=Paper=
{{Paper
|id=Vol-2748/Paper10
|storemode=property
|title=Existence of Mild Solution For Stochastic Differential Equations With Fractional Derivative Driven by Multiplicative Noise
|pdfUrl=https://ceur-ws.org/Vol-2748/IAM2020_paper_10.pdf
|volume=Vol-2748
|authors=Habib Djourdem,Noureddine Boutera
}}
==Existence of Mild Solution For Stochastic Differential Equations With Fractional Derivative Driven by Multiplicative Noise==
https://ceur-ws.org/Vol-2748/IAM2020_paper_10.pdf
Existence of mild solution for stochastic differential
equations with fractional derivative driven by
multiplicative noise
Habib Djourdema , Noureddine Bouterab
a
University Center Ahmed-Zabana, Relizane, Algeria
d
Oran Higher School of Economics, Oran, Algeria
Abstract
This paper focuses on the study of the existence of a mild solution to time and space-fractional stochas-
tic equation perturbed by multiplicative white noise. The required results are obtained by means of
Krasnoselskiiβs fixed point theorem.
Keywords
stochastic equation, mild solution, Expectation, Krasnoselskiiβ s fixed point theorem.
1. Introduction
In this paper, we are interested in the existence of solutions for nonlinear fractional difference
equations
π πΌ
π·0+ [π’ β β (π’)] = Ξπ’ (π‘) + π’ β
βπ’ + π (π’) π (π‘) , π₯ β π·, π‘ > 0, (1)
subject to the initial condition
π’ (π₯, 0) = π’0 (π₯) , π₯ β π·, π‘ = 0, (2)
and the Dirichlet boundary conditions
π’ (π₯, π‘) = 0, π₯ β ππ·, (3)
where π· β βπ , π’ (π₯, π‘) represents the velocity field of the fluid, the state π’ (β
) takes values in a
separable real Hilbert space π» with inner product β¨β
, β
β©, the term π (π’) π (π‘) = ππ‘π π (π‘) describes
a state dependent random noise, where π (π‘)π‘β[0,π ] is a πΉπ‘ βadapted Wiener process defined
in completed probability space (Ξ©, πΉ , π) with expectaction πΈ and associate with the normal
filtration πΉπ‘ = π {π (π ) βΆ 0 β€ π β€ π‘}. The operator Ξ is the Laplacian. Here, π π·π‘πΌ denotes the
Caputo type derivative of order πΌ (0 < πΌ < 1) for the function π’ (π₯, π‘)with respect time π‘ which
is defined by {
π π·πΌ π’ 1 π‘ ππ’(π₯,π ) ππ
π‘ (π₯, π‘) = Ξ(1βπΌ) β«0 ππ (π‘βπ )πΌ , 0 < πΌ < 1,
ππ’(π₯,π‘) (4)
ππ‘ , πΌ = 1,
IAMβ20: Third conference on informatics and applied mathematics, 21β22 October 2020,Guelma, ALGERIA
" djourdem.habib7@gmail.com (H. Djourdem); bouteraa-27@hotmail.fr (N. Boutera)
0000-0002-7992-581X (H. Djourdem); 0000-0002-8772-1315 (N. Boutera)
Β© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)
where Ξ (β
) stands for the Gamma function.
The existence and non-existence of solutions for the Navier-Stokes equations (NSEs) have
been discussed in [13]. Chemin et al. [5] studied the global regularity for the large solutions
to the NSEs. Miura [19] focused on the uniqueness of mild solutions to the NSEs. Germain
[9] presented the uniqueness criteria for the solutions of the Cauchy problem associated to the
NSEs. However, The existence and uniqueness of solutions for the stochastic Navier-Stokes
equations (SNSEs) with multiplicative Gaussian noise were proved in [8], [20]. The large de-
viation principle for SNSEs with multiplicative noise had been established in [23], [27], [31],
[32]. Just mention a few, the study of time-fractional Navier-Stokes equations has become a
hot topic of research due to its significant role in simulating the anomalous diffusion in fractal
media [12], [16], [33].
There has been a widespread interest during the last decade in constructing a stochastic
integration theory with respect to fractional Brownian motion (FBM) and solving stochastic
differential equations driven by FBM. On the other hand, time-fractional differential equations
are found to be quite effective in modelling anomalous diffusion processes as its can charac-
terize the long memory processes [22], [25], [28], [29], [30]. Hence, Burgers equation with
time-fractional can be adapted to describe the memory effect of the wall friction through the
boundary layer [35]. Furthermore, the analytical solutions of the time- and space-fractional
Burgers equations have been investigated by variational iteration method [11] and Adomian
decomposition method [21].
The existence of solution for partial neutral integro-differential equation with infinite delay
in infinite dimensional spaces has been extensively studied by many authors. Ezzinbi and al.
[7] investigated the existence and regularity of solutions for some partial functional integro-
differential equations in Banach spaces. Cui and Yan [4] investigated the existence of mild
solutions for a class of fractional neutral stochastic integro-differential equations with infinite
delay in Hilbert spaces by means of Sadovskiiβs fixed point theorem and in another paper [1],
Balasubramaniam et al. discussed the existence of mild and strong solutions of semilinear neu-
tral functional differential evolution equations with nonlocal conditions by using fractional
power of operators and Krasnoselskii fixed point theorem, Djourdem and Bouteraa [6] studied
the existence of a mild solution to time and space-fractional stochastic equation perturbed by
multiplicative with noise via Sadovskiiβs fixed point theorem. In particular, the stability theory
of stochastic differential equations has been popularly applied in variety fields of science and
technology. Several authors have established the stability results of mild solutions for these
equations by using various techniques, we refer the reader to [2], [10], [17], [26].
