SQUARE-MEAN ALMOST PERIODIC SOLUTIONS TO SOME CLASSES OF NONAUTONOMOUS STOCHASTIC EVOLUTION EQUATIONS WITH FINITE DELAY PAUL BEZANDRY AND TOKA DIAGANA

Abstract. In this paper, using the well-known Krasnoselskii fixed point theorem, we obtain the existence of a square-mean almost periodic solution to some classes of nonautonomous stochastic evolution equations with finite delay.

1. Introduction Let (H, k·k, h·, ·i) be a real separable Hilbert space and let (Ω, F, P) be a complete probability space equipped with a normal filtration {Ft : t ∈ R}, that is, a rightcontinuous, increasing family of sub σ-algebras of F. In recent years, the existence of almost periodic (respectively, periodic) solutions to semilinear evolution equations has received much attention by many authors (see, e.g. [3], [8], [16], and [33] and the references therein) due to their significance and applications in physics, mathematical biology, control theory, and others. This interest also arises from a need to extend well-known results on stochastic ordinary differential equations to a class of stochastic partial differential equations. In their recent paper [9], the authors studied the existence of square-mean almost periodic solutions to the class of nonautonomous stochastic differential equations (1.1)

dX(t) = A(t)X(t) dt + M (t, X(t)) dt + N (t, X(t)) dW(t),

t ∈ R,

where (A(t))t∈R is a family of densely defined closed linear operators, M : R × L2 (Ω, H) → L2 (Ω, H) and N : R × L2 (Ω, H) → L2 (Ω, L02 ) are jointly continuous satisfying some additional conditions, and W is a Wiener process. For that, Bezandry and Diagana assumed that the family of linear operators A(t) satisfy the well-known Acquistapace-Terreni conditions [3], which in fact do guarantee the existence of an evolution family T = {V (t, s)}t≥s associated with the family of linear operators A(t). The main result in [9] was then subsequently utilized to study the existence of square-mean almost periodic solutions to some parabolic stochastic partial differential equations. In this paper, our approach to this problem is somewhat different from that used in [9]. We consider a more general setting, that is, we make extensive use of intermediate space techniques and the Krasnoselskii fixed point theorem to study and obtain the existence of square-mean almost periodic solutions to the class of 1991 Mathematics Subject Classification. 34K50; 35R60; 43A60; 34B05; 34C27; 42A75; 47D06; 35L90. Key words and phrases. stochastic evolution equation; stochastic processes; square-mean almost periodic; Wiener process; sectorial operator; evolution family; exponential dichotomy; Acquistapace and Terreni conditions; Krasnoselskii fixed point theorem. 1

2

PAUL BEZANDRY AND TOKA DIAGANA

nonautonomous stochastic differential equations with finite delay given by h i (1.2) d X(t) + F1 (t, Xt ) = A(t)X(t)dt + F2 (t, Xt )dt + F3 (t, Xt )dW(t) where (A(t))t∈R is a family of closed linear operators on H satisfying AcquistapaceTerreni conditions, the history Xt defined by Xt (θ) := X(t + θ) for each θ ∈ [−τ, 0], and the functions F1 : R × L2 (Ω, Cτ ) → L2 (Ω, Htβ ) (0 ≤ α < 0.5 < β < 1), Fi (i = 2, 3) : R × L2 (Ω, Cτ ) 7→ L2 (Ω, H) are jointly continuous satisfying some additional conditions, and W is a R-valued Brownian motion with the real number line as time parameter. (Here Htβ is an interpolation space and for 0 ≤ α < 1, Cτ, α := C([−τ, 0], Hα ) is the Banach space of all continuous functions from [−τ, 0] in Hα .) In view of the above, there exists an evolution family U = {U (t, s)}t≥s associated with the family of closed linear operators A(t). Assuming that the evolution family U = {U (t, s)}t≥s is exponentially dichotomic (hyperbolic) and under some additional assumptions, it will be shown that Eq. (1.2) has a square-mean almost periodic solution. It is worth mentioning that the main result of this paper generalizes, to some extent, most of known results on square-mean almost periodic solutions to autonomous and nonautonomous differential equations, especially those in [8], [9], and [10]. The paper is organized as follows: Section 2 is devoted to preliminaries facts related to the existence of an evolution family. Some preliminary results on intermediate spaces are also stated there. In Section 3, basic definitions and results on the concept of square-mean almost periodic functions are given. In Section 4, we first state a key technical lemma due to Diagana [14] and next make use of it to prove the main result.

