Existence of S p Almost Periodic Mild Solutions- 123docz.net

6.3 Existence Results Through the Schauder Fixed Point Theorem

6.3.2 Existence of S p Almost Periodic Mild Solutions

In this subsection, we introduce and develop another notion of almost periodicity known as the concept of Stepanov almost periodicity. This notion is weaker thanp- th almost periodicity. Basic results on Stepanov almost periodic processes will be, subsequently, utilized to study the existence and uniqueness of Stepanov almost periodic solutions to the nonautonomous differential equations, (5.3), where(A(t))t∈R

is a family of closed linear operators onLp(Ω;H)satisfying Acquistapace–Terreni conditions, and the forcing termsF,Gare Stepanov almost periodic.

6.3.2.1 SpAlmost Periodic Processes

Definition 6.3.The Bochner transformXb(t,s),t∈R,s∈[0,1], of a stochastic pro- cessX:R→Lq(Ω;B)is defined by

Xb(t,s):=X(t+s).

Remark 6.2.A stochastic processZ(t,s),t∈R,s∈[0,1], is the Bochner transform of a certain stochastic processX(t),

Z(t,s) =Xb(t,s), if and only if

Z(t+τ,s−τ) =Z(s,t) for allt∈R,s∈[0,1], andτ∈[s−1,s].

Definition 6.4.Let p,q≥1. The space BSp(Lq(Ω;B))of all Stepanov bounded stochastic processes consists of all stochastic processes X on R with values in Lq(Ω;B)such thatXb∈L∞

R;Lp((0,1),Lq(Ω;B))

. This is a Banach space with the norm

kXkSp=kXbkL∞(R,Lp)=sup

t∈R

Z t+1 t

E X(τ)

pdτ 1/p

Definition 6.5.Let p,q≥1. A stochastic process X ∈BSp(Lq(Ω;B)) is called

p b∈AP R;Lp((0,1),Lq(Ω;B))

, that is, for eachε>0 there existsl(ε)>0 such that any interval of lengthl(ε)contains at least a numberτfor which

sup

t∈R Z t+1

X(s+τ)−X(s)

pds<ε.

The collection of such functions will be denoted bySpAP(R;Lq(Ω;B)).

Throughout this section, we supposep=q.

The proof of the next theorem is straightforward and hence omitted.

Theorem 6.5.If X:R→Lp(Ω;B)is a p-th mean almost periodic stochastic process, then X is Spalmost periodic, that is, AP(R;Lp(Ω;B))⊂SpAP(R;Lp(Ω;B)).

Lemma 6.13.Let(Xn(t))n∈N be a sequence of Spalmost periodic stochastic processes such that

Stepanov almost periodic (orS almost periodic) ifX

sup

t∈R Z t+1

Xn(s)−X(s)

pds→0, as n→∞.

Then X∈SpAP(R;Lp(Ω;B)).

Proof. For eachε>0, there existsN(ε)such that Z t+1

EkXn(s)−X(s)kpds≤ ε

3p, ∀t∈R, n≥N(ε).

From theSpalmost periodicity of XN(t), there existsl(ε)>0 such that every interval of lengthl(ε)contains a numberτwith the following property:

Z t+1 t

XN(s+τ)−XN(s)

pds< ε

3p, ∀t∈R. Now

X(t+τ)−X(t)

p≤3p−1E

X(t+τ)−XN(t+τ)

p+3p−1E

XN(t+τ)−XN(t)

+3p−1E

XN(t)−X(t)

and hence

sup

t∈R Z t+1

X(s+τ)−X(s)

pds<ε 3+ε

3+ε 3 =ε, which completes the proof.

Similarly,

Lemma 6.14.Let(Xn(t))n∈Nbe a sequence of p-th mean almost periodic stochastic processes such that

sup

s∈R

Xn(s)−X(s)

p→0, as n→∞. Then X∈AP(R;Lp(Ω;B)).

Using the inclusionSpAP(R;Lp(Ω;B))⊂BSp(R;Lp(Ω;B))and the fact that (BSp(R;Lp(Ω;B)),

Sp)is a Banach space, one can easily see that the next theorem is a straightforward consequence of Lemma 6.13.

Theorem 6.6.The space SpAP(R;Lp(Ω;B))equipped with the norm X

Sp=sup

t∈R

Z t+1 t

E X(s)

pds 1/p

is a Banach space.

Let (B1,k ã kB1) and(B2,k ã kB2)be Banach spaces and let Lp(Ω;B1) and Lp(Ω;B2)be their correspondingLpspaces, respectively.

