2)
(2) where
is a probability density function defined on
.
We need here to impose the following assumptions to prove the well-posedness of problem (Equation1(1)
(1) ). For
:
- The linear operator
is an infinitesimal generator of
-semigroup
, satisfies: for all
and all
, there exist
such that
(3)
(3)
and
(4)
(4)
- The operator
(not necessarily linear) satisfies the global Lipschitz and the linear growth conditions, i.e.
(5)
(5)
and
(6)
(6)
for some positive constant
.
We need here to reformulate Assumption in the random context. To this end, let x and y be two H-valued random variables. Then, we have
(7)
(7)
and
(8)
(8)
- The operator
is linear and bounded, i.e.
, for some positive constant
.
- The initial condition
is an
-measurable random variable, satisfies
, i.e.
.
Remark 2.1
In the rest of this paper, when we need to use estimations in the random context as it has been proved above for Assumption , we will do it without proof in order to avoid the repetitions.
Theorem 2.1
Under the Assumptions (Est.(Equation3
(3)
(3) ) with
and Est.(Equation4
(4)
(4) ) with
),
,
and
, the problem (Equation1
(1)
(1) ) admits a unique mild solution
, provided that
for
and
.
To prove this theorem, we need the following useful result.
Lemma 2.1
see [Citation24]
Let and
. It is true that
where
is a probability density function defined on
and Γ is Gamma function.
Proof
Proof of Theorem 2.1
Let and
. We define the mapping
as follows
(9)
(9)
First we have,
(10)
(10)
To estimate
, we use Assumption
; Est.(Equation3
(3)
(3) ) (with
), Assumption
and Lemma 2.1 (with
) to get
(11)
(11)
To estimate the second term in the RHS of (Equation10
(10)
(10) ), let
. Then, by using Assumption
; Est.(Equation3
(3)
(3) ) (with
), Lemma 2.1(with
) and Assumption
, we obtain
(12)
(12)
Now, to deal with the stochastic term in the RHS of (Equation10
(10)
(10) ), we use Burkholder–Davis–Gundy inequality (see [Citation25, Proposition 2.12, p.24]) as follows
where
.
First, we need to estimate . To do this, let
and
, by using the fact that
, for every
and every
, we obtain
The use of Assumption
; Est.(Equation3
(3)
(3) ) (with
) and Est.(Equation4
(4)
(4) ) (with
), Assumption
and Lemma 2.1 (with
, which is possible since
) helps us to get
And so,
Thanks to the condition
, we can easily obtain
(13)
(13)
Coming back to Est.(Equation10
(10)
(10) ), we replace Est.(Equation11
(11)
(11) ), Est.(Equation12
(12)
(12) ) and Est.(Equation13
(13)
(13) ) in it, to get
(14)
(14)
For all
. And so Ψ is well defined.
Now, we prove that Ψ is a contraction mapping. To do this, let . From Est.(Equation9
(9)
(9) ) we infer that
(15)
(15)
To estimate the term in the RHS of Est.(Equation15
(15)
(15) ), we follow the same steps to lead to Est.(Equation12
(12)
(12) ). Thus
(16)
(16)
Consequently,
(17)
(17)
If
, then ψ is a contraction, and so it has a unique fixed point that coincides with a unique mild solution of problem (Equation1
(1)
(1) ).
3. Spatial regularity of problem (1)
In this section, we study the spatial regularity of the mild solution. Its main result is the following.
Theorem 3.1
Let be the mild solution of problem (Equation1
(1)
(1) ) with an initial condition
satisfies
, for all
. According to Assumptions
,
and
, the solution u satisfies
for some positive constant
.
Before proving this result, we first need to recall the Grönwall lemma (see [Citation26, Lemma 7.1.1] and [Citation25, Lemma A.2]).
Lemma 3.1
Let be a positive and continuous function, for a fixed T>0. If there exists
such that
for some two positive constants
and
. Then, there exists
, such that
Proof
Proof of Theorem 3.1
Let ,
and
(which is possible thanks to the condition
). First, from Equation (Equation2
(2)
(2) ), we have
(18)
(18)
for all
. Then, the use of Est.(Equation3
(3)
(3) ) (with
) and the application of Lemma 2.1 (with
) lead to
(19)
(19)
Again, the use of Est.(Equation3
(3)
(3) ) (with
), the Assumption
and Lemma 2.1 (with
) yields
The fact
,
and Est.(Equation4
(4)
(4) ) (with
) yield
(20)
(20)
An application of Burkholder-Davis-Gundy inequality enables us to estimate
as follows
The use of Est.(Equation3
(3)
(3) ) (with
), Est.(Equation4
(4)
(4) ) (with
), Lemma 2.1 (with
) and Assumption
, helps us to get
(21)
(21)
By replacing Est.(Equation19
(19)
(19) ), Est.(Equation20
(20)
(20) ) and Est.(Equation21
(21)
(21) ) in Est.(Equation18
(18)
(18) ), we obtain
where
and
.
An application of Grönwall Lemma 3.1 with (which is possible thanks to the condition
) yields
(22)
(22)
Then, the desired result is obtained.
4. Temporal regularity of problem (1)
We deal with the temporal regularity of the mild solution in this section, where its main result is the following.
Theorem 4.1
According to Assumptions ,
and
, the mild solution u of (Equation1
(1)
(1) ) with an initial condition
satisfies
, for all
, is time Hölder continuous with exponent
, i.e.
for all
, with
and T be fixed and for some positive constant C.
In order to prove Theorem 4.1, we need the following useful lemma besides Lemma 2.1.