2. Preliminaries In this section, we introduce some notations and collect some preliminary results from Diagana [14] that will be used later. Throughout the rest of this paper, (H, k · k, h·, ·i) stands for a real separable Hilbert space, A(t) for t ∈ R is a family of closed operators on D(A(t)) satisfying the Acquistapace-Terreni conditions (see (H.1)). Moreover, the operators A(t) are not necessarily densely defined. The functions F1 : R × Cτ → Htβ (0 ≤ α < 0.5 < β < 1), and Fi (i = 2, 3) : R × Cτ → H are respectively jointly continuous satisfying some additional assumptions. If L is a linear operator on H, then ρ(L), σ(L), D(L), ker(L), R(L) stand respectively for the resolvent set, spectrum, domain, kernel, and range of L. Moreover, one sets R(λ, L) = (λI − L)−1 for all λ ∈ ρ(L). Throughout the rest of the paper, we set Q(t) = I − P (t), if P (t) is a family of projections. If B1 , B2 are Banach spaces, then the notation B(B1 , B2 ) stands for the Banach space of bounded linear operators from B1 into B2 . When B1 = B2 , this is simply denoted B(B1 ). In the present work we study operators A(t) on H subject to the following hypotheses. Hypothesis (H.1). The family of closed linear operators A(t) for t ∈ R on H with domain D = D(A(t)) (possibly not densely defined) satisfy the so-called Acquistapace and Terreni conditions, that is, there exist constants ζ ∈ R, θ ∈ ( π2 , π),

SQUARE-MEAN ALMOST PERIODIC SOLUTIONS

L > 0 and µ, ν ∈ (0, 1] with µ + ν > 1 such that




(2.1) Σθ ∪ 0 ⊂ ρ(A(t) − ζ) 3 λ, R λ, A(t) − ζ ≤

3

K 1 + |λ|

and

µ −ν





(2.2) A(t) − ζ R λ, A(t) − ζ R ζ, A(t) − R ζ, A(s) ≤ L t − s λ n o for t, s ∈ R, λ ∈ Σθ := λ ∈ C\{0} : argλ ≤ θ . In the particular case when A(t) has a constant domain D = D(A(t)), Eq. (2.2) can be replaced with the following: there exist constants L and 0 < µ ≤ 1 such that

µ



A(t) − A(s) R ζ, A(r) ≤ L t − s , s, t, r ∈ R. It should be mentioned that (H.1) was introduced in the literature by Acquistapace and Terreni [3] for ζ = 0. among other things, it ensures that there exists a unique evolution family U = U (t, s) on H satisfying (a) U (t, s)U (s, r) = U (t, r) for t ≥ s ≥ r ∈ R; (b) U (t, t) = I (c) (t, s) → U (t, s) ∈ B(H) is continuous for t > s;

∂U (t, s) = A(t)U (t, s) and A(t)k U (t, s) ≤ (d) U (·, s) ∈ C 1 ((s, ∞), B(H)), ∂t C(t − s)−k for 0 < t − s ≤ 1, k = 0, 1; and (e) ∂s+ U (t, s)x = −U (t, s)A(s)x for t > s and x ∈ D(A(s)) with A(s)x ∈ D(A(s). It should also be mentioned that the above-mentioned proprieties were mainly established in [1, Theorem 2.3] and [35, Theorem 2.1], see also [3, 34]. In this case we say that A(·) generates the evolution family U (·, ·). Definition 2.1. We say that an evolution family U(t, s) has an exponential dichotomy (or is hyperbolic) if there are projections P (t), t ∈ R, being uniformly bounded and strongly continuous in t and constants δ > 0 and N ≥ 1 such that (1) U (t, s)P (s) = P (t)U (t, s), (2) the restriction UQ (t, s) : Q(s)H → Q(t)H of U (t, s) is invertible (and we set −1 ˜ U ),

Q (s, t) := U Q (t, s) −δ(t−s)

˜Q (s, t)Q(t) ≤ N e−δ(t−s) for t ≥ s and (3) U (t, s)P (s) ≤ N e and U t, s ∈ R. Here and below Q = I − P for a projection P . Under Acquistpace-Terreni conditions, the family of operators defined by  if t ≥ s, t, s ∈ R  U (t, s)P (s), Γ(t, s) =  ˜ −UQ (t, s)Q(s), if t < s, t, s ∈ R are called Green function corresponding to U and P (·). If P (t) = I for t ∈ R, then U is exponentially stable. This setting requires some estimates related to U (t, s). For that, we introduce the interpolation spaces for A(t). For details, we refer the reader to Lunardi [24].

