Malaya J. Mat. 3(1)(2015) 1–13
Existence of mild solution result for fractional neutral stochastic
integro-differential equations with nonlocal conditions and infinite delay
N. Ait ouali
a,∗and A. Kandouci
ba,bLaboratory of Stochastic Models, Statistic and Applications, Tahar Moulay University PO.Box 138 En-Nasr, 20000 Saida, Algeria.
Abstract
We investigate in this paper the existence of mild solutions for the fractional differential equations of neutral type with nonlocal conditions and infinite delay in Hilbert spaces by employing fractional calculus and Krasnoselski-Schaefer fixed point theorem. Finally an example is provided to illustrate the application of the obtained results.
Keywords: Infinite delay, Stochastic fractional differential equations, mild solution, fixed point theorem.
2010 MSC:35G20. c2012 MJM. All rights reserved.
1
Introduction
The main purpose of this paper is to prove the Existence of the mild solution for fractional differential equations of neutral type with infinite delay in Hilbert spaces of the form.
cDα
t[x(t)−h(t, xt)] =A[x(t)−h(t, xt)] +f(t, xt) +
Rt
−∞σ(t, s, xs)dW(s) t∈J = [0, b]
x(0) +µ(x) =x0=φ(t) t∈(−∞,0],
(1.1)
Here,x(.)takes value in a real separable Hilbert spaceHwith inner product(·,·)Hand the normk·kH. The fractional derivativecDα, α ∈(0,1), is understood in the Caputo sense. The operator A generates a strongly
continuous semigroup of bounded linear operatorsS(t), t≥0, onH. LetKbe another separable Hilbert space
with inner product(., .)Kand the normk·k
K. W is a givenK-valued Wiener process with a finite trace nuclear covariance operatorQ ≥ 0defined on a filtered complete probability space(Ω,F,(Ft)t≥0,P). The histories
xt : Ω → Cυdefined byxt={x(t+θ), θ ∈(−∞,0]}belong to the phase spaceCυ, which will be defined in
section 2. The initial dataφ={φ(t), t∈(−∞,0]}is anF0- measurable,Cυ-valued random variable
indepen-dent of W with finite second moments, andh:J× Cϑ →H,h:J× Cυ →H,σ:J ×J1× Cυ → L02(K,H)are
appropriate functions, whereJ1= (−∞, b]andL02(K,H)denotes the space of all Q-Hilbert Schmidt operators
fromK. intoH.µ:C(J,H)→His bounded and the initial datax0is anFadaptedH-valued random variable
independent of Wiener process W.
The fractional differential equations arise in many engineering and scientific disciplines as the mathemat-ica modeling of systems and processes in the fields of physics,
chemistry, aerodynamics, electrodynamics of a complex medium, polymer rheology,etc.,
involves derivatives of fractional order. It is worthwhile mentioning that several important problems of the theory of ordinary and delay differential equations lead to investigations of functional differential equations
∗Corresponding author.
of various types (see the books by Hale and Verduyn Lunel [16] , Wu [31] , Liang et al [17], Liang and Xiao [18], and the references therein).
In particular the nonlocal condition problems for some fractional differential equations have been attrac-tive to many researchers Mophou et al [23] studied existence of mild solution for some fractional differential equations with nonlocal condition. Chang et al [7] investigate the fractional order integro-differential equa-tions with nonlocal condiequa-tions in the Riemann-Liouville fractional derivative sense.
In this paper, we prove the existence theorem of mild solution for neutral differential equation with non-local conditions and infinite delay by using the Krasnoselski-Schaefer fixed point theorem. An example is provided to illustrate the application of the obtained results.
2
Preliminaries
Next we mention a few results and notations needed to establish our results.
Let(H,k·kH)and(K,k·kK)be two real separable Hilbert spaces. We denote byL(K,H)the set of all linear
bounded operators fromKintoH, equipped with the usual operator normk.k. In this article, we use the
symbolk.kto denote norms of operators regardless of the spaces involved when no confusion possibly arises.