The main contribution of this paper is to establish the existence of mild solution for the prob-
lem (1)-(3). Using mainly the Krasnoselskiiβs fixed point theorem. The rest f paper organised as
follows, In Section 2, we will introduce some notations and preliminaries, which play a crucial
role in our theorem analysis. In Section 3, the existence results on a mild solutions are derived.
2. Preliminaries
In this section, we give some notions and certain important preliminaries, which will be used
in the subsequent discussions. Let (Ξ©, πΉ , π, {πΉ }π‘β₯0 ) be a filtered probability space with a normal
filtration, where π is a probability measure on (Ξ©, πΉ ) and πΉ is the Borel πβalgebra . Let {πΉ }π‘β₯0
satisfying that πΉ0 contains all π-null sets. Theoperator π΄ is the infinitesimal generator of a
strongly continuous semigroup on a separable real Hilbert space π» .
Denote the basic functional space πΏπ (π·) , 1 β€ π < β and π» π (π·) by the usual Lebesgue and
Sobolev space, respectively. We assume that π΄ is the negative Laplacian βΞ in a bounded
domain with zero Dirichlet boundary conditions in Hilbert space π» = πΏ2 (π·), which are given
by
π΄ = ββ³, π· (π΄) = π»01 (π·) β© π» 2 (π·) ,
since the operator π΄ is self-adjoint, i.e., there exist the eigenvectors ππ corresponding to eigen-
values ππ such that
β
π΄ππ = ππ ππ , ππ = 2π ππ (ππ) , ππ = π 2 π 2 , π β β+ .
π
For any π > 0, let π» π be the domain of the fractional power π΄ 2 = (βΞ) 2 , which can be
π
defined by
π π
π > 0, π΄ 2 ππ = πΎπ2 ππ , π = 1, 2, β¦
and { }
β π
π
π 2
π» = π· (π΄ ) = π£ β πΏ
2 (π·) , π .π‘. βπ£β2π» π = βπΎπ2 π£π2 < β ,
π=1
β π β
where π£π = β¨π£, ππ β© with the inner product β¨β
, β
β© in πΏ2 (π·), the norm βπ» π π£β = βπ΄ 2 π£ β, the bilinear
β β
operator π΅ (π’, π£) = π’ β
βπ£ and ξ° (π΅) = π»01 (π·) with the slight abuse of notation π΅ (π’) = π΅ (π’, π’).
Then we can rewrite the equation (1)-(3) as follows in the abstract form
{
π π·πΌ π (π‘)
π‘ [π’ (π‘) β β (π’ (π‘))] = π΄π’ (π‘) + π΅ (π’ (π‘)) + π (π’ (π‘)) ππ‘ , π‘ > 0,
(5)
π’ (0) = π’0 ,
where {π (π‘) , π‘ β₯ 0} is a π-Wiener process with linear bounded covarience operator π such
β
that a trace class operator π denote π π (π) = β ππ < β, which satisfies that πππ = ππ ππ , π =
π=1
1, 2, β¦, then the Wiener process is given by
β β
π (π‘) = β ππ π½π (π‘) ππ ,
π=1
where {π½π }β
π=1 is a sequence of real-valued standard Brownian motions.
Let πΏ20 = πΏ2 (π 2 (π» ) , π» ) be a Hilbert-Schmidt space of operators from π 2 (π» ) to π» with the
1 1
norm 1
β 2
β 1β 1
βπβπΏ20 = βππ 2 β π = βππ 2 ππ ,
β βπ» ( π=1 )
i.e., { }
β β β
2 1 1
πΏ20 = π β πΏ (π» ) βΆ β ββππ ππ ππ ββ < β , 2 2
π=1 β β
where πΏ (π» ) is the space of bounded linear operators from π» to π» .
For an arbitrary Banach space π΅, we denote
1
π
βπ£βπΏπ (Ξ©,π΅) = (πΈ βπ£βπ΅ ) π , βπ£ β πΏπ (Ξ©, πΉ , π, π΅) , π ππ πππ¦ π β₯ 2.
We shall also need the following result with respect to the operator π΄ (see [28]).
For any π > 0, an analytic semigroup π (π‘) = π βπ‘π΄ , π‘ β₯ 0 is generated by the operator π΄ on
πΏπ , there exists a constant πΆπ dependent on π such that
βπ΄π (π‘)βπΏ(πΏπ ) β€ πΆπ π‘ βπ , π‘ > 0,
in which πΏ (π΅) denotes the Banach space of all bounded operators from π΅ to itself.
Next we will introduce the following lemma to estimate the stochastic integrals, which con-
tains the Burkhoder-Davis-Gundyβs inequality.
[15] For any 0 β€ π‘1 < π‘2 β€ π and π β₯ 2 and for any predictable stochastic process π£ βΆ
[0, π ] Γ Ξ© β πΏ20 which satisfies
π
β‘β π β 2 β€β₯
β’β
πΈ β’ β« βπ£ (π )β2πΏ2 ππ β β₯ < β,
β’ββ 0 β β₯
0
β£ β β¦
then, we have
π
β‘β π‘2 βπ β€ β‘β π‘2 β 2 β€β₯
β
β’β β β₯ β’β
β 2
πΈ β’ββ« π£ (π ) ππ (π )β β₯ < πΆ (π) πΈ β’ β« βπ£ (π )βπΏ2 ππ β β₯ .
β β
β’ββπ‘1 β β₯
0
β’βπ‘1 β β₯ β β¦
β£β β β¦ β£
Ispired by the definition of the mild solution to the time-fractional differential equations (see
[24], [34]), we give the following definition of mild solution for our time-fractional stochastic
equation.