4

PAUL BEZANDRY AND TOKA DIAGANA

Definition 2.2. A linear operator A : D(A) ⊂ H → H (not necessarily densely defined)  π is said to be sectorial if the following hold: there exist constants ζ ∈ R, θ∈ , π , and M > 0 such that Sθ,ζ ⊂ ρ(A), 2  Sθ,ζ := λ ∈ C : λ 6= ζ, arg(λ − ζ) < θ

M , λ ∈ Sθ,ζ and R(λ, A) ≤ λ − ζ where R(λ, A) = (λI − A)−1 for each λ ∈ ρ(A). Let A be a sectorial operator on H and let α ∈ (0, 1). Define the real interpolation space n o

A

: sup rα (A − ζ)R(r, A − ζ)x < ∞ , HA α := x ∈ H : x α

r>0

A which is a Banach space when endowed with the norm · α . For convenience we further write

A



A

x := x , HA

x := (ζ − A)x . HA 0 := H, 1 := D(A) and 0

1

ˆ A := D(A) of H. In particular, we will frequently be using the Moreover, let H following continuous embedding
α ˆA (2.3) D(A) ,→ HA ,→ HA α ,→ H ⊂ H, β ,→ D (ζ − A) for all 0 < α < β < 1, where the fractional powers are defined in the usual way. In general, D(A) is not dense in the spaces HA α and H. However, we have the following continuous injection

A

· α HA ,→ (2.4) D(A) β for 0 < α < β < 1. Given the family of linear operators A(t) for t ∈ R, satisfying (H.1), we set Htα := HA(t) , α

ˆ t := H ˆ A(t) H

for 0 ≤ α ≤ 1 and t ∈ R, with the corresponding norms. The embedding in (2.3) hold with constants independent of t ∈ R. These interpolation spaces are of class Jα (see [24]) and hence there is a constant c(α) such that

t



y ≤ c(α) y 1−α A(t)y α , y ∈ D(A(t)). (2.5) α we have the following estimates for the evolution family U. Lemma 2.3. [7, 14] For x ∈ H, 0 ≤ α ≤ 1, and t > s, the following hold (i) There is a constant c(α) such that

t

δ

(2.6)

U (t, s)P (s)x ≤ c(α)e− 2 (t−s) (t − s)−α x . α

(ii) There is a constant m(α) such that

s

δ

˜

(2.7)

UQ (s, t)Q(s)x ≤ m(α)e− 2 (t−s) x . α

SQUARE-MEAN ALMOST PERIODIC SOLUTIONS

5

In addition to above, we also need the following assumptions: Hypothesis (H.2). The domain D(A(t) = D is constant in t ∈ R. Moreover, the evolution family U generated by A(·) has an exponential dichotomy with constants N , δ > 0 and dichotomy projections P (t) for t ∈ R. Furthermore, 0 ∈ ρ(A(t)) for each t ∈ R and the following holds



sup A(s)A−1 (t) (2.8) < c0 . B(H,Hα )

t, s∈R

If 0 ≤ µ < α < β < 1, then the nonnegative constants k(α), k, k 0 ,, and k2 denote respectively the bounds of the embeddings Hβ ,→ Hα , Hα ,→ H, Hβ ,→ H, and Hµ ,→ H. 3. Square-Mean Almost periodic Stochastic Processes For details on this section, we refer the reader to [8, 11] and the references therein. In this paper, we assume that (Ω, F, P) is a probability space. Let W be a Brownian motion on R. It is worth mentioning that W can be obtained as follows: let {Wi (t), t ∈ R+ }, i = 1, 2, be independent R-valued Brownian motions, then  if t ≥ 0,  W1 (t) W(t) =  W2 (−t) if t ≤ 0, is a Brownian motion with the real number line as time parameter. We then let Ft = σ{W(s), s ≤ t}. Let (B, k · k) be a Banach space. This setting requires the following preliminary definitions. Definition 3.1. A stochastic process X : R → L2 (Ω; B) is said to be continuous whenever

2

lim E X(t) − X(s) = 0. t→s

Definition 3.2. A continuous stochastic process X : R → L2 (Ω; B) is said to be square-mean almost periodic if for each ε > 0 there exists l(ε) > 0 such that any interval of length l(ε) contains at least a number τ for which

2

sup E X(t + τ ) − X(t) < ε. t∈R

The collection of all stochastic processes X : R → L2 (Ω; B) which are quadratic mean almost periodic is then denoted by AP (R; L2 (Ω; B)). The next lemma provides some properties of square-mean almost periodic processes. Lemma 3.3. If X belongs to AP (R; L2 (Ω; B)), then

2 (i) the mapping t → E X(t) is uniformly continuous;

2 (ii) there exists a constant M > 0 such that E X(t) ≤ M , for all t ∈ R. Let CUB(R; L2 (Ω; B)) denote the collection of all stochastic processes X : R → L (Ω; B), which are continuous and uniformly bounded. It is then easy to check that CUB(R; L2 (Ω; B)) is a Banach space when it is equipped with the norm:  1



X = sup EkX(t)k2 2 . ∞ 2

t∈R

6

PAUL BEZANDRY AND TOKA DIAGANA

Lemma 3.4. AP (R; L2 (Ω; B)) ⊂ CUB(R; L2 (Ω; B)) is a closed subspace. 2 In view of the above, the space AP

(R;

L (Ω; B)) of square-mean almost periodic

processes equipped with the norm · ∞ is a Banach space.