Definition 6.6.A functionF:R×Lp(Ω;B1)→Lp(Ω;B2),(t,Y)7→F(t,Y)is said to beSpalmost periodic int∈Runiformly inY ∈KwhereK⊂Lp(Ω;B1)is a compact if for anyε>0, there existsl(ε,K)>0 such that any interval of length l(ε,K)contains at least a numberτfor which

sup

t∈R Z t+1

F(s+τ,Y)−F(s,Y)

B2ds<ε for each stochastic processY:R→K.

Theorem 6.7.Let F:R×Lp(Ω;B1)→Lp(Ω;B2),(t,Y)7→F(t,Y)be an Spal- most periodic process in t∈Runiformly in Y∈K, whereK⊂Lp(Ω;B1)is compact. Suppose that F is Lipschitz in the following sense:

F(t,Y)−F(t,Z)

B2≤ME Y−Z

p B1

for all Y,Z∈Lp(Ω;B1)and for each t∈R, where M>0. Then for any Spalmost periodic processΦ:R→Lp(Ω;B1), the stochastic process t7→F(t,Φ(t))is Sp almost periodic.

Proof. The proof is left as an exercise.

Theorem 6.8.Let F:R×Lp(Ω;B1)→Lp(Ω;B2),(t,Y)7→F(t,Y)be an Spal- most periodic process in t∈Runiformly in Y ∈K, where K⊂Lp(Ω;B1)is any compact subset. Suppose that F(t,ã)is uniformly continuous on bounded subsets K0⊂Lp(Ω;B1)in the following sense: for allε>0there existsδε>0such that X,Y ∈K0andE

X−Y

1<δε, then E

F(t,Y)−F(t,Z)

2<ε, ∀t∈R.

Then for any Spalmost periodic processΦ:R→Lp(Ω;B1), the stochastic process t7→F(t,Φ(t))is Spalmost periodic.

Proof. SinceΦ :R→Lp(Ω;B1)is an Sp almost periodic process, for allε>0 there existslε>0 such that every interval of lengthlε >0 contains aτ with the property that

Z t+1 t

Φ(s+τ)−Φ(s)

1ds<ε, ∀t∈R. (6.10) In addition,Φ:R→Lp(Ω;B1)is bounded, that is, sup

t∈R

E Φ(t)

1<∞. LetK00⊂ Lp(Ω;B1)be a bounded subset such thatΦ(t)∈K00for allt∈R.

Now

Z t+1 t

F(s+τ,Φ(s+τ))−F(s,Φ(s))

p 2ds

≤2p−1 Z t+1

F(s+τ,Φ(s+τ))−F(s+τ,Φ(s))

p 2ds +2p−1

Z t+1 t

F(s+τ,Φ(s))−F(s,Φ(s))

p 2ds.

Taking into account Eq. (6.10) (takeδε=ε) and using the uniform continuity ofF on bounded subsets ofLp(Ω;B1)it follows that

sup

t∈R Z t+1

F(s+τ,Φ(s+τ))−F(s+τ,Φ(s))

p 2ds< ε

2p. (6.11) Similarly, using theSpalmost periodicity ofFit follows that

sup

t∈R Z t+1

F(s+τ,Φ(s))−F(s,Φ(s))

p 2ds< ε

2p. (6.12)

Combining Eqs. (6.11) and (6.12) one obtains that sup

t∈R Z t+1

F(s+τ,Φ(s+τ))−F(s,Φ(s))

p 2ds<ε, and hence the stochastic process t7→F(t,Φ(t))isSpalmost periodic.

6.3.2.2 Existence ofSpAlmost Periodic Mild Solutions

To studySpalmost periodic solutions to Eq. (5.3), we first study the existence ofSp almost periodic solutions to the stochastic nonautonomous differential equations

dX(t) =A(t)X(t)dt+f(t)dt+g(t)dW(t), t∈R, (6.13) whereA(t)fort∈Ris a family of closed linear operators where the family of linear operatorA(t):D(A(t))⊂Lp(Ω;H)→Lp(Ω;H)satisfies the above-mentioned assumptions and the forcing terms f ∈SpAP(R,Lp(Ω;H))∩C(R,Lp(Ω;H))and g∈SpAP(R,Lp(Ω;L02))∩C(R,Lp(Ω;L02)). In addition to (5H)3 and(6H)6, we require the following assumptions.

(6H)12 R(ζ,A(ã))∈SpAP(Lp(Ω;H)).

(6H)13 The functionF:R×Lp(Ω,H)→Lp(Ω,H)isSpalmost periodic in the first variable uniformly in the second variable. Furthermore,X→F(t,X)is uniformly continuous on any bounded subsetOofLp(Ω,H)for eacht∈R. Finally,

sup

t∈R

F(t,X)

p≤M1

X ∞

whereM1:R+→R+is a continuous, monotone increasing function satisfying

r→∞lim M1(r)

r =0.