Let(Ω,F,(Ft)t≥0,P)be a filtered complete probability space satisfying the usual condition, which means
that the filtration is a right continuous increasing family andF0 contains allP-null sets. W = (Wt)t≥0 be
a Q-Wiener process defined on(Ω,F,(Ft)t≥0,P) with the covariance operator Q such thattrQ < ∞. We
assume that there exists a complete orthonormal system{ek}k≥1inK, a bounded sequence of nonnegative
real numbersλksuch thatQek=λkek, k= 1,2, ..., and a sequence of independent Brownian motions{βk}k≥1
such that
(W(t), e)K=
∞ X
k=1 p
λk(ek, e)Kβk(t) e∈K t≥0
LetL0
2=L2(Q
1
2K,H)be the space of all Hilbert Schmidt operators fromQ 1
2KtoHwith the inner product
hϕ, φiL0
2 =tr[ϕQφ
∗].
The semigroupS(·)is uniformly bounded. That is to say,kS(t)k ≤M for some constantM ≥1and every t≥0.
Assume thatυ: (−∞,0]→(0,+∞)withl=R0
−∞υ(t)dt <+∞a continuous function.
Recall that the abstract phase spaceCυis defined by
Cυ={ϕ: (−∞,0]→H, for anya >0,(E|ϕ(θ)|2)1/2is bounded and measurable
function on[−a,0]and R0
−∞υ(s) sup
s≤θ≤0
(E|ϕ(θ)|2)1/2ds <+∞}.
IfCυis endowed with the norm
kϕkC
υ= Z 0
−∞
υ(s) sup s≤θ≤0
(E|ϕ(θ)|2)12ds, ϕ∈ Cυ
then(Cυ,k.kCυ)is a Banach space (see[20]).
Let us now recall some basic definitions and results of fractional calculus.
Definition 2.1. [21]The fractional integral of orderαwith the lower limit 0 for a function f is defined as
Iαf = 1 Γ(α)
Z t
0
f(s)
(t−s)1−αds t >0 α >0
provided the right-hand side is pointwise defined on[0,∞), whereΓ(.)is the gamma function.
Definition 2.2. The Caputo derivative of orderαwith the lower limit 0 for a function f can be written as
cDα
f(t) = 1 Γ(n−α)
Z t
0
fn(s)
(t−s)α+1−nds=I
The Caputo derivative of a constant equal to zero. Iff is an abstract function with values inH, then the
integrals appearing in the above definitions are taken in Bochner’s sense (see[21]).
Lemma 2.1. 5 Let H be a Hilbert space andΦ1,Φ2two operators on H such that
i) Φ1is a contraction and
ii) Φ2is completely continuous.
Then either
a) the operator equationΦ1x+ Φ2x=xhas a solution or
b) G={x∈H:λΦ1(xλ) +λΦ2x=x}is unbounded forλ∈(0,1).
Lemma 2.2. [15]Letv(.), w(.) : [0, b] →[0,∞)be continuous function. Ifw(.)is nondecreasing and there exist two constantsθ≥0and0< α <1such that
v(t)≤w(t) +θ
Z t
0
v(s)
(t−s)1−αds, t∈J
then
v(t)≤eθn(Γ(α))ntnα/Γ(nα)
n−1 X
j=0 θbα
α
j
w(t),
for everyt∈[0, b]and everyn∈Nsuch thatnα >1.
3
Existence results
Definition 3.3. AnH- valued stochastic process{x(t), t∈(−∞, b]}is a mild solution of the system1.1ifx(0)+µ(x) =
x0=φ(t)on(−∞,0]satisfyingkφk 2
Cυ <+∞, the process x satisfies the following integral equation
x(t) =Sα(t)[φ(0)−µ(x)−h(0, φ)] +h(t, xt) +
Z t
0
(t−s)α−1Tα(t−s)f(s, xs)ds
+
Z t
0
(t−s)α−1Tα(t−s)
Z s
−∞
σ(s, τ, xτ)dW(τ)
ds
where
Sα(t)x=
Z ∞
0
ζα(θ)S(tαθ)xdθ, Tα(t)x=α
Z ∞
0
θζα(θ)S(tαθ)xdθ
andζαis a probability density function defined on(0,∞)
The following properties ofSα(t)andTα(t)appeared in[34]are useful.