An πΉπ‘ -adapted stochastic process (π’ (π‘) , π‘ β [0, π ]) is called a mild solution to (5) if the fol-
lowing integral equation is satisfied
π‘
π’ (π‘) = πΈπΌ (π‘) π’0 + β (π’ (π‘)) + β« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π΅ (π’ (π )) ππ
0
π‘
+ β« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π (π’ (π )) ππ (π ) , (6)
0
where the generalized Mittag-Leffler operators πΈπΌ (π‘) and πΈπΌ,πΌ (π‘) are defined, respectively, by
β
πΈπΌ (π‘) = β« ππΌ (π) π (π‘ πΌ π) ππ,
0
and
β
πΈπΌ,πΌ (π‘) = β« πΌπππΌ (π) π (π‘ πΌ π) ππ,
0
where π (π‘) π‘ β₯ 0 is an analytic semi group generated by the operator βπ΄ and the
= π βπ‘π΄ ,
Mainardiβs Wright-type function with πΌ β (0, 1) is given by
β
(β1)π π π
ππΌ (π) = β .
π=0
π!Ξ (1 β πΌ (1 + π))
[3] For any πΌ β (0, 1) and β1 < π < β, it is not difficult to verity that
β
Ξ (1 + π)
ππΌ (π) β₯ 0 πππ β« π π ππΌ (π) ππ = , (7)
0 Ξ (1 + πΌπ)
for all π β₯ 0.
The operators and {πΈπΌ (π‘)}π‘β₯0 and {πΈπΌ,πΌ (π‘)}π‘β₯0 in (7) have the following properties.
For any πΈπΌ (π‘) and πΈπΌ,πΌ (π‘) are linear and bounded operators. Moreover, for 0 < πΌ < 1 and
0 β€ π < 2, there exists a constant πΆ > 0 such that πΈπΌ (π‘) and πΈπΌ,πΌ (π‘) are defined, respectively, by
(8)
πΌπ
βπΈπΌ (π‘) π βπ» π β€ πΆπ‘ β 2 βπ β , βπΈπΌ,πΌ (π‘) π β π β€ πΆπ‘ β πΌπ2 βπ β .
β βπ»
For π > 0 and 0 β€ π < 2, by means of Lemma 2 and Lemma 2, we have
β
βπΈπΌ (π‘) π βπ» π β€ β« ππΌ (π) βπ΄π π (π‘ πΌ π) π β ππ
0
β
πΌπ
β€ β« πΆπ π‘ β 2 π βπ ππΌ (π) βπ β ππ
0
πΆπ Ξ (1 β π) β πΌπ
= π‘ 2 βπ β , π β πΏ2 (π·) ,
Ξ (1 β πΌπ)
and
β
βπΈπΌ,πΌ (π‘) π β π β€ πΌ
β βπ» β« πΌπππΌ (π) βπ΄π π (π‘ π) π β ππ
0
β
πΌπ
β€ β« πΆπ πΌπ‘ β 2 π 1βπ ππΌ (π) βπ β ππ
0
πΆπ πΌΞ (2 β π) β πΌπ
= π‘ 2 βπ β , π β πΏ2 (π·) ,
Ξ (1 β πΌπ)
so, πΈπΌ (π‘) and πΈπΌ,πΌ (π‘) are linear and bounded operators. The proof is completed.
For any π‘ > 0, the operators πΈπΌ (π‘) and πΈπΌ,πΌ (π‘) are strongly continuous. Moreover, for 0 <
πΌ < 1 and 0 β€ π < 2 and 0 β€ π‘1 < π‘2 β€ π , there exists a constant πΆ > 0 such that
πΌπ
β(πΈπΌ (π‘2 ) β πΈπΌ (π‘1 )) π βπ» π β€ πΆ (π‘2 β π‘1 ) 2 βπ β , (9)
and
β(πΈπΌ,πΌ (π‘2 ) β πΈπΌ,πΌ (π‘1 )) π β π β€ πΆ (π‘2 β π‘1 ) πΌπ2 βπ β . (10)
β βπ»
For any 0 < π0 β€ π‘1 < π‘2 β€ π , it iseasy to deduce that
π‘2
ππ (π‘ πΌ π)
β« ππ‘ = π (π‘2πΌ π) β π (π‘1π )
ππ‘
π‘1
π‘2
= β« πΌπ‘ πΌβ1 ππ΄π (π‘ πΌ π) ππ‘,
π‘1
and by (7) and Lemma 2, we have
β(πΈπΌ (π‘2 ) β πΈπΌ (π‘1 )) π βπ» π = βπ΄π (πΈπΌ (π‘2 ) β πΈπΌ (π‘1 )) π β
β β β
β πΌ π
β
= ββ« ππΌ (π) π΄π (π (π‘2 π) β π (π‘1 )) π ππ ββ
β
β β
β0 β
β π‘2
β€ β« πΌπππΌ (π) β« π‘ πΌβ1 βπ΄2+π π (π‘ πΌ π) π βπΏ2 ππ‘ππ
0 π‘1
β
β π‘2 πΌπ β
β π2
β€ β« πΆπ πΌπ ππΌ (π) ββ« π‘ β 2 β1 ππ‘ β βπ β ππ
β β
0 βπ‘1 β
2πΆπ Ξ (1 β π2 ) 2β πΌπ2 β πΌπ
=
πΞ (1 β πΌπ (π‘1 β π‘2 ) βπ β
2 )
2πΆπ Ξ (1 β π2 ) πΌπ
β€ (π‘2 β π‘1 ) 2 βπ β , π β πΏ2 (π·) .