(6H)14 The functionG:R×Lp(Ω,H)→Lp(Ω,L02)isSpalmost periodic in the first variable uniformly in the second variable. Furthermore,X→G(t,X)is uniformly continuous on any bounded subsetO0ofLp(Ω,H)for eacht∈R. Finally,

sup

t∈R

G(t,X)

p≤M2

X ∞

whereM2:R+→R+is a continuous, monotone increasing function satisfying

r→∞lim M2(r)

r =0.

Theorem 6.9.Assume that(2.38),(2.39), and(5H)3hold. Then(6.13)has a unique bounded solution X∈SpAP(R,Lp(Ω;H)).

We need the following lemmas. For the proofs of Lemmas 6.15 and 6.16, see the proof of Theorem 6.10.

Lemma 6.15.Under the assumptions of Theorem 6.9, the integral defined by Xn(t) =

Z n n−1

U(t,t−ξ)f(t−ξ)dξ belongs to SpAP(R,Lp(Ω;H))for each for n=1,2, ....

Lemma 6.16.Under the assumptions of Theorem 6.9, the integral defined by Yn(t) =

Z n

n−1U(t,t−ξ)g(t−ξ)dW(ξ) belongs to SpAP(R,Lp(Ω;L02))for each for n=1,2, ....

Proof. (Theorem 6.9) By assumption there exist some constantsM,δ >0 such that kU(t,s)k ≤Me−δ(t−s) for every t≥s.

Let us first prove uniqueness. Assume thatX:R→Lp(Ω;H)is a bounded stochastic process that satisfies the homogeneous equation

dX(t) =A(t)X(t)dt, t∈R. (6.14) ThenX(t) =U(t,s)X(s)for anyt≥s. Hence

X(t)

≤MDe−δ(t−s)with X(s)

≤ Dfors∈Ralmost surely. Take a sequence of real numbers(sn)n∈Nsuch thatsn→

−∞asn→∞. For anyt∈Rfixed, one can find a subsequence(snk)k∈N⊂(sn)n∈N

such thatsnk<tfor allk=1,2, .... By lettingk→∞, we getX(t) =0 almost surely.

Now, ifX1,X2:R→Lp(Ω;H)are bounded solutions to Eq. (6.13), thenX = X1−X2is a bounded solution to Eq. (6.14). In view of the above,X=X1−X2=0 almost surely, that is,X1=X2almost surely.

Now let us investigate the existence. Consider for eachn=1,2, ..., the integrals Xn(t) =

Z n n−1

U(t,t−ξ)f(t−ξ)dξ and

Yn(t) = Z n

n−1

U(t,t−ξ)g(t−ξ)dW(ξ).

n p p

Moreover, note that Z t+1

E Xn(s)

pds≤ Z t+1

Z n

n−1U(s,s−ξ)f(s−ξ)dξ

≤Mp Z n

n−1e−pδ ξ Z t+1

f(s−ξ)

pds

dξ

≤Mp f

p Sp

Z n n−1

e−pδ ξdξ

≤ Mp pδ

Spe−pδn(epδ+1). Since the series

pδ (epδ+1)

∞ n=2∑

e−pδn

is convergent, it follows from the Weierstrass test that the sequence of partial sums defined by

Ln(t):=

∑

k=1

Xk(t) converges in sense of the norm

Sp uniformly onR. Now let

l(t):=

∞ n=1∑

Xn(t) for eacht∈R.

Observe that

l(t) = Z t

−∞

U(t,ξ)f(ξ)dξ, t∈R, and hencel∈C(R;Lp(Ω,H)).

Similarly, the sequenceYnbelongs toSpAP(R,Lp(Ω;L02)). Moreover, note that First, we know by Lemma 6.15 that the sequenceX belongs toS AP(R,L (Ω;H)).

Z t+1 t

E Yn(s)

pds≤Cp Z t+1

EhZ n n−1

U(s,s−ξ)

g(s−ξ)

2dξip/2

≤CpMp Z n

n−1e−pδ ξ Z t+1

g(s−ξ)

pds

dξ

≤CpMp pδ

Spe−pδn(epδ+1).

Proceeding as before we can show easily that the sequence of partial sums defined by

Mn(t):=

n k=1∑

Yk(t) converges in sense of the norm

Sp uniformly onR. Now let

m(t):=

∞

∑

n=1

Yn(t) for eacht∈R.