Let (B1 , · B ) and (B2 , · B ) be Banach spaces and let L2 (Ω; B1 ) and L2 (Ω; B2 ) 1 2 be their corresponding L2 -spaces, respectively.

Definition 3.5. A function F : R × L2 (Ω; B1 ) → L2 (Ω; B2 )), (t, Y ) 7→ F (t, Y ), which is jointly continuous, is said to be square-mean almost periodic in t ∈ R uniformly in Y ∈ K where K ⊂ L2 (Ω; B1 ) is any compact subset if for any ε > 0, there exists l(ε, K) > 0 such that any interval of length l(ε, K) contains at least a number τ for which

2

sup E F (t + τ, Y ) − F (t, Y ) < ε B2

t∈R

for each stochastic process Y : R → K. Theorem 3.6. Let F : R × L2 (Ω; B1 ) → L2 (Ω; B2 ), (t, Y ) 7→ F (t, Y ) be a squaremean almost periodic process in t ∈ R uniformly in Y ∈ K, where K ⊂ L2 (Ω; B1 ) is compact. Suppose that F is Lipschitz in the following sense:

2

2



E F (t, Y ) − F (t, Z) ≤ M E Y − Z B2

B1

2

for all Y, Z ∈ L (Ω; B1 ) and for each t ∈ R, where M > 0. Then for any squaremean almost periodic process Φ : R → L2 (Ω; B1 ), the function t 7→ F (t, Φ(t)) is square-mean almost periodic. The present setting requires the following composition of square-mean almost periodic processes. Theorem 3.7. Let F : R × L2 (Ω; B1 ) → L2 (Ω; B2 ), (t, Y ) 7→ F (t, Y ) be a squaremean almost periodic process in t ∈ R uniformly in Y ∈ K, where K ⊂ L2 (Ω; B1 ) is any compact subset. Suppose that F (t, ·) is uniformly continuous on bounded subsets K 0 ⊂ L2 (Ω; B1 ) in the following sense: for all ε > 0 there exists δε > 0 such

2

that X, Y ∈ K 0 and E X − Y < δε , then B1

2

E F (t, Y ) − F (t, Z) < ε, ∀t ∈ R. B2

Then for any square-mean almost periodic process Φ : R → L2 (Ω; B1 ), the function t 7→ F (t, Φ(t)) is square-mean almost periodic. Proof. Since Φ : R → L2 (Ω; B1 ) is a square-mean almost periodic process, for all ε > 0 there exists lε > 0 such that every interval of length lε > 0 contains a τ with the property that

2

(3.1) E Φ(t + τ ) − Φ(t) < ε, ∀t ∈ R. B1

2

In addition, Φ : R → L2 (Ω; B1 ) is bounded, that is, sup E Φ(t) < ∞. Let t∈R

B1

K 00 ⊂ L2 (Ω; B1 ) be a bounded subset such that Φ(t) ∈ K 00 for all t ∈ R. Now

SQUARE-MEAN ALMOST PERIODIC SOLUTIONS

2

E F (t + τ, Φ(t + τ )) − F (t, Φ(t))

B2

7

2

≤ E F (t + τ, Φ(t + τ )) − F (t + τ, Φ(t)) B2

2

+ E F (t + τ, Φ(t)) − F (t, Φ(t)) . B2

Taking into account Eq. (3.1) (take δε = ε) and using the uniform continuity of F on bounded subsets of L2 (Ω; B1 ) it follows that

2 ε

(3.2) sup E F (t + τ, Φ(t + τ )) − F (t + τ, Φ(t)) < . 2 B2 t∈R Similarly, using the square-mean almost periodicity of F it follows that

2 ε

(3.3) sup E F (t + τ, Φ(t)) − F (t, Φ(t)) < . 2 B2 t∈R Combining Eq. (3.2) and Eq. (3.3) one obtains that

2

sup E F (t + τ, Φ(t + τ )) − F (t, Φ(t)) < ε, B2

t∈R

and hence the function t 7→ F (t, Φ(t)) is square-mean almost periodic.