ππ0πΌπ Ξ (1 β πΌπ
2 )
Also
β(πΈπΌ,πΌ (π‘2 ) β πΈπΌ,πΌ (π‘1 )) π β π = βπ΄π (πΈπΌ,πΌ (π‘2 ) β πΈπΌ,πΌ (π‘1 )) π β
β βπ» β β
β β β
β β
= βββ« πΌπππΌ (π) π΄π (π (π‘2πΌ π) β π (π‘1π )) π ππ ββ
β β
β0 β
β π‘2
β€ β« πΌ π ππΌ (π) β« π‘ πΌβ1 βπ΄2+π π (π‘ πΌ π) π βπΏ2 ππ‘ππ
2 2
0 π‘1
β
β π‘2 πΌπ β
2 1β π2
β€ β« πΆπ πΌ π ππΌ (π) ββ« π‘ β 2 β1 ππ‘ β βπ β ππ
β β
0 βπ‘1 β
2πΌπΆπ Ξ (2 β π2 ) β πΌπ β πΌπ
= π ( π‘1 2 β π‘2 2 ) βπ β
πΞ (1 + πΌ (1 β 2 ))
2πΆπ Ξ (2 β π2 ) πΌπ
β€ (π‘2 β π‘1 ) 2 βπ β , π β πΏ2 (π·) .
ππ0πΌπ Ξ (1 + πΌ (1 β π2 ))
It is obviously to see that the term
β(πΈπΌ (π‘2 ) β πΈπΌ (π‘1 )) π βπ» π β 0,
and
β(πΈπΌ,πΌ (π‘2 ) β πΈπΌ,πΌ (π‘1 )) π β π β 0,
β βπ»
as π‘1 β π‘2 which mean that the operators πΈπΌ (π‘) and πΈπΌ,πΌ (π‘) are strongly continuous.
3. Existence results
In this section, we present our main results on the existence of mild solutions of problem (5)
and we define the following space
{ }
πΎ = π’ βΆ π’ β πΆ ([0, π ] , π» π ) , sup βπ’β < β .
π‘β[0,π ]
To do this, we make the following hypotheses:
(π»1 ) π΄ is the infinitesimal generator of a strongly continuous semigroup {π (π‘) , π‘ β₯ 0} on π» .
We will also suppose that the operator πΈπΌ (π‘) , π‘ > 0 is compact.
(π»2 ) The function π βΆ Ξ©Γπ» β πΏ20 satisfies the following global Lipshitz and growth conditions:
βπ (π£)βπΏ20 β€ πΆ βπ’β , βπ (π’) β π (π£)βπΏ20 β€ πΆ βπ’ β π£β ,
for any π’ β π» , π£ β π» .
(π»3 ) The initial value π’0 βΆ Ξ© β π» π is a πΉ0 β measurable random variable, it hold that
βπ’0 βπΏπ (Ξ©,π» π ) < β, π ππ πππ¦ 0 β€ π < πΌ < 2.
(π»4 ) The function β βΆ πΏ20 β πΏ20 is continuous and there exists πΏβ > 0 such that
π π
πΈ ββ (π’1 (π‘)) β β (π’2 (π‘))βπΏ2 β€ πΏβ βπ’1 (π‘) β π’2 (π‘)βπΏ2 , π‘ β [0, π ] , π’1 , π’2 β πΏ20 ,
0 0
and
π π
πΈ ββ (π’ (π‘))βπΏ2 β€ πΏβ πΈ βπ’ (π‘)βπΏ2 , π‘ β [0, π ] , π’ β πΏ20 .
0 0
(π»5 ) Let πΆ > 0 be a real number, then the bounded bilinear operator π΅ βΆ πΏ2 (π·) β π» β1 (π·)
satisfies the following properties
βπ΅ (π’)βπ» β1 β€ πΆ βπ’β2 ,
and
βπ΅ (π’) β π΅ (π£)βπ» β1 β€ πΆ (βπ’β + βπ£β) βπ’ β π£β ,
for any π’, π£ β πΏ2 (π·).
Our main results is based on the following Krasnoselskii fixed points theorem [14]. [14] Let
π be a Banach space, πΆ a closed, bounded, convex and nonemty subset of π . Consider the
operators πΉ1 and πΉ2 such that
(π) πΉ1 π’ + πΉ2 π£ β πΆ whenever π’, π£ β πΆ,
(πππ) πΉ1 is a contraction mapping.
(ππ) πΉ2 is compact and continuous.
Then there exists π§ β πΆ such that π§ = πΉ1 π§ + πΉ2 π§.
In the proof of main result, we need the following Lemmas. Assume that conditions (π»1 ) and
(π»2 ) hold. Let Ξ¦1 and Ξ¦2 be two operators defined respectively for each π’ β πΎ by
π‘
Ξ¦1 (π’) = β« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π΅ (π’ (π )) ππ ,
(11)
0
π‘
Ξ¦2 (π’) = β« ππΌ (π‘ β π ) π (π’ (π )) ππ (π ) .
0
Then Ξ¦1 and Ξ¦2 are continuous and map πΎ into itself.
It is obvious that Ξ¦1 is continuous. Next we show that Ξ¦1 (πΎ ) β πΎ . By (π»1 ) and (π»2 ), from the
equation (11) and by applying Holder inequality, we have
β π‘ βπ
π β πΌβ1
β
πΈ β(Ξ¦1 π’) (π‘)βπ» π = πΈ βββ« (π‘ β π ) π΄1 πΈπΌ,πΌ (π‘ β π ) π΄πβ1 π΅ (π’ (π )) ππ ββ
β β
β0 βπ» π
π‘ πβ1
β π πΌβ1
( 2 ) β π‘
β€ πΆπΌπ ββ« (π‘ β π ) πβ1 ππ β β« πΈ [βπ΄πβ1 π΅ (π’ (π ))βπ ] ππ
β β 0
β0 β
πβ1 π‘
2 (π β 1)
π
πβ2 π
β€ πΆ πΆπΌ
[ πβ2 ]
(π ) 2
β« πΈ [βπ’ (π‘)βπ» π ]
0
π‘
(12)
π
= πΎ1 β« πΈ [βπ’ (π )βπ» π ] ππ ,
0
πβ1 πβ2
where πΎ1 = πΆ π πΆπΌ [ 2(πβ1)
πβ2 ] (π ) 2 . This complete the proof.