Observe that

m(t) = Z t

−∞

U(t,ξ)g(ξ)dW(ξ), t∈R, and hencem∈C(R,Lp(Ω;L02)).

Setting

X(t) = Z t

−∞

U(t,ξ)f(ξ)dξ+ Z t

−∞

U(t,ξ)g(ξ)dW(ξ), one can easily see thatXis a bounded solution to Eq. (6.13). Moreover,

Z t+1 t

X(s)−(Ln(s) +Mn(s))

pds→0 as n→∞

uniformly int∈R, and hence using Lemma 6.13, it follows thatX is aSpalmost periodic solution. In view of the above, it follows thatX is the only boundedSp almost periodic solution to Eq. (6.13).

Definition 6.7.AnFt-progressively process{X(t)}t∈Ris called a mild solution of (5.3)onRif

X(t) =U(t,s)X(s) + Z t

U(t,σ)F(σ,X(σ))dσ (6.15) +

Z t s

U(t,σ)G(σ,X(σ))dW(σ) for allt≥sfor eachs∈R.

Now, define the nonlinear integral operatorsΓ onSpAP(R,Lp(Ω,H))as follows:

Γ1X(t) =Γ1X(t) +Γ2X(t) where

(Γ1X)(t):=

Z t

−∞

U(t,s)F(s,X(s))ds and

(Γ2X)(t):=

Z t

−∞

U(t,s)G(s,X(s))dW(s). Throughout this section we assume thatα ∈ 0,12−1p

if p>2 andα ∈ 0,12 if p=2. Moreover, we suppose that

2β>α+1.

Lemma 6.17.Under assumptions(5H)3,(6H)6,(6H)9,(6H)13, and(6H)14, the mappingsΓi(i=1,2):BC(R,Lp(Ω,H))→BC(R,Lp(Ω,Hα))are well defined and continuous.

Proof. The proof follows along the same lines as that of Lemma 6.7 and hence is omitted.

Lemma 6.18.Under assumptions(5H)3,(6H)6,(6H)9,(6H)13,(6H)14, the integral operatorΓi(i=1,2)maps SpAP R,Lp(Ω,H)

into itself.

Proof. Consider for eachn=1,2, . . ., the integral Rn(t) =

Z n n−1

U(t,t−ξ)f(t−ξ)dξ+ Z n

n−1

U(t,t−ξ)g(t−ξ)dW(ξ), wheref(σ) =F(σ,X(σ))andg(σ) =G(σ,X(σ)).

Set

Xn(t) = Z n

n−1

U(t,t−ξ)f(t−ξ)dξ and

Yn(t) = Z n

n−1

U(t,t−ξ)g(t−ξ)dW(ξ).

Let us first show thatXn(ã)isSpalmost periodic wheneverX is. Indeed, assuming thatXisSpalmost periodic and using(6H)13, Theorem 6.8, and Lemma 5.1, given ε>0, one can findl(ε)>0 such that any interval of lengthl(ε)contains at leastτ with the property that

U(t+τ,s+τ)−U(t,s)

≤εe−δ2(t−s) for allt−s≥ε, and

Z t+1 t

f(s+τ)−f(s)

pds<η(ε) for eacht∈R,whereη(ε)→0 asε→0.

For theSpalmost periodicity ofXn(ã), we need to consider two cases.

Case 1:n≥2 Z t+1

Xn(s+τ)−Xn(s)

pds

= Z t+1

Z n n−1

U(s+τ,s+τ−ξ)f(s+τ−ξ)dξ

− Z n

n−1

U(s,s−ξ)f(s−ξ)dξ

≤2p−1 Z t+1

t Z n

n−1

U(s+τ,s+τ−ξ)

f(s+τ−ξ)−f(s−ξ)

pdξds

+2p−1 Z t+1

t Z n

n−1

U(s+τ,s+τ−ξ)−U(s,s−ξ)

f(s−ξ)

pdξds

≤2p−1Mp Z t+1

t Z n

n−1e−pδ ξE

f(s+τ−ξ)−f(s−ξ)

pdξds

+2p−1εp Z t+1

t Z n

n−1

e−p2δ ξE

f(s−ξ)

pdξds

≤2p−1Mp Z n

n−1e−pδ ξ Z t+1

f(s+τ−ξ)−f(s−ξ)

pds

dξ +2p−1εp

Z n n−1e−p2δ ξ

Z t+1

f(s−ξ)

pds

dξ. Case 2:n=1

We have Z t+1

X1(s+τ)−X1(s)

pds

= Zt+1

Z 1

U(s+τ,s+τ−ξ)f(s+τ−ξ)dξ− Z 1

U(s,s−ξ)f(s−ξ)dξ

≤3p−1 Z t+1

Z 1 0

U(s+τ,s+τ−ξ)