4. Existence of Square-Mean Almost Periodic Solutions This section is devoted to the existence of a square-mean almost periodic solution to the nonautonomous stochastic differential equation Eq. (1.2). In the sequel, we denote for brevity that for 0 ≤ α < 1, Cτ, α = C([−τ, 0], Hα ), the space of all continuous functions from [−τ, 0] into Hα , equipped with the sup norm given by h



i1/2

z

z(θ) 2 := sup . τ, α α −τ ≤θ≤0

Definition 4.1. A continuous random function, X : R 7→ L2 (Ω; Cτ, α ) is said to be a bounded solution of Eq.(1.2) on R provided that s → A(s)Γ(t, s)F1 (s, Xs ) is integrable on (−∞, t) and, s → A(s)Γ(t, s)F1 (s, Xs ) is integrable on (t, ∞), and Z t Z ∞ X(t) = −F1 (t, Xt ) − A(s)Γ(t, s) F1 (s, Xs ) ds + A(s)Γ(t, s) F1 (s, Xs ) ds −∞

Z

t

t

Z

∞

Γ(t, s) F2 (s, Xs ) ds −

+ −∞ Z t

Γ(t, s) F2 (s, Xs ) ds t

Z Γ(t, s) F3 (s, Xs ) dW(s) −

+ −∞

∞

Γ(t, s) F3 (s, Xs ) dW(s) t

for each t ≥ s and for all t, s ∈ R. In this paper, Γ1 , Γ2 , Γ3 , Γ4 , Γ5 , and Γ6 stand respectively for the nonlinear integral operators defined by Z t Z ∞ (Γ1 X)(t) := A(s)Γ(t, s) Ψ1 (s) ds, (Γ2 X)(t) := A(s)Γ(t, s) Ψ1 (s) ds, −∞

t

Z

t

(Γ3 X)(t) :=

Z

−∞

∞

Γ(t, s) Ψ2 (s) ds,

Γ(t, s) Ψ2 (s) ds, (Γ4 X)(t) := t

8

PAUL BEZANDRY AND TOKA DIAGANA

Z

t

(Γ5 X)(t) :=

Z

∞

Γ(t, s) Ψ3 (s) dW(s), (Γ6 X)(t) := −∞

Γ(t, s) Ψ3 (s) dW(s), t

where Ψ1 (s) = F1 (s, Xs ), Ψ2 (s) = F2 (s, Xs ), and Ψ3 (s) = F3 (s, Xs ). To discuss the existence of square-mean almost periodic solution to Eq. (1.2) we need some additional assumptions. First of all, note that for 0 < α < β < 1, then L2 (Ω, Hβ ) ,→ L2 (Ω, Hα ) ,→ L2 (Ω; H) are continuously embedded and hence therefore exist constants k1 > 0, k(α) > 0 such that

2

2 E X ≤ k1 E X α for each X ∈ L2 (Ω, Hα ) and

2

2 E X α ≤ k(α)E X β for each X ∈ L2 (Ω, Hβ ).   (H.3) R(ζ, A(·)) ∈ B AP (R; L2 (Ω; Hα )) . Moreover, there exists a function γ : [0, ∞) → [0, ∞) with γ ∈ L1 [0, ∞) such that for every ε > 0 there exists l(ε) such that every interval of length l(ε) contains a τ with the property



≤ εγ( t − s )

A(t + τ )Γ(t + τ, s + τ ) − A(t)Γ(t, s) B(Hα ,H)

for all s, t ∈ R. (H.4) Let 0 ≤ α < 0.5 < β < 1. Let F1 : R × L2 (Ω; Cτ,α ) → L2 (Ω, Hβ ) be squaremean almost periodic in t ∈ R uniformly in X ∈ O (O ⊂ L2 (Ω; Cτ,α ) being any compact subset). Moreover, F1 is Lipschitz in the following sense: there exists K1 > 0 for which

2

2



E F1 (t, X) − F1 (t, Y ) ≤ K1 E X − Y β

τ,α

2

for all random variables X, Y ∈ L (Ω; Cτ,α ) and t ∈ R; (H.5) Let Fi (i = 2, 3) : R × L2 (Ω; Cτ,α ) → L2 (Ω, H) be square-mean almost periodic in t ∈ R uniformly in X ∈ Oi (Oi ⊂ L2 (Ω; Cτ,α )) being any compact subset). Moreover, Fi (R × B) is precompact for each bounded subset B of L2 (Ω; Cτ,α ), and locally uniformly continuous, that is, for each r, ε > 0, there is η(r, ε) such that EkFi (t, X) − Fi (t, Y )k2 ≤ ε whenever t ∈ R and X, Y ∈ L2 (Ω; Cτ,α ) with EkXk2τ,α < r, EkY k2τ,α < r and EkX − Y k2τ,α < η; (H.6) For i = 2, 3 and for any ε > 0, there is a > 0 such that EkFi (t, X)k2 ≤ εEkXk2τ,α for all t ∈ R and X ∈ L2 (Ω; Cτ,α ) with EkXk2τ,α ≥ a. The main result of the present paper will be based upon the use of the well-known fixed point theorem of Krasnoselskii given as follows: Theorem 4.2. Let C be a closed, convex, and nonempty subset of a Banach space B. Suppose the (possibly nonlinear) operators L and M map C into B satisfying (a) for all u, v ∈ C, then Lu + M v ∈ C; (b) the operator L is a contraction; (c) the operator M is continuous and M (C) is contained in a compact set. Then there exists u ∈ C such that u = Lu + M u. To prove the main result (Theorem 4.9) we need the technical Lemma 4.3 due to Diagana. For the sake of clarity and completeness, we reproduce its proof here.