Lemma 4.1
see [Citation24, Lemma 2.5]
Let T>0 be fixed. For any and any
. It is true that
where
.
Proof
Proof of Theorem 4.1
Let ,
,
,
and
(which is possible thanks to the condition
). From Equation (Equation2
(2)
(2) ) we have
(23)
(23)
where
and
To estimate
, let
. First, the semigroup property enables us to write
By using Est.(Equation3
(3)
(3) ) (with
) and Lemma 2.1 (with
), we obtain
The use of Lemma 4.1 (with
), which is possible since
yields
(24)
(24)
where
, with
.
Let . To estimate
we have
The use of Assumption
(Est.(Equation3
(3)
(3) ) with
and Est.(Equation4
(4)
(4) ) with
), Assumption
and Lemma 2.1 (with
) lead to
An application of Lemma 4.1 (with
) yields
(25)
(25)
where
and
.
Let . We estimate
according to Assumption
, Assumption
(Est.(Equation3
(3)
(3) ) with
and Est.(Equation4
(4)
(4) ) with
), Lemma 2.1 (with
) and Lemma 4.1 (with
), as follows
(26)
(26)
where
.
To estimate , let
. We use Assumption
, Assumption
(Est.(Equation3
(3)
(3) ) with
and Est.(Equation4
(4)
(4) ) with
) and Lemma 2.1 (with
) and argue as above. Then,
(27)
(27)
where
.
To estimate we first use Burkholder–Davis–Gundy inequality as follows
(28)
(28)
To deal with the term
, let
. The use of Assumption
(Est.(Equation3
(3)
(3) ) with
and Est.(Equation4
(4)
(4) ) with
) and Lemma 4.1 (with
) yields
Coming back to Est.(Equation28
(28)
(28) ), an application of Lemma 2.1 (with
) and the use of Assumption
give us
(29)
(29)
where
.
To estimate , we have
(30)
(30)
Let
. The use of Assumption
(Est.(Equation3
(3)
(3) ) with
and Est.(Equation4
(4)
(4) ) with
) enables us to write
(31)
(31)
From (Equation30
(30)
(30) ) and (Equation31
(31)
(31) ) with an application of Lemma 2.1 (with
), Assumption
and Lemma 4.1 (with
), we arrive at
(32)
(32)
where
.
To deal with , we argue as above with the use of Est.(Equation31
(31)
(31) ). Then,
(33)
(33)
where
.
Finally, we replace Est.(Equation24(24)
(24) ), Est.(Equation25
(25)
(25) ), Est.(Equation26
(26)
(26) ), Est.(Equation27
(27)
(27) ), Est.(Equation29
(29)
(29) ), Est.(Equation32
(32)
(32) ) and Est.(Equation33
(33)
(33) ) in Est.(Equation23
(23)
(23) ) to get
(34)
(34)
where
, where
,
, and
. By this the proof is completed.
5. Example: stochastic space-time fractional integro-differential equation
We give in this section an important example of problem (Equation1(1)
(1) ) in order to confirm and support the validity of our theoretical results. Namely; Theorems 2.1, 3.1 and 4.1.
Example 5.1
Let where
. We consider the following stochastic space-time fractional integro-differential equation in the Hilbert space
with an initial condition
.
(35)
(35)
where
is the fractional Laplacian and
is Laplace operator endowed with the Dirichlet boundary conditions. We know that (see for more details [Citation10,Citation27]), the operator
is linear generates an analytic semigroup
.
Further, it satisfies the following estimates.
Lemma 5.1
– Let
and
for T be fixed. For all
,
such that
(36)
(36)
– For all
, there exists
such that
(37)
(37)
Proof.
The first estimate already exists, see, e.g. [Citation10, Lemma 2.3].
About the second estimate, let
The operatorand let
are the eigenpairs of the operator
, where
and
. The use of the definition of the Hilbert-Schmidt norm, with the fact that
, lead to
(38)
(38) The integral
converges, thanks to
, and so
satisfies Assumption
; Est.(Equation3
(3)
(3) ) (with
) and Est.(Equation4
(4)
(4) ) (with
).
Indeed, let
and
(which is possible thanks to the condition
). From Est.(Equation36
(36)
(36) ) (with
) and Est.(Equation37
(37)
(37) ) (with
, which is possible since
) respectively, we have
and
The operator
is given by
, for any
. It is easy to check that F satisfies the global Lipschitz and the linear growth conditions imposed in this paper.
In fact, let
, we have
and
Hence, the operator F satisfies Assumption
.
The operator
is the identity operator
defined on the Hilbert space H. It is well known that,
. Thus, G satisfies Assumption
.
6. Conclusion
Stochastic fractional integral (or integro-differential) equations have been used as mathematical models of many physical processes as the anomalous diffusions. Nevertheless, we can find in the literature a few works concerned with this type of equations. Motivated by this fact, we considered in this paper a class of stochastic time-fractional integral equations in a Hilbert space H. We used the fixed point theorem in order to prove the well-posedness of the problem by imposing the global Lipschitz and the linear growth conditions. Moreover, we achieved not only the spatial regularity of the mild solution but the temporal regularity as well. Precisely, by imposing a regularity condition on the initial value (i.e.
), we proved that such solution satisfies
, where
,
and it is time Hölder continous with exponent
, for
and
. In general, it is not easy to solve this kind of equations analytically, for this, the numerical study plays an important role by providing a numerical approximations of the analytic solutions with respect to time, space or to both simultaneously. Motivated by this fact, some numerical approximations for the mild solution of the problem are interesting directions for our future research.
Acknowledgments
The authors would like to thank the reviewers for giving them constructive comments and suggestions which would help them in order to improve the quality of the paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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