By using also the Holder inequality and Lemma 2, we obtain
β π‘ βπ
π β πΌβ1
β
πΈ β(Ξ¦2 π’) (π‘)βπ» π = πΈ βββ« (π‘ β π ) πΈπΌ,πΌ (π‘ β π ) π (π’ (π )) ππ (π )ββ
β β
β0 βπ» π
π
β‘β π‘ β 2 β€β₯
β’β β πΌβ1 β2 2
β€ πΆ (π) πΈ β’ β« β(π‘ β π ) πΈπΌ,πΌ (π‘ β π )β βπ΄π π (π’)βπΏ2 ππ β β₯
β β
β’ββ 0 β β₯
0
β£ β β¦
πβ2
β π‘ 2π(πΌβ1)
β 2 π‘
πβ π
β€ πΆ (π) πΆπΌ β« (π‘ β π ) πβ2 β β« πΈ βπ΄π π (π’)βπΏ2 ππ
β β 0
β0 β 0
πβ2 π‘
2
πβ2 π
β€ πΆ (π) πΆπΌπ πΈ βπ΄π π (π’)βπΏ2 ππ
( π (2πΌ β 1) β 2 ) β« 0
0
π‘
(13)
π
= πΎ2 β« πΈ [βπ’ (π )βπ» π ] ππ ,
0
πβ2
where πΎ2 = πΆ (π) πΆπΌ πΆ π [ π(2πΌβ1)β2 ] .
π πβ2 2
That is Ξ¦2 (πΎ ) β πΎ .
Assume that conditions (π»1 ) and (π»4 ) hold. Let Ξ¦3 be the operator defined by for each π’ β πΎ
(Ξ¦3 π’) (π‘) = πΈπΌ (π‘) π’0 + β (π’ (π‘)) .
Then Ξ¦3 is continuous and maps πΎ into πΎ . The continuity of Ξ¦3 follows from (π»4 ).
Next, we show that Ξ¦3 (π ) β π . By (π»1 ), (π»5 ) and from (13), we have
π π π
πΈ β(Ξ¦3 π’) (π‘)βπΏ2 β€ πΈ ββ (π’ (π‘))βπΏ2 β€ πΏβ πΈ βπ’ (π‘)βπΏ2 .
0 0 0
So, we conclude Ξ¦3 (πΎ ) β πΎ .
Assume that conditions (π»1 ) and (π»2 ) hold. Then
πΈ [βπΈπΌ (π‘) π’0 βπ» π ] β€ πΈ [βπ’0 βπ» π ] .
By Lemma 2, we have
β‘ β 1 β€
πΈ [βπΈπΌ (π‘) π’0 βπ» π ] β€ πΈ β’β« ππΌ (π) (βπ΄π π (π‘ πΌ π) π’0 β2 ) 2 ππ β₯
β’ β₯
β£0 β¦
1
β‘ β β 2 β€
βπ‘ πΌ ππ΄ 2
β’
β€ πΈ β« ππΌ (π) β β¨π΄π π π’0 , ππ β© ππ β₯
β’ ( π=1 ) β₯
β£0 β¦
1
β‘ β β
πΌ
π
2
2 2 β€
β€ πΈ β’β« ππΌ (π) β π΄π π’0 , π βπ‘ πππ , ππ ππ β₯
β’ ( π=1 β¨ β©) β₯
β£0 β¦
β‘ β β€
β€ πΈ β’β« ππΌ (π) βπ’0 βπ» π ππ β₯ = πΈ [βπ’0 βπ» π ] .
β’ β₯
β£0 β¦
First, we define a map πΉ βΆ πΎ β πΆ ([0, π ] , π» π ) in the following manner: for any π’ β πΎ ,
π‘ π‘
(πΉ π’) (π‘) = β« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π΅ (π’ (π )) ππ + β« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π (π’ (π )) ππ (π )
0 0
+πΈπΌ (π‘) π’0 (π ) + β (π’ (π‘)) .
Now, we set πΉ = πΉ1 + πΉ2 , where
(πΉ1 π’) (π‘) = πΈπΌ (π‘) π’0 (π ) + β (π’ (π‘)) ,
and
π‘ π‘
πΌβ1
(πΉ2 π’) (π‘) = β« (π‘ β π ) πΈπΌ,πΌ (π‘ β π ) π΅ (π’ (π )) ππ + β« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π (π’ (π )) ππ (π ) ,
0 0
for π‘ β [0, π ].
Assume (π»2 ) , (π»4 ), (π»5 ) hold and 0 < π < πΌ β€ 2, π β₯ 2, Then
π πΌπ
π
πΈ βπΈπΌ (π‘2 ) β πΈπΌ (π‘1 )βπ» π β€ πΆπΌ,π (π‘2 β π‘1 ) 2 πΈ βπ’0 βπ .
We set
πΌ1 = πΉ1 (π‘2 ) β πΉ1 (π‘1 ) = πΈπΌ (π‘2 ) π’0 β πΈπΌ (π‘2 ) π’0
For any π β₯ 2, by vertue of Lemma 2, it follows that
π
πΈ [βπΌ1 βπ» π ] = πΈ [π΄ βπΈπΌ (π‘2 ) π’0 β πΈπΌ (π‘2 ) π’0 βπ ]
πΌπ
π
β€ πΆπΌ,π (π‘2 β π‘1 ) 2 πΈ βπ’0 βπ .