SQUARE-MEAN ALMOST PERIODIC SOLUTIONS

9

Lemma 4.3. [14, Diagana] Under assumptions (H.1)—(H.2), if 0 ≤ µ < α < β < 1 with 2α > µ + 1, then there exist two constants m(α, β), n(α, µ) > 0 such that



eQ (t, s)Q(s)x (4.1) for t ≤ s,

A(s)U

≤ m(α, β)eδ(s−t) x α β



δ



(4.2) for t > s.

A(s)U (t, s)P (s)x ≤ n(α, µ)(t − s)−α e− 4 (t−s) x , β

α

Proof. Let x ∈ Hβ . Since the restriction of A(s) to R(Q(s)) is a bounded linear operator it follows that

eQ (t, s)Q(s)x

A(s)U



e

≤ ck(α) U Q (t, s)Q(s)x β

δ(s−t) ≤ ck(α)m(β)e

x



≤ m(α, β)eδ(s−t) x

α

β

for t ≤ s by using Eq. (2.7). Similarly, for each x ∈ Hβ , using (H.2), we obtain



A(s)U (t, s)P (s)x

α



= A(s)A(t)−1 A(t)U (t, s)P (s)x α





−1 ≤ A(s)A(t)

A(t)U (t, s)P (s)x B(H,Hα )



≤ c0 A(t)U (t, s)P (s)x



≤ c0 k A(t)U (t, s)P (s)x α

for t ≥ s.



First of all, note that A(t)U (t, s)

≤ N 0 (t − s)−(1−α) for all t, s such

B(H,Hα )

that 0 < t − s ≤ 1 and α ∈ [0, 1]. Letting t − s ≥ 1, we obtain



A(t)U (t, s)P (s)x

α



= A(t)U (t, t − 1)U (t − 1, s)P (s)x α





≤ A(t)U (t, t − 1)

U (t − 1, s)P (s)x B(H,Hα )



≤ N N 0 eδ e−δ(t−s) x



≤ K1 e−δ(t−s) x β

δ

− 3δ (t−s) = K1 e 4 (t − s)α (t − s)−α e− 4 (t−s) x . β

− 3δ 4 (t−s)

Now since e (t − s)α → 0 as t → ∞ it follows that there exists c4 (α) > 0 such that

δ



A(t)U (t, s)P (s)x ≤ c4 (α)(t − s)−α e− 4 (t−s) x α

β

and hence

δ



A(s)U (t, s)P (s)x ≤ c0 kc4 (α)(t − s)−α e− 4 (t−s) x α

β

10

PAUL BEZANDRY AND TOKA DIAGANA

for all t, s ∈ R such that t − s > 1. Now, let 0 < t − s ≤ 1. Using Eq. (2.6) and the fact 2α > µ + 1, we obtain



A(t)U (t, s)P (s)x

α

t+s t+s

)U ( , s)P (s)x = A(t)U (t, 2 2 α

t+s

t+s



) , s)P (s)x ≤ A(t)U (t,

U ( 2 2 B(H,Hα )

t+s

t+s



) , s)P (s)x ≤ k2 A(t)U (t,

U ( 2 2 B(H,Hα ) µ

 t − s −µ δ  t − s α−1

c(µ) e− 4 (t−s) x ≤ k2 N 0 2 2

 t − s −µ δ  t − s α−1

c(µ) e− 4 (t−s) x ≤ k2 k 0 N 0 2 2 β

α−1−µ − δ4 (t−s) ≤ c5 (α, µ)(t − s) e

x β

−α − δ4 (t−s) ≤ c5 (α, µ)(t − s) e

x . β

Therefore there exists n(α, µ) > 0 such that

δ



A(t)U (t, s)P (s)x ≤ n(α, µ)(t − s)−α e− 4 (t−s) x α

β

for all t, s ∈ R with t ≥ s.