Lemma 3.3. The operatorsSα(t)andTα(t)have the following properties
i) For any fixedt≥0,Sα(t)andTα(t)are linear and bounded operators such that for anyx∈H
kSα(t)xkH≤MkxkH and kTα(t)xkH≤
Mα
Γ(1 +α)kxkH
ii) Sα(t)andTα(t)are strongly continuous and compact.
To study existence of mild solutions of1.1, we introduce the following hypotheses.
(H1) : The functionh, f:J× Cυ→Hare continuous and there exist some constantsMh, Mf, such that
Ekh(t, x)−h(t, y)k2
H≤Mhkx−yk
2
Cυ, x, y∈ Cυ, t∈J
Ekh(t, x)kH2 ≤Mh(1 +kxk2C
υ)
Ekf(t, x)−f(t, y)k2
H≤Mfkx−yk
2
Cυ, x, y∈ Cυ, t∈J
Ekf(t, x)k2
H≤Mf(1 +kxk
(H2) :µis continuous and there exists some positive constantsMµsuch that
Ekµ(x)−µ(y)k2H≤Mµkx−yk
2
Cυ, x, y∈ Cυ, t∈J
Ekµ(x)k2
H≤Mµ(1 +kxk
2 Cυ)
(H3) : For eachϕ∈ Cυ,
k(t) = lim
a→∞
Z 0
−a
σ(t, s, ϕ)dW(s)
exists and is continuous. Further, there exists a positive constantMksuch that
Ekk(t)k2H≤Mk
(H4) The functionσ:J×J1× Cυ→L(K,H)satisfies the following:
i) for each(t, s)∈J×J×, σ(t, s, .) :Cυ→L(K,H)is continuous and for eachx∈ Cυ,σ(., ., x) :J×J →L(K,H)
is strongly measurable;
ii) there is a positive integrable function m ∈ L1([0, b]) and a continuous nondecreasing function M
σ :
[0,∞)→(0,∞)such that for every(t, s, x)∈J×J× Cυ, we have
Z t
0
Ekσ(t, s, x)k2L0
2ds≤m(t)Mσ(kxk
2
Cυ), lim infr→∞
Mσ(r)
r ds= ∆<∞
iii) For anyx, y∈ Cυ, t≥0, there exists a positive constantLσsuch that
Z t
0
Ekσ(t, s, x)−σ(t, s, y)k2L0
2ds≤Lσkx−yk
2 Cυ
(H5) :
N0= 2l2{12M2Mµ+ 4Mh} (3.1)
N1= 2kφk2Cυ+ 2l2F¯ (3.2)
N2= 8l2
M
α Γ(1 +α)
2bα
αMf (3.3)
N3= 16bl2
M
α Γ(1 +α)
2bα
αT r(Q) (3.4)
K1=
N1
1−N0
, K2=
N2
1−N0
, K3=
N3
1−N0
(3.5)
¯
F = 12M2(C
1+C2) + 12M2Mµ+ 4Mh+ 4
M
α Γ(1+α)
2
b2α
α2Mf+ 8b
M
α Γ(1+α)
2
b2α
α2Mk (3.6)
Now, we consider the space,
Cυ0 ={x: (−∞, b]→H, x0=φ∈ Cυ}
Setk.kbbe a seminorm defined by
kxkb=kx0kCυ+ sup
s∈[0,b]
(E|x(s)|2)12, x∈ C
0
υ
Lemma 3.4. [6]Assume thatx∈ Cυ0, then for allt∈J,xt∈ Cυ, Moreover,
l(E|x(t)|2)12 ≤ kxtk
Cυ ≤l sup
s∈[0,t]
(E|x(s)|2)12 +kx0k
Cυ
wherel=R0
−∞υ(s)ds <∞
The main object of this paper is to explain and prove the following theorem.
Theorem 3.1. Assume that assumptions(H0)−(H5)hold.Then there exists a mild solution Proof Consider the mapΠ :Cυ0 → Cυ0 defined by
(Πx)(t) =
φ(t) t∈(−∞,0]
Sα(t)[φ(0)−µ(x)−h(0, φ)] +h(t, xt) +Rt 0(t−s)
α−1Tα(t−s)f(s, xs)ds
+Rt 0(t−s)
α−1T
α(t−s)
h Rs
−∞σ(s, τ, xτ)dW(τ)
i
ds t∈J
(3.7)
In what follows, we shall show that the operatorΠhas a fixed point, which is then a mild solution for system1.1.