f(s+τ−ξ)−f(s−ξ)

pdξds +3p−1

Zt+1

Z 1

U(s+τ,s+τ−ξ)−U(s,s−ξ)

f(s−ξ)

pdξds +3p−1

Zt+1 t

Z ε

U(s+τ,s+τ−ξ)−U(s,s−ξ)

f(s−ξ)

pdξds

≤3p−1Mp Z t+1

t Z 1

e−pδ ξE

f(s+τ−ξ)−f(s−ξ)

pdξds

+3p−1εp Z t+1

t Z 1

e−p2δ ξE

f(s−ξ)

pdξds

+6p−1Mp Z t+1

t Z ε

e−pδ ξE

f(s−ξ)

pdξds

≤3p−1Mp Z 1

e−pδ ξ Z t+1

f(s+τ−ξ)−f(s−ξ)

pds

dξ +3p−1εp

Z 1 ε

e−2pδ ξ Z t+1

f(s−ξ)

pds

dξ +6p−1Mp

Z ε 0

e−pδ ξ Z t+1

f(s−ξ)

pds

dξ which implies thatXn(ã)isSpalmost periodic.

Similarly, assuming thatXisSpalmost periodic and using(6H)14, Theorem 6.8, and Lemma 5.1, givenε>0, one can findl(ε)>0 such that any interval of length l(ε)contains at leastτwith the property that

U(t+τ,s+τ)−U(t,s)

≤εe−δ2(t−s) for allt−s≥ε, and

Z t+1 t

g(s+τ)−g(s)

L02ds<η(ε) for eacht∈R,whereη(ε)→0 asε→0.

The next step consists in proving theSpalmost periodicity ofYn(ã). Here again, we need to consider two cases.

Case 1:n≥2

Forp>2, we have Z t+1

Yn(s+τ)−Yn(s)

pds

= Z t+1

Z n n−1

U(s+τ,s+τ−ξ)g(s+τ−ξ)dW(ξ)

− Z n

n−1U(s,s−ξ)g(s−ξ)dW(ξ)

≤2p−1Cp

Z t+1 t

E hZ n

n−1

U(s+τ,s+τ−ξ)

g(s+τ−ξ)−g(s−ξ)

2 L02dξ

ip/2

ds +2p−1Cp

Z t+1 t

EhZ n

n−1

U(s+τ,s+τ−ξ)−U(s,s−ξ)

g(s−ξ)

2 L02dξ

ip/2

≤2p−1Mp Z t+1

EhZ n n−1

e−2δ ξ

g(s+τ−ξ)−g(s−ξ)

2 L02dξ

ip/2

ds +2p−1εp

Z t+1 t

EhZ n

n−1e−δ ξE

g(s−ξ)

L02dξip/2

≤2p−1Mp Z n

n−1

e−pδ ξ Z t+1

g(s+τ−ξ)−g(s−ξ)

p L02ds

dξ +2p−1εp

Z n n−1

e−p2δ ξ Z t+1

g(s−ξ)

p L02ds

dξ.

Forp=2, a simple computation using It ˆo isometry identity shows that Z t+1

Yn(s+τ)−Yn(s)

2ds

≤2M2 Z n

n−1e−2δ ξnZ t+1

g(s+τ−ξ)−g(s−ξ)

2 L02dso

dξ +2ε2

Z n n−1

e−δ ξnZ t+1

g(s−ξ)

2 L02

o . Case 2:n=1

Forp>2, we have Zt+1

Y1(s+τ)−Y1(s)

pds

= Z t+1

Z 1

U(s+τ,s+τ−ξ)g(s+τ−ξ)dW(ξ)

− Zn+1

U(s,s−ξ)g(s−ξ)dW(ξ)

≤3p−1Cp

Z t+1 t

EhZ 1 0

U(s+τ,s+τ−ξ)

g(s+τ−ξ)−g(s−ξ)

2 L02dξ

ip/2

ds +3p−1Cp

Z t+1

E hZ 1

U(s+τ,s+τ−ξ)−U(s,s−ξ)

2Ekg(s−ξ)k2

L02dξ ip/2

ds +3p−1Cp

Z t+1 t

EhZ ε 0

U(s+τ,s+τ−ξ)−U(s,s−ξ)

g(s−ξ)

2 L02dξ

ip/2

≤3p−1MpCp Zt+1

E hZ 1

e−2δ ξE

g(s+τ−ξ)−g(s−ξ)

2 L02dξ

ip/2

ds +3p−1εpCp

Z t+1 t

E hZ 1

e−δ ξE

g(s−ξ)