Lemma 4.4. Under assumptions (H.1)-(H.2)-(H.3)-(H.4)-(H.5), the integral operators Γ1 and Γ2 defined above map AP (R; L2 (Ω, Hα )) into itself. Proof. Let X ∈ AP (R; L2 (Ω; Cτ, α )). Since t 7→ Xt is square-mean almost periodic, using Theorem 3.6 it follows that Ψ1 (·) = F1 (·, Xt (·)) is in AP (R; L2 (Ω; Hβ )) whenever X ∈ AP (R; L2 (Ω; Cτ,α )). Let us show that Γ1 X ∈ AP (R; L2 (Ω; Hα )). Indeed, since Ψ1 ∈ AP (R; L2 (Ω; Hβ )), for every ε > 0 there exists l(ε) > 0 such that for all ξ there is t ∈ [ξ, ξ + l(ε)] such that EkΨ1 (t + τ ) − Ψ1 X(t)k2β < ν 2 ε for each t ∈ R, where ν =

δ 1−α (Γ(·) being the classical gamma function). n(α)Γ(1 − α)21−α

Now

2

E Γ1 X(t + τ ) − Γ1 X(t) α

2

Z t

A(s + τ )Γ(t + τ, s + τ )[Ψ1 (t − s + τ ) − Ψ1 (t − s)] ds ≤ 2E −∞ t

Z

+2E

−∞

≤ 2L1 + 2L2 .

α

2  

A(s + τ )Γ(t + τ, s + τ ) − A(s)Γ(t, s) Ψ1 (s) ds

α

SQUARE-MEAN ALMOST PERIODIC SOLUTIONS

11

Using the estimate in Eq. (4.2) yields L1

Z ≤ E

t

−∞

2

A(s + τ )Γ(t + τ, s + τ )[Ψ1 (t − s + τ ) − Ψ1 (t − s)] ds

Z ≤ n(α, µ) E

t −α

2

(t − s)

2

Ψ1 (t − s + τ ) − Ψ1 (t − s) ds

− δ4 (t−s)

e

−∞

Z

≤ n(α, µ)2

t

δ

(t − s)−α e− 4 (t−s) ds

2

≤ n(α, µ)2

β

t

−∞



2

sup E Ψ1 (t + τ ) − Ψ1 (t)

1−α

Γ(1 − α)4 δ 1−α

2

2

sup E Ψ1 (t + τ ) − Ψ1 (t) < ε. β

t

Similarly, using assumption (H.3), it follows that L2

2





A(s + τ )Γ(t + τ, s + τ ) − A(s)Γ(t, s) Ψ1 (s) ds α −∞ Z t  Z t

2 

≤ ε2 γ(t − s) ds γ(t − s)E Ψ1 (s) ds t

Z ≤ E

≤ ε2

Z

−∞ t

γ(t − s) ds −∞

α

−∞

2

2

2

sup E Ψ1 (s) = ε2 k(α) γ L1 KΨ1 . α

t

Therefore,  

2

2

E Γ1 X(t + τ ) − Γ1 X(t) ≤ 1 + k(α) γ L1 KΨ1 ε, α

for each t ∈ R, and hence Γ1 X ∈ AP (R; L2 (Ω; Hα )). The proof for Γ2 X(·) is omitted as it follows along the same line as that of Γ1 X.  Lemma 4.5. Under assumptions (H.1)-(H.2)-(H.3)-(H.4)-(H.5), the integral operators Γ3 and Γ4 defined above map AP (R; L2 (Ω; Hα )) into itself. Proof. The proof for the square-mean almost periodicity of Γ4 X is similar to that of Γ3 X and hence will be omitted. Note, however, that for Γ4 X, we make use of Eq. (2.6 ) rather than Eq. (2.7 ). Let X ∈ AP (R; L2 (Ω, Cτ,α )). Clearly, Xt ∈ AP (R; L2 (Ω, Cτ,α )). Setting Ψ2 (t) = F2 (t, Xt ) and using Theorem 3.7 it follows that Ψ2 ∈ AP (R; L2 (Ω, H)). We now show that Γ3 X ∈ AP (R; L2 (Ω, Hα )). Indeed, since Ψ2 ∈ AP (R; L2 (Ω, H)), for every ε > 0 there exists l(ε) > 0 such that for all ξ there is τ ∈ [ξ, ξ + l(ε)] with

2

E Ψ2 (t + τ ) − Ψ2 (t) < κ2 · ε for each t ∈ R, where κ =

δ 1−α (Γ(·) being the classical gamma function). c(α)21−α Γ(1 − α)