It is obviously to see that the term β(πΉ1 (π‘2 ) β πΉ1 (π‘1 ))βπ β 0 as π‘1 β π‘2 which mean that the
operators πΉ1 is strongly continuous.
Assume (π»2 ) , (π»4 ), (π»5 ) hold and 0 < π < πΌ β€ 2, π β₯ 2, then the operator πΉ2 is uniformly
bounded.
From Lemma 11, using the estimate (12) and by means of extension of Gronwallβs lemma,
we have
π
π π’π πΈ [βπΉ2 (π’ (π‘))βπ» π ] β€ β,
π‘β[0,π ]
that is the operator πΉ2 is uniformly bounded.
Assume (π»2 ) , (π»4 ), (π»5 ) hold and 0 < π < πΌ β€ 2, π β₯ 2. Then the operator πΉ2 is equicontinu-
ous.
For any 0 β€ π‘1 < π‘2 β€ π , from
π‘2
(πΉ2 π’) (π‘2 ) β (πΉ2 π’) (π‘1 ) = β« (π‘2 β π )πΌβ1 πΈπΌ,πΌ (π‘2 β π ) π΅ (π’ (π )) ππ
0
π‘1 π‘2
πΌβ1
β β« (π‘1 β π ) πΈπΌ,πΌ (π‘1 β π ) π΅ (π’ (π )) ππ + β« (π‘2 β π )πΌβ1 πΈπΌ,πΌ (π‘2 β π ) π (π’) ππ (π ) .
0 0
π‘1
β β« (π‘1 β π )πΌβ1 πΈπΌ,πΌ (π‘1 β π ) π (π’) ππ (π ) = πΌ2 + πΌ3 , (14)
0
where
π‘2 π‘1
πΌβ1
πΌ2 = β« (π‘2 β π ) πΈπΌ,πΌ (π‘2 β π ) π΅ (π’ (π )) ππ β β« (π‘1 β π )πΌβ1 πΈπΌ,πΌ (π‘1 β π ) π΅ (π’) π (π )
0 0
π‘1
= β« (π‘1 β π )πΌβ1 [πΈπΌ,πΌ (π‘2 β π ) β πΈπΌ,πΌ (π‘1 β π )] π΅ (π’ (π )) ππ
0
π‘1
+ β« [(π‘2 β π )πΌβ1 β (π‘1 β π )πΌβ1 ] πΈπΌ,πΌ (π‘2 β π ) π΅ (π’ (π )) ππ
0
π‘2
+ β« (π‘2 β π )πΌβ1 πΈπΌ,πΌ (π‘2 β π ) π΅ (π’ (π )) ππ
π‘1
= πΌ21 + πΌ22 + πΌ23 , (15)
and
π‘2 π‘1
πΌβ1
πΌ3 = β« (π‘2 β π ) πΈπΌ,πΌ (π‘2 β π ) π (π’ (π )) πππ β β« (π‘1 β π )πΌβ1 πΈπΌ,πΌ (π‘1 β π ) π (π’) ππ (π )
0 0
π‘1
= β« (π‘1 β π )πΌβ1 [πΈπΌ,πΌ (π‘2 β π ) β πΈπΌ,πΌ (π‘1 β π )] π (π’ (π )) ππ (π )
0
π‘1
+ β« [(π‘2 β π )πΌβ1 β (π‘1 β π )πΌβ1 ] πΈπΌ,πΌ (π‘2 β π ) π (π’ (π )) ππ (π )
0
π‘2
+ β« (π‘2 β π )πΌβ1 πΈπΌ,πΌ (π‘2 β π ) π (π’ (π )) ππ (π )
π‘1
= πΌ31 + πΌ32 + πΌ33 . (16)
For the first term πΌ21 in (15), applying the assumptions (π»5 ) and Lemma 2 and Holder inequality,
we have
β‘β π‘1 βπ β€
π β πΌβ1
β
πΈ [βπΌ21 βπ» π ] = πΈ β’βββ« (π‘1 β π ) [πΈπΌ,πΌ (π‘2 β π ) β πΈπΌ,πΌ (π‘1 β π )] π΅ (π’ (π )) ππ ββ β₯
β’β β β₯
β£β 0 β β¦
πβ1
β π‘1 β π‘
π
β€ πΆπΌπ (π‘2 β π‘1 )
ππΌ(π+1)
2 β (π‘ β π )
π(πΌβ1)
πβ1 ππ β π
πΈ [βπ΄β1 π΅ (π’ (π ))βπ» 1 ] ππ (17)
ββ« 1 β β«
β0 β 0
πβ1
πβ1 2π ππΌ(π+1)
β€ πΆ π πΆπΌπ
π
π ππΌ π π’π πΈ [βπ’ (π )βπ» 1 ] (π‘2 β π‘1 ) 2 .