Forφ∈ Cυ, define
˜
φ(t) =
φ(t) t∈(−∞,0]
Sα(t)φ(0) t∈J
(3.8)
Thenφ˜ ∈ Cυ0. Letx(t) = ˜φ(t) +z(t),−∞< t ≤b. It is easy to see thatxsatisfies 1.1 if and only if z satisfies z0= 0and
z(t) =Sα(t)h−µ( ˜φ+z)−h(0, φ)i+h(t,φ˜t+zt) +
Z t
0
(t−s)α−1Tα(t−s)f(s,φ˜s+zs)ds
+
Z t
0
(t−s)α−1Tα(t−s)
Z s
−∞
σ(s, τ,φ˜τ+zτ)dW(τ)
ds
Let
Cυ00={z∈ C
0
υ, z0= 0∈ Cυ}
For anyz∈ C00υ, we have
kzkb=kz0kCυ+ sup
s∈[0,b]
(Ekz(s)k2)12 = sup
s∈[0,b]
(Ekz(s)k2)12
Thus(Cυ00,k.kb)is a Banach space, set
Bq={z∈ C
00
υ,kzk
2
b≤q}, for someq≥0
then,Bq⊂ C
00
υ is uniformly bounded.
then, for each q,Bq is clearly a bounded closed convex set inC
00
. Forz∈Bq, from Lemma3.3, we have
zt+ ˜φt
2
Cυ
≤2(kztk
2 Cυ+
˜
φt
2
Cυ
)
≤4(l2 sup s∈[0,t]
Ekz(s)k2+kz0k 2 Cυ+l
2 sup
s∈[0,t]
E
˜
φ(s)
2
+
˜
φ0 2
Cυ
)
≤4l2(q+M2Ekφ(0)k2
H) + 4kφk
2 Cυ
= `q
(Φz)(t) =
0 t∈(−∞,0]
Sα(t)[−µ( ˜φ+z)−h(0, φ)] +h(t,φ˜t+zt) +R t
0(t−s)
α−1Tα(t−s)f(s,φ˜s+zs)ds +Rt
0(t−s)
α−1T
α(t−s)
h Rs
−∞σ(s, τ,φ˜τ+zτ)dW(τ)
i
ds t∈J
Observe thatΦis well defined onBqfor eachq >0.
Now we will show that the operator Φhas a fixed point onBq, which implies that E.q 1.1 has a mild
solution. To this end, we decomposeΦasΦ = Φ1+ Φ2, where the operatorsΦ1andΦ2are defined onBq,
respectively, by
(Φ1z)(t) =Sα(t)[−µ( ˜φ+z)−h(0, φ)] +h(t,φ˜t+zt)
(Φ2z)(t) = Z t
0
(t−s)α−1Tα(t−s)f(s,φ˜s+zs)ds
+
Z t
0
(t−s)α−1Tα(t−s)
Z s
−∞
σ(s, τ,φ˜τ+zτ)dW(τ)
ds
Thus, the theorem follows from the next theorem
Theorem 3.2. If assumption(H1)−(H5)hold, thenΦ1is a contraction andΦ2is completely continuous. ProofTo prove thatΦ1is a contraction onC
00
υ, we takeu, v∈ C
00
υ. Then for eacht∈Jwe have
EkΦ1u(t)−Φ1v(t)k2H≤2E
Sα(t)(µ( ˜φ+u)−µ( ˜φ+v)) 2
H
+ 2E
h(t,
˜
φt+ut)−h(t,φ˜t+vt)
2
H
≤2M2Mµku−vk
2
Cυ+ 2Mhkut−vtk 2 Cυ
≤2(M2Mµ+Mh)kut−vtk
2 Cυ
≤2(M2Mµ+Mh) [2l2 sup
s∈[0,t]
Eku(s)−v(s)k2+ 2ku0k 2
Cυ+ 2kv0k 2 Cυ]
≤4l2(M2Mµ+Mh)Eku(s)−v(s)k2
≤ sup s∈[0,b]
L0Eku(s)−v(s)k2
where we have used the fact thatku0k 2
Cυ = 0,kv0k 2 Cυ = 0.