2 L02dξ

ip/2

ds +6p−1MpCp

Z t+1 t

E hZ ε

e−2δ ξE

g(s−ξ)

2 L02dξ

ip/2

≤3p−1MpCp Z 1

e−pδ ξ Zt+1

g(s+τ−ξ)−g(s−ξ)

p L02ds

dξ

+3p−1εpCp Z 1

e−2pδ ξ Z t+1

g(s−ξ)

p L02ds

dξ +6p−1MpCp

Z ε 0

e−pδ ξ Z t+1

g(s−ξ)

p L02ds

dξ. Forp=2, a simple calculation shows that

Z t+1 t

Y1(s+τ)−Y1(s)

2ds≤3M2 Z1

e−2δ ξnZt+1

g(s+τ−ξ)−g(s−ξ)

2 L02dso

dξ +3ε2

Z 1 ε

e−δ ξ nZt+1

g(s−ξ)

2 L02

o dξ +6M2

Zε

e−δ ξnZt+1

g(s−ξ)

2 L02

o dξ

which implies thatYn(ã)isSpalmost periodic.

Setting ΓX(t):=

Z t

−∞

U(t,σ)F(σ,X(σ))dσ+ Z t

−∞

U(t,σ)G(σ,X(σ))dW(σ) and proceeding as in the proof of Theorem 6.9, one can easily see that

Z t+1 t

X(s)−(Xn(s) +Yn(s))

pds→0 as n→∞

uniformly int∈R, and hence using Lemma 6.13, it follows thatΓXis anSpalmost periodic solution.

Letγ∈(0,1]and letBCγ R,Lp(Ω,Hα)

X∈BC R,Lp(Ω,Hα) :

X α,γ<

∞ o

, where

α,γ=sup

t∈R

h E

X(t)

p α

i1p

+γ sup

t,s∈R,s6=t

h E

X(t)−X(s)

p α

i1p t−s|γ . Clearly, the spaceBCγ R,Lp(Ω,Hα)

equipped with the norm ã

α,γis a Banach space, which is in fact the Banach space of all bounded continuous H¨older functions fromRtoLp(Ω,Hα)whose H¨older exponent isγ.

Lemma 6.19.Under assumptions(5H)3,(6H)6,(6H)9,(6H)13, and(6H)14, the mappingΓi(i=1,2)defined previously map bounded sets of BC R,Lp(Ω,H)

into bounded sets of BCγ(R,Lp(Ω,Hα))for some0<γ<1.

Proof. The proof is almost identical to that of Lemmas 6.9 and 6.10 and may be omitted.

i p

into bounded sets of BCγ(R,Lp(Ω,Hα))∩SpAP(R,Lp(Ω,H))for0<γ<α. Proof. The proof follows along the same lines as that of Lemma 6.9 and hence is omitted.

Similarly, the next lemma is a consequence of [80, Proposition 3.3]. Note in this context thatX=Lp(Ω,H)andY=Lp(Ω,Hα).

Lemma 6.21.For0<γ<α, BCγ(R,Lp(Ω,Hα))is compactly contained in BC(R,Lp(Ω,H)), that is, the canonical injection

id:BCγ(R,Lp(Ω,Hα))→BC(R,Lp(Ω,H)) is compact, which yields

id:BCγ(R,Lp(Ω,Hα))∩SpAP(R,Lp(Ω,H))→AP(R,Lp(Ω,H)) is compact, too.

Theorem 6.10.Suppose assumptions (5H)3, (6H)6, (6H)9, (6H)13, and (6H)14 hold, then the nonautonomous differential equation(5.3)has at least one Spalmost periodic solution.

Proof. Let us recall that in view of Lemmas 6.7 and 6.8, we have

Γ1+Γ2 X

α,∞≤d(β,δ) M1

X ∞

+M2

X ∞

and

Γ1+Γ2

X(t2)− Γ1+Γ2 X(t1)

p α

≤s(α,β,δ) M1

X ∞

+M2

X ∞

t2−t1

for allX∈BC(R,Lp(Ω,Hα)),t1,t2∈Rwitht16=t2, whered(β,δ)ands(α,β,δ) are positive constants. Consequently,X ∈BC(R,Lp(Ω,H))and

∞<Ryield (Γ1+Γ2)X∈BCγ(R,Lp(Ω,Hα))and

Γ1+Γ2 X

α,∞<R1where R1=c(α,β,δ)

M1(R) +M2(R)

. Since M(R)/R→0 as R→∞, and since E

p≤cE X

αfor allX∈Lp(Ω,Hα), it follows that there exists anr>0 such that for allR≥r, the following holds:

Γ1+Γ2

BSpAP(R,Lp(Ω,H))(0,R)

⊂BBCγ(R,Lp(Ω,Hα))∩BSpAP(R,Lp(Ω,H))(0,R). In view of the above, it follows that Γ1+Γ2

:D→Dis continuous and compact, whereDis the ball inSpAP(R,Lp(Ω,H))of radiusRwithR≥r. Using the Schauder fixed point it follows that Γ1+Γ2

has a fixed point, which is obviously ap-th mean almost periodic mild solution to Eq. (5.3).