( ππΌ β 1 ) (π‘β[0,π ] )
Using the assumptions (π»5 ) and Lemma 2 and Holder inequality, we have
β‘ββ π‘1 βπ β€
π β
πΈ [βπΌ22 βπ» π ] = πΈ β’βββ« [(π‘2 β π )πΌβ1 β (π‘1 β π )πΌβ1 ] [π΄π πΈπΌ,πΌ (π‘2 β π )] π΅ (π’ (π )) ππ ββ β₯
β’β β β₯
β£β 0 β β¦
πβ1
β π‘1 { βπΌ(π+1)
} πβ1
π β
β€ πΆπΌ β« [(π‘2 β π )πΌβ1 β (π‘1 β π )πΌβ1 ] Γ (π‘2 β π ) 2
πβ
ππ β
β β
β0 β
π‘ (18)
π
Γβ« πΈ [βπ΄β1 π΅ (π’ (π ))βπ» 1 ] ππ
0
πβ1
β β
π π β πβ1 β 2π ππΌ(1βπ)β2
β€ πΆ πΆπΌ π β β π π’π πΈ βπ’ (π )βπ» 1 ] (π‘2 β π‘1 ) 2 ,
β π (πΌ β πΌ(π+1) (π‘β[0,π ] [ )
2 )β
β β
and
β‘β π‘2 βπ β€
β’β β β₯
π β β
πΈ [βπΌ23 βπ» π ] = πΈ β’ββ« (π‘2 β π )πΌβ1 π΄π πΈπΌ,πΌ (π‘2 β π ) π΅ (π’ (π )) ππ β β₯
β β
β’βπ‘1 β β₯
β£β β β¦
πβ1
β π‘2 β π‘2
πΌβ1β πΌ(π+1)
(19)
πβ β π
β€ πΆπΌ β« (π‘2 β π ) 2 ππ β« πΈ [βπ΄β1 π΅ (π’ (π ))βπ» 1 ] ππ
β β
βπ‘1 β π‘1
πβ1
β β
π πβ πβ1 β 2π ππΌ(1βπ)
β€ πΆ πΆπΌ β β π π’π πΈ βπ’ (π )βπ» 1 ] (π‘2 β π‘1 ) 2 .
πΌ(π+1)
β π (πΌ β 2 ) β 1 β (π‘β[0,π ] [ )
β β
Next, by following similar arguments as in the proof of (17)-(19) and using Lemma 2 there
holds,
β‘β π‘1 βπ β€
π β πΌβ1
β
πΈ [βπΌ31 βπ» π ] = πΈ β’βββ« (π‘1 β π ) [πΈπΌ,πΌ (π‘2 β π ) β πΈπΌ,πΌ (π‘1 β π )] π (π’ (π )) πππ ββ β₯
β’ β β β₯
β£β 0 β β¦
π
β‘β π‘1 β 2 β€β₯
β’β β πΌβ1 β2 2
β€ πΆ (π) πΈ β’ β« β(π‘1 β π ) π΄π [πΈπΌ,πΌ (π‘2 β π ) β πΈπΌ,πΌ (π‘1 β π )]β βπ (π’ (π ))βπΏ2 ππ β β₯
β β
β’ββ 0 β β₯
0
β£ β β¦
πβ2
ππΌπ
β π‘1 2π(πΌβ1)
β 2 π‘1
π β (π‘1 β π ) πβ2 ππ β π
β€ πΆ (π) πΆπΌπ (π‘2 β π‘1 ) 2 πΈ βπ (π’ (π ))βπΏ2 ππ
ββ« β β« 0
β0 β 0
πβ1
2ππΌβπβ1 πβ1 ππΌπ
β€ πΆ π πΆπΌπ
π
π 2 π π’π πΈ [βπ’ (π )βπ ] (π‘2 β π‘1 ) 2 , (20)
( 2ππΌ β π β 2 ) (π‘β[0,π ] )
and
β‘ββ π‘1 βπ β€
π β
πΈ [βπΌ32 βπ» π ] = πΈ β’βββ« [(π‘2 β π )πΌβ1 β (π‘1 β π )πΌβ1 ] [π΄π πΈπΌ,πΌ (π‘2 β π )] π (π’ (π )) πππ ββ β₯
β’β β β₯
β£β 0 β β¦
π
β‘β π‘1 β 2 β€β₯
β’β β πΌβ1 πΌβ1 β2 2
β€ πΆ (π) πΈ β’ β« β[(π‘2 β π ) β (π‘1 β π ) ] [π΄π πΈπΌ,πΌ (π‘2 β π )]β βπ (π’ (π ))βπΏ2 ππ β β₯
β β
β’ββ 0 β β₯
0
β£ β β¦
πβ2
β π‘1 { βπΌπ
} πβ2
2π β 2
πβ πΌβ1 πΌβ1
β€ πΆ (π) πΆπΌ β« [(π‘2 β π ) β (π‘1 β π ) ] Γ (π‘2 β π ) 2 ππ β
β β
β0 β
π‘
π
Γ β« πΈ [βπ (π’ (π ))βπΏ2 ] ππ
0
0
πβ2
2
2 (π β 2)
β€ πΆ (π) πΆ πΆπΌπ π
π
( 2ππΌ (2 β π) β 2 (π + 2) )
2ππΌ(2βπ)β2(π+2)
Γ π π’π πΈ [βπ’ (π‘)βπ ] (π‘2 β π‘1 ) 4 , (21)
(π‘β[0,π ] )
and
β‘β π‘2 βπ β€
β’ β β β₯
π β β
πΈ [βπΌ33 βπ» π ] = πΈ β’ββ« (π‘2 β π )πΌβ1 π΄π πΈπΌ,πΌ (π‘2 β π ) π΅ (π’ (π )) ππ β β₯
β β
β’βπ‘1 β β₯
β£β β β¦
π
β‘β π‘1 β 2 β€β₯
β’β β πΌβ1 β2 2
β€ πΆ (π) πΈ β’ β« β(π‘2 β π ) π΄π πΈπΌ,πΌ (π‘2 β π )β βπ (π’ (π ))βπΏ2 ππ β β₯
β β
β’ββ 0 β β₯
0
β£ β β¦
πβ2
β π‘2 β 2 π‘2
πβ πΌβ1β πΌπ
2 β
π
β€ πΆ (π) πΆπΌ β« (π‘2 β π ) Γ β« πΈ [βπ (π’ (π ))βπΏ2 ] ππ
β β 0
βπ‘1 β π‘1
πβ2
2
2 (π β 2) 2ππΌ(2βπ)β2π
β€ πΆ (π) πΆ πΆπΌπ
π
π π’π πΈ [βπ’ (π‘)βπ ] (π‘2 β π‘1 ) 4 . (22)
( 2ππΌ (2 β π) β 2 (π + 2) ) (π‘β[0,π ] )
Taking expectation on the both side of (14) and in view of estimates (15) and (17) β (22), we
conclude that
β(πΉ2 π’) (π‘2 ) β (πΉ2 π’) (π‘1 )βπΏπ (Ξ©,π» π ) β€ πΆ (π‘2 β π‘1 )πΎ ,
{ }
where πΎ = πππ πΌπ 2 , πΌπ(1βπ)β2 2ππΌ(2βπ)β2(π+2)
2π , 4π when 0 < π‘2 β π‘1 < 1.