Thus,
kΦ1u−Φ1vk ≤L0ku−vk
and by assumption0≤L0≤1it is clear thatΦ1is contraction.
Now, we show that the operatorΦ2is completely continuous, firstly we prove thatΦ2:C 00
h → C
00
h is
contin-uous.
Let{zn(t)}∞
n=0,withzn → zinC 00
h. Then, there is a numberq ≥0such that|zn(t)| ≤ q, for all n and a.e.
t∈J. Soz(n)∈Bqandz∈Bq.
f(t, zt(n)+ ˜φt)→f(t, zt+ ˜φt)
fort∈J, and since
E [f(t, z
(n)
t + ˜φt)−f(t, zt+ ˜φt)]
2
≤2Mq0(t)
E
[σ(s, τ, z (n)
τ + ˜φτ)−σ(s, τ, zτ(n)+ ˜φτ)]
2
≤2m(t)Mσ(q0)
By the dominated convergence theorem we obtain continuity ofΦ2
E
Φz
(n)
t −Φtz
2
≤2 sup t∈J
E Z t 0
(t−s)α−1Tα(t−s)[f(t, zs(n))−f(t, zs)]ds
2
+ 2bsup t∈J
E Z t 0
(t−s)α−1Tα(t−s)
Z s
−∞
[σ(s, τ, z(τn))−σ(s, τ, zτ)]dw(τ)
ds 2 ≤2 M α Γ(1 +α)
2b2α
α2 Z t
0
E [f(t, z
(n)
s )−f(t, zs)]
2
ds
+ 2b
M
α Γ(1 +α)
2b2α
α2 Z t 0 E Z s −∞
[σ(s, τ, zτ(n))−σ(s, τ, zτ)]dw(τ)
ds 2
→0asn→ ∞
Next, we prove thatΦ2maps bounded sets into bounded sets inC 00
υ.
For eachz∈Bqfrom[3.4], we have
zt+ ˜φt
2
Cυ
≤4l2(q+M2Ekφ(0)k2H) + 4kφk2C
υ =q 0
EkΦ2z(t)k2H≤2E (t−s)
α−1Tα(t−s)f(s,φ˜
s+zs)
2
H
+ 2E
(t−s)α−1Tα(t−s)[
Z s
−∞
σ(s, τ,φ˜τ+zτ)dW(τ)]ds
2 H ≤2 M α Γ(1 +α)
2bα
α
Z t
0
(t−s)α−1Mf(1 +
˜
φs+zs
2 Cυ )ds + M α Γ(1 +α)
2bα
α
Z t
0
(t−s)α−1(2Mk+ 2T r(Q)m(s)Mσ(
˜
φs+zs
2 Cυ )ds. ≤2 M α Γ(1 +α)
2b2α
α2Mf(1 +q 0
)
+ 2
M
α Γ(1 +α)
2b2α
α2(Mk+T r(Q)Mσ(q 0
) sup t∈J
m(s))
≤r
Which implies that for eachz∈Bq,|Φ2zk 2
b ≤r.
Next, we establish the compactness ofΦ2. We employ the Arzela-Ascoli theorem to show the setV(t) =
{(Φ2z)(t), z∈Bq}is relatively compact inH. Le0< t≤bbe fixed andbe a real number satisfying0< ≤t.