Lemma 6.20.The integral operatorsΓ(i=1,2)map bounded sets of AP(Ω,L (Ω,H))

The next result is weaker than Theorem 6.10 although we require thatG be bounded in some sense.

Theorem 6.11.Under assumptions (5H)3, (6H)6, (6H)9, (6H)13, (6H)14, if we assume that there exists L>0 such that EkG(t,Y)kp

L02 ≤L for all t ∈R and Y ∈Lp(Ω;H), then Eq.(5.3)has a unique p-th mean almost periodic mild solution, which can be explicitly expressed as follows:

X(t) = Z t

−∞

U(t,σ)F(σ,X(σ))dσ+ Z t

−∞

U(t,σ)G(σ,X(σ))dW(σ) for each t∈R whenever K and K0are small enough.

Proof. We use the same notations as in the proof of Theorem 6.10. Let us first show that Xn(ã) is p-th mean almost periodic upon the Sp almost periodicity of f =F(ã,X(ã)). Indeed, assuming thatX isSpalmost periodic and using (6H)13, Theorem 6.8, and Lemma 5.1, givenε>0, one can find l(ε)>0 such that any interval of lengthl(ε)contains at leastτwith the property that

U(t+τ,s+τ)−U(t,s)

≤εe−δ2(t−s) for allt−s≥ε, and

Z t+1 t

f(s+τ)−f(s)

pds<η(ε) for eacht∈R,whereη(ε)→0 asε→0.

The next step consists in proving thep-th mean almost periodicity ofXn(ã). Here again, we need to consider two cases.

Case 1:n≥2 E

Xn(t+τ)−Xn(t)

Z n

n−1U(t+τ,t+τ−ξ)f(t+τ−ξ)dξ

− Z n

n−1

U(t,t−ξ)f(s−ξ)dξ

≤2p−1 Z n

n−1

U(t+τ,t+τ−ξ)

f(t+τ−ξ)−f(t−ξ)

pdξ +2p−1

Z n n−1

U(t+τ,t+τ−ξ)−U(t,t−ξ)

f(t−ξ)

pdξ

≤2p−1Mp Z n

n−1e−pδ ξE

f(t+τ−ξ)−f(t−ξ)

pdξ +2p−1εp

Z n n−1

e−p2δ ξE

f(t−ξ)

pdξ

≤2p−1Mp Z t−n

t−n+1

f(r+τ)−f(r)

pdr+2p−1εp Z t−n

t−n+1

E f(r)

pdr Case 2:n=1

X1(t+τ)−X1(t)

Z 1 0

U(t+τ,t+τ−ξ)f(t+τ−ξ)dξ− Z 1

U(t,t−ξ)f(t−ξ)dξ

≤3p−1E Z 1

U(t+τ,t+τ−ξ)

f(t+τ−ξ)−f(t−ξ) dξ

+3p−1E Z 1

U(t+τ,t+τ−ξ)−U(t,t−ξ)

f(t−ξ) dξ

+3p−1E Z ε

U(t+τ,t+τ−ξ)−U(t,t−ξ)

f(t−ξ) dξ

≤3p−1MpE Z 1

e−δ ξ

f(t+τ−ξ)−f(t−ξ) dξ

+3p−1εpE Z 1

e−δ2ξ

f(t−ξ) dξ

+6p−1MpE Z ε

e−δ ξ

f(t−ξ) dξ

. Now, using H¨older’s inequality, we have

≤3p−1Mp Z 1

e−pδ ξE

f(t+τ−ξ)−f(t−ξ)

pdξ

+3p−1εp Z 1

e−pδ2ξE

f(t−ξ)

pdξ

+6p−1Mp Z ε

e−pδ ξE

f(t−ξ)

pdξ

≤3p−1Mp Z t

t−1

f(r+τ)−f(r)

pdr

+3p−1εp Z t−ε

t−1

E f(r)

pdr+6p−1Mpε Z t

t−ε

E f(r)

pdr, which implies thatXn(ã)isp-th mean almost periodic.