{ }
Otherwise, if π‘2 β π‘1 β₯ 1, then we set πΎ = πππ₯ πΌ(π+1) 2 , πΌ(2βπβ1) 2ππΌ(2βπ)β2π
2 , 4π .
Assume the conditions (π»1 ) and (π»2 ) hold. Then πΉ maps πΎ into itself.
Let the nonlinear operator πΉ defined by, for π‘ β₯ 0,
π‘
(πΉ π’) (π‘) = πΈπΌ (π‘) π’0 + β (π’ (π‘)) + β« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π΅ (π’ (π )) ππ
0
π‘
+ β« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π (π’) ππ (π ) .
0
We prove that the operator πΉ has a fixed point, which is a mild solution of the problem (1)-(2).
We shall employ Theorem 3. For better readability, we divide the proof into two steps.
Step 1. πΉ βΆ π β πΆ ([0, π ] , π» π ) is continuous. Let {π’π (π‘)}πβ₯0 with π’π β π’ (π β β) in
π . Then there is
{ a number π > 0 such } that πΈ βπ’π (π‘)βπ» π β€ π for all π and π.π. π‘ β [0, π ], so
2
π’π β π΅π (0, π ) = π’ β π βΆ π π’π βπ’βπ» π and π’ β π΅π (0, π ). By the assumptions (π»2 ) and similar
π‘β[0,π ]
argument to obtain (12) and (13), we have
π
πΈ β(πΉ π’π ) (π‘) β (πΉ π’) (π‘)βπ» π
π
β€ 3πβ1 ββ (π’π (π‘)) β β (π’ (π‘))βππ» π + 3πβ1 πΈ βΞ¦1 (π’π (π‘) β π’ (π‘))βπ» π
π
+ 3πβ1 πΈ βΞ¦2 (π’π (π‘) β π’ (π‘))βπ» π
π‘
β β
πβ1 π π
β€3 ββ (π’π (π‘)) β β (π’ (π‘))βπ» π + 3πβ1 (πΊπΎ1 + πΎ πΎ2 ) ββ« πΈ βπ’π β π’βπ» π ππ β .
β β
β0 β
Then, we have for all π‘ β [0, π ] ,
π
βπΉ π’π β πΉ π’βπ βΆ 0, ππ π βΆ β.
Therefore πΉ is continuous.
Step 2. We decompose πΉ as πΉ = πΉ1 + πΉ2 where πΉ1 and πΉ2 defined above.
(1) πΉ1 is a contraction on π . Let π’, π£ β π . It follows from Lemma 3 that
π π
πΈ βπΉ1 π’ β πΉ1 π£βπ» π β€ πΏβ πΈ βπ’ (π ) β π£ (π )βπ» π
π
β€ πΏβ π π’π πΈ βπ’ (π ) β π£ (π ) ππ βπ» π
π β[0,π ]
π
β€ πΏβ βπ’ (π ) β π£ (π ) ππ βπ
Taking supremum over π‘
π π
βπΉ1 π’ β πΉ1 π£βπ β€ πΏ0 βπ’ (π ) β π£ (π )βπ ,
where πΏ0 = πΏβ < 1.
Hence πΉ1 is a contraction on π .
(2) πΉ2 is compact operator. Let π’, π£ β π . It follows from (π»2 ) , (π»5 ) and Lemma 3 that
β π‘ β2
β β
πΈ βπΉ2 π’ β πΉ2 π£β2π» π β€ 2πβ1 πΈ βββ« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π΄π [π (π’ (π )) β π (π£ (π ))] ππ (π )ββ
β β
β0 βπ» π
β π‘ βπ
β β
+2πβ1 πΈ βββ« (π‘ β π )πΌβ1 πΈπΌ,πΌ (π‘ β π ) π΄π [π΅ (π’ (π )) β π΅ (π£ (π ))] ππ ββ
β β
β0 βπ» π
β π‘ β
β€ (πΎ1 + πΎ2 ) πΈ ββ« βπ’ β π£β2π» π ππ β ,
β β
β0 β
which implies
π π’π πΈ βπΉ2 π’ β πΉ2 π£β2π» π = (πΎ1 + πΎ2 ) π π’π πΈ βπ’ β π£β2π» π .
π‘β[0,π ] π‘β[0,π ]
Since 0 < πΏ = πΎ1 + πΎ2 < 1, then πΉ is contraction maping on π .
From Lemma 3 and Lemma 3, the operator πΉ2 is relatively compact. together with Ascoliβs
theorem, we conclude that the operator πΉ2 is compact.
In view of Theorem 3, we conclude that πΉ has at least one fixed point, which is a mild solution
of the problem (1)-(2).
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