(Φ,δ2 z)(t) =α
Z t−
0
Z ∞
δ
θ(t−s)α−1ηα(θ))S((t−s)αθ)f(s,φ˜s+zs)ds
+α
Z t−
0
Z ∞
δ
θ(t−s)α−1ηα(θ))S((t−s)αθ)[
Z s
−∞
σ(s, τ,φ˜τ+zτ)dW(τ)]ds
=S(αδ)α
Z t−
0
Z ∞
δ
θ(t−s)α−1ηα(θ))S((t−s)θ−αδ)f(s,φ˜s+zs)ds
+S(αδ)α
Z t−
0
Z ∞
δ
θ(t−s)α−1ηα(θ))S((t−s)θ−αδ)[
Z s
−∞
σ(s, τ,φ˜τ+zτ)dW(τ)]ds
SinceS(t), t > 0, is a compact operator, the setV,δ = {Φ ,δ
2 (t), z ∈ Bq}is relatively compact inHfor every
∈(0, t),δ >0. Moreover, for eachz∈Bq, we have
E
(Φ2z)(t)−(Φ
,δ
2 z)(t) 2
H
≤4α2E
Z t 0 Z δ 0
θ(t−s)α−1ηα(θ)S((t−s)αθ)f(s, zs+ ˜φs)dtθds
2 H
+ 4α2E
Z t t− Z ∞ δ
θ(t−s)α−1ηα(θ)S((t−s)αθ)f(s, zs+ ˜φs)dθds
2 H
+ 4α2E
Z t 0 Z δ 0
θ(t−s)α−1ηα(θ)S((t−s)αθ)
Z s
−∞
σ(s, τ, zτ+ ˜φτ)dW(τ)
dθds 2 H
+ 4α2E
Z t t− Z δ ∞
θ(t−s)α−1ηα(θ)S((t−s)αθ)
Z s
−∞
σ(s, τ, zτ+ ˜φτ)dW(τ)
dθds 2 H
≤4M2b2αMf(1 +q
0
)
Z δ
0
θηα(θ)dθ
!2
+4M
22αM
f(1 +q
0
) Γ2(1 +α)
+ 4αM2bα
Z t
0
(t−s)α−1(2Mk+ 2T r(Q)Mσ(q
0
)m(s))ds
Z δ
0
θηα(θ)dθ
!2
+ 4αM
2α
Γ2(1 +α) Z t
t−
(t−s)α−1(2Mk+ 2T r(Q)Mσ(q
0
)m(s))ds
where we have used the equality (see [22, 29])
Z ∞
0
θςηα(θ) = Γ(1 +ς) Γ(1 +ας)
We see that for eachz∈Bq
E
(Φ2z)(t)−(Φ
,δ 2 ) 2 H
→0as+→0, δ−→0.
Since the right-hand side of the above inequality can be made arbitrarily small, there is relatively compact V,δarbitrarily close to the setV(t). Hence, the setV(t)is relatively compact inBq. It remains to showt hatΦ2
maps is bounded set into equicontinuous sets ofCυ00.
with|s1−s2|< δ. Forz∈Bq, we have
EkΦ2z(t+h)−Φ2z(t)k 2
H
≤6E
Z t 0
[(t+h−s)α−1−(t−s)α−1]Tα(t+h−s)f(s,φ˜s+zs)ds
2 H
+ 6E
Z t+h
t
(t+h−s)α−1Tα(t+h−s)f(s,φ˜s+zs)ds
2 H
+ 6E
Z t 0
(t−s)α−1[Tα(t+h−s)−Tα(t−s)]f(s,φ˜s+zs)ds
2 H
+ 6E
Z t 0
[(t+h−s)α−1−(t−s)α−1]Tα(t+h−s)[
Z s
−∞
σ(s, τ,φ˜τ+zτ)dW(τ)ds]
2 H
+ 6E
Z t+h
t
(t+h−s)α−1Tα(t+h−s)[
Z s
−∞
σ(s, τ,φ˜τ+zτ)dW(τ)ds]
2 H .
+ 6E
Z t 0
(t−s)α−1[Tα(t+h−s)−Tα(t−s)][
Z s
−∞
σ(s, τ,φ˜τ+zτ)dW(τ)ds]
2 H ≤6 M α Γ(1 +α)
2Z t
0
(t+h−s)α−1−(t−s)α−1 2
Mf(1 +q
0
)ds
+ 6
M
α Γ(1 +α)
2Z t+h
t
(t+h−s)α−1 2
Mf(1 +q
0
)ds
+ 62
Z t
0
(t−s)α−1 2
Mf(1 +q
0
)ds+ 6
M
α Γ(1 +α)
2Z t
0
(t+h−s)α−1−(t−s)α−1 2
×(2Mk+ 2T r(Q)m(s)Mσ(q
0
))ds
+ 6
Mα Γ(1 +α)
2Z t
0
(t+h−s)α−1 2
(2Mk+ 2T r(Q)m(s)Mσ(q
0
))ds
+ 62
Z t
0
(t−s)α−1
(2Mk+ 2T r(Q)m(s)Mσ(q
0
))ds
It is known that the compactness ofTα(t), t >0implies the continuity in the uniform operator topology.