Similarly, using(6H)14, Theorem 6.8, and Lemma 5.1, givenε>0, one can find l(ε)>0 such that any interval of lengthl(ε)contains at leastτwith the property that

U(t+τ,s+τ)−U(t,s)

≤εe−δ2(t−s) for allt−s≥ε, and

Z t+1 t

g(s+τ)−g(s)

p L02ds<η

for eacht∈R,whereη(ε)→0 asε→0. Moreover, there exists a positive constant L>0 such that

sup

σ∈R

E g(σ)

p L02 ≤L.

The next step consists in proving thep-th mean almost periodicity ofYn(ã).

Case 1:n≥2

Forp>2, we have E

Yn(t+τ)−Yn(t)

Z n n−1

U(t+τ,t+τ−ξ)g(s+τ−ξ)dW(ξ)−

Z n n−1

U(t,t−ξ)g(t−ξ)dW(ξ)

≤2p−1CpEhZ n

n−1

U(t+τ,t+τ−ξ)

g(t+τ−ξ)−g(t−ξ)

L02dξip/2

+2p−1CpEhZ n

n−1

U(t+τ,t+τ−ξ)−U(t,t−ξ)

g(t−ξ)

L02dξip/2

≤2p−1CpMpEhZ n

n−1e−2δ ξ

g(t+τ−ξ)−g(t−ξ)

2 L02dξ

ip/2

+2p−1CpεpEhZ n n−1

e−δ ξE

g(t−ξ)

L02dξip/2

≤2p−1Mp Z t−n+1

t−n

g(r+τ)−g(r)

L02dr+2p−1Cpεp Z t−n+1

t−n

E g(r)

p L02dr.

Forp=2, a simple calculation using It ˆo isometry identity shows E

Yn(t+τ)−Yn(t)

2≤2M2 Zt−n+1

t−n E

g(r+τ)−g(r)

L02dr+2ε2 Zt−n+1

t−n E

g(r)

2 L02dr.

Case 2:n=1

Forp>2, we have E

Y1(t+τ)−Y1(t)

Z 1 0

U(t+τ,t+τ−ξ)g(s+τ−ξ)dW(ξ)−

Z 1 0

U(t,t−ξ)g(t−ξ)dW(ξ)

≤3p−1CpEhZ 1

U(t+τ,t+τ−ξ)

g(t+τ−ξ)−g(t−ξ)

L02dξip/2

+3p−1Cp Z t+1

Z 1

+ Z ε

U(t+τ,t+τ−ξ)−U(t,t−ξ)

g(t−ξ)

2 L02dξip/2

≤3p−1CpMpEhZ 1

e−2δ ξ

g(t+τ−ξ)−g(t−ξ)

L02dξip/2

+3p−1CpεpEhZ 1

e−δ ξ

g(t−ξ)

L02dξip/2

+6p−1CpMpEhZ ε

e−2δ ξ

g(t−ξ)

L02dξip/2

≤3p−1CpMp Z 1

g(t+τ−ξ)−g(t−ξ)

p L02dξ +3p−1Cpεp

Z 1 ε

g(t−ξ)

L02dξ+6p−1CpMp Z ε

g(t−ξ)

p L02dξ

≤3p−1CpMp Z t

t−1

g(r+τ)−g(r)

p L02dr +3p−1Cpεp

Z t t−1

E g(r)

L02dr+6p−1CpMp Z ε

g(t−ξ)

p L02dξ

≤3p−1CpMp Z t

t−1

g(r+τ)−g(r)

p L02dr +3p−1Cpεp

Z t t−1

E g(r)

L02dr+6p−1εCpMpL.

Forp=2, we have E

Y1(t+τ)−Y1(t)

3M2 Z t

t−1

g(r+τ)−g(r)

2 L02dr +3ε2

Z t t−1

E g(r)

2 L02+6M2

Z ε 0

g(t−ξ)

2 L02dξ,

which implies thatYn(ã)isp-th mean almost periodic. Moreover, setting ΓX(t) =

Z t

−∞

U(t,σ)F(σ,X(σ))dσ+ Z t

−∞

U(t,σ)G(σ,X(σ))dW(σ) for eacht∈Rand proceeding as in the proofs of Theorems 6.4 and 6.10, one can easily see that

sup

s∈R

X(s)−(Xn(s) +Yn(s))

p→0 as n→∞

and it follows thatΓXis ap-th mean almost periodic solution to Eq. (5.3).

In view of the above, the nonlinear operatorΓ mapsAP(R;Lp(Ω;B))into itself.

Consequently, using the Schauder fixed-point principle it follows thatΓhas a unique fixed point {X1(t), t∈R}, which in fact is the only p-th mean almost periodic solution to Eq. (5.3).