Therefore, forsufficiently small, the right-hand side of the above inequality tends to zero ash→ 0. Thus, the set{Φ2z, z∈Bq}is equicontinuous.
This completes the proof thatΦ2is completely continuous.
To apply the Krasnoselski-Schaefer theorem, it remains to show that the set
G={x∈H:λΦ1(
x
λ) +λΦ2x=x}is bounded forλ∈(0,1) .
We consider the following nonlinear operator equation,
x(t) =λ(Sα(t)[φ(0)−µ(x)−h(0, φ)]) +λh(t, xt) +λ
Z t
0
(t−s)α−1Tα(t−s)f(s, xs)ds
+λ
Z t
0
(t−s)α−1Tα(t−s)
Z s
−∞
σ(s, τ, xτ)dW(τ)
Ekx(t)k2≤EkSα(t)(φ(0)−µ(x)−h(0, φ))k2
H+ 4kh(t, xt)k
2
H
+ 4E
Z t
0
(t−s)α−1Tα(t−s)f(s, xs)ds
2
H
+ 4E
Z t
0
(t−s)α−1Tα(t−s)
Z s
−∞
σ(s, τ, xτ)dW(τ)
ds
2
H
≤12M2(C1+C2+Mµ) + 12M2(1 +kxk
2 Cυ)
+ 4
M
α Γ(1 +α)
2b2α
α2Mf(1 +kxk 2 Cυ)
+ 4b
M
α Γ(1 +α)
2bα
α
Z t
0
(t−s)α−1(2Mk+ 2T r(Q)m(s)Mσ(kxsk
2 Cυ)ds
Now, we consider the functionνdefined by
ϑ(t) = sup{Ekx(s)k2,0≤s≤t},0≤t≤b From lemma [3.4] and the above inequality, we have
Ekx(t)k2= 2kφk2C
υ+ 2l 2 sup
0≤s≤t
(Ekx(s)k2)
Therefore, we get
ϑ(t)≤2kφk2C
υ+ 2l
2{F¯+ 12M2ϑ(t) + 4
M
α Γ(1 +α)
2b2α
α2Mfϑ(t)
+ 8b
M
α Γ(1 +α)
2bα
α
Z t
0
(t−s)α−1T r(Q)m(s)Mσ(ϑ(s))ds}
whereF¯is given in (3.6). Thus, we have
ϑ(t)≤K1+K2 Z t
0
ϑ(s)
(t−s)1−αds+K3
Z t
0
m(s)Mσ(ϑ(s))ds
whereK1, K2, K3are given in (3.5). By Lemma [2.2], we have
ϑ≤B0(K1+K3 Z t
0
m(s)Mσ(ϑ(s))ds)
Where
B0=eK n
2(Γ(α))
nbnα/Γ(nα)
n−1 X
j=0
K
2bα
α
j
Denoting byν(t)the right hand side of the last inequality, we haveν(0) =B0K1
`
ν(t)≤B0K3m(t)Mσϑ(t) `
ν(t)≤B0K3m(t)Mσ(ϑ(t))
This implies
Z ν(t)
ν(0)
ds Mσ(s) ≤
Z b
0
π(s)ds <
Z ∞
B0K1
ds Mσ(s)
This inequality implies that there is a constantρsuch thatν(t) ≤ ρ,t ∈ J and henceϑ(t) ≤ ρ, t ∈ J. Furthermore, we getkxtk
2
Cυ ≤ϑ(t)≤ν(t)≤ρ,t ∈J, whereρdepends only on b and on the functionsπ(s)
andMσ(s).
Proof. Let us take the set
D(Φ) ={z∈ Cυ00:z=λΦ1(
z
x) +λΦ2zfor someλ∈[0,1]} (3.9) Then, for anyz∈D(Φ), we have by theorem ... thatkxk2C
υ ≤K, t∈J, and hence
kzk2b =kz0k 2
Cυ+ sup{Ekz(t)k 2
; 0≤t≤b}
= sup{Ekz(t)k2: 0≤t≤b}
≤sup{Ekx(t)k2: 0≤t≤b}+ sup{E
˜
φ(t) 2
: 0≤t≤b} ≤sup{l−kx(t)k2C
υ : 0≤t≤b}+ sup{ksα(t)φ(0)k: 0≤t≤b}
≤l−ρ+M1kφ(0)k 2
This implies that D is bounded on J. Consequently by Lemma 2.1 , the operatorΦhas a fixed pointz∈ Ch00. So Eq.(1.1) has a mild solution.Theorem is proved.
Example 3.1. As an application of the above result,consider the following fractional order neutral stochastic partial differential system with non local conditions and infinite delay in Hilbert space.
cDα
t[z(t, x)−
Rt −∞e
4(s−t)z(s, x)ds] = ∂2
∂x2[z(t, x)−
Rt −∞e
4(s−t)z(s, x)ds] +η(t, x)
+R0
−∞ˆa(s) sinz(t+s, x)ds+ Rt
−∞ Rt
−∞σ(t, x, s−t)dsdβ(s, x) t∈J = [0, b]
z(t,0) =z(t, π) = 0 t∈J
z(0, x) +Rπ
0 k1(x, y)z(t, y)dy=x0=ϕ(t, x) t∈(−∞,0],
(3.10)
WherecDαis a Caputo fractional partial derivative of orderα∈ (0,1), andK
1(x, y)∈ H=L2([0, π]×[0, π])and
R0
−∞|ˆa(s)|ds <+∞.β(t)is a one-dimensional standard Wiener process on filtered probability space(Ω,F,(Ft)t≥0,P).
To rewrite this system into the abstract form (1.1), let H = L2([0, π])with the normk.k. Define A : H → Hby
A(t)z=z00with the domain
D(A) =nz∈H:z, z0 are absolutely continuous, z00∈H, z(0) =z(π) = 0o
It is well known that A generates a strongly continuous semigroup T(.),which is compact,analytic and self adjoint. Then
Az=
∞ X
n=1
n2hz, znizn, z∈D(A)
where zn(s) =qπ2sin(ns), n= 1,2, ....is the orthonormal set of eigenvector of A. It is well known that A is thein infinitesimal generator of ananalytic semigroup T(t)in H and is given by
T(t)z=
∞ X
n=1
e−n2thz, znizn
Then the operatorA−1
2 is given by
A−12z=
∞ X
n=1
nhz, znizn
on the spaceD(A−1
2) ={z(.)∈H:
∞ X
n=1
Now, we present a specialCυspace. Letϑ(s) =e2s, s <0, thenl=R
0
−∞ϑ(s)ds= 1 2.
Let
kϕkC
υ = Z 0
−∞
h(s) sup s≤θ≤0
Ekϕ(θ)k2
1 2
ds
Then(Cυ,k.kCυ)is a Banach space.
For(t, ϕ) ∈ J × Cυ where ϕ(θ)(x) = ϕ(θ, x),(θ, x) ∈ (−∞,0]×[0, π], and define the Lipschitz continuous
functionsh, f :J× Cυ→H,σ:J× Cυ→LQ(H), for the infinite delay as follows
h(t, ϕ)(x) =
Z 0
−∞
e−4θϕ(θ)(x)dθ
f(t, ϕ)(x) =
Z 0
−∞
ˆ
a(θ) sin(ϕ(θ)(x))dθ
σ(t, ϕ)(x) =
Z 0
−∞
ς(t, x, θ)σ(ϕ(θ)(x))dθ
Then, the equation (3.10) can be rewritten as the abstract form as the system 1.1. Thus, under the appropriate condition so the functions h,f, andσare satisfies the hypotheses(H1)−(H5). All conditions of the Theorem 3.2 are
satisfied, therefore the system (3.10) has a mild solution.
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Received: July 1, 2014;Accepted: October 29, 2014
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