12

PAUL BEZANDRY AND TOKA DIAGANA

Now

2

E (Γ3 X)(t + τ ) − (Γ3 X)(t) α

Z t Z

Γ(t + τ, s + τ )Ψ2 (s + τ ) ds − = E

2

Γ(t, s)Ψ2 (s) ds

−∞ −∞ α

Z ∞

2

≤ 3E Γ(t + τ, t − s + τ )[Ψ2 (t − s + τ ) − Ψ2 (t − s)] ds

0 α

Z ∞

2

  +3E Γ(t + τ, t − s + τ ) − Γ(t, t − s) Ψ2 (t − s) ds

t

ε

Z

+3E

α

ε

0

2

  Γ(t + τ, t − s + τ ) − Γ(t, t − s) Ψ2 (t − s) ds

α

≤ 3L01 + 3L02 + 3L03 . Using Eq. (2.6), it follows that 2 Z ∞

0 L1 ≤ E

Γ(t + τ, t − s + τ )[Ψ2 (t − s + τ ) − Ψ2 (t − s)] ds α

0

Z ≤ c(α)2 E 2

Z

≤ c(α)

∞

0 ∞

2 δ

s−α e− 2 s Ψ2 (t − s + τ ) − Ψ2 (t − s) ds 2

−α − δ2 s

ds

1−α

2

s

e

t

0

≤ c(α)2 For

L02 ,





sup E Ψ2 (t + τ ) − Ψ2 (t)

Γ(1 − α)2 δ 1−α



sup E Ψ2 (t + τ ) − Ψ2 (t) ≤ ε . t

we use [25, Proposition 4.4] to obtain 2 Z ∞





L02 ≤ E Γ(t + τ, t − s + τ ) − Γ(t, t − s) Ψ (t − s) ds

2 α

ε

2 4

≤ 2 ε2 sup E Ψ2 (t) . δ t∈R The evaluation of the last term is straightforward. We obtain: 2 Z ε





L03 ≤ E

Γ(t + τ, t − s + τ ) − Γ(t, t − s) Ψ2 (t − s) ds α

0

2

≤ 4 M 2 ε2 sup E Ψ2 (t) . t∈R

Combining these evaluations, we conclude that Γ3 X ∈ AP (R; L2 (Ω; Hα )).



Lemma 4.6. Under assumptions (H.1)-(H.2)-(H.3)-(H.4)-(H.5), the integral operators Γ5 and Γ6 defined above map AP (R; L2 (Ω; Hα )) into itself. Proof. Let X ∈ AP (R; L2 (Ω; Cτ,α )). Setting Ψ3 (t) = F3 (t, Xt ) and using Theorem 3.7 it follows that Ψ3 ∈ AP (R; L2 (Ω; H)). We claim that Γ5 X ∈ AP (R; L2 (Ω; Hα )). Indeed, since Ψ3 ∈ AP (R; L2 (Ω; H)), for every ε > 0 there exists l(ε) > 0 such that for all ξ there is τ ∈ [ξ, ξ + l(ε)] with (4.3)

EkΨ3 (t + τ ) − Ψ3 (t)k2 < ζ · ε for each t ∈ R,

SQUARE-MEAN ALMOST PERIODIC SOLUTIONS

δ 1−2α . C(α) Γ(1 − 2α) Now using the expression Z

13

where ζ =

t

(Γ5 X)(t+τ )−(Γ5 X)(t) =

Z Γ(t+τ, s+τ )Ψ3 (s+τ )dW(s)−

−∞

t

Γ(t, s)Ψ3 (s)dW(s)

−∞

it follows that

2

E (Γ5 X)(t + τ ) − (Γ5 X)(t) α

Z ∞

2

Γ(t + τ, t − s + τ )[Ψ3 (t − s + τ ) − Ψ3 (t − s)] dW(s) ≤ 3E α 0

Z ∞ 

2 

+3E Γ(t + τ, t − s + τ ) − Γ(t, t − s) Ψ3 (t − s) dW(s) α ε

Z ε 

2 

Γ(t + τ, t − s + τ ) − Γ(t, t − s) Ψ3 (t − s) dW(s) +3E α

0

≤ 3L001 + 3L002 + 3L003 . Now, L001

Z

= E

∞

2  

Γ(t + τ, t − s + τ ) Ψ3 (t − s + τ ) − Ψ3 (t − s) dW(s)

Z

∞

α

0

≤ c(α)2

0 2

≤ c(α)

Z 0

2

s−2α e−δs E Ψ3 (t − s + τ ) − Ψ3 (t − s) ds

∞

2 

s−2α e−δs ds sup E Ψ3 (t + τ ) − Ψ3 (t) t∈R

2 Γ(1 − 2α)

sup E (t + τ ) − Ψ (t) ≤ c(α)2

Ψ

<ε. 3 3 δ 1−2α t∈R For L002 , using [25, Proposition 4.4], it follows that L002

Z

= E

ε

≤

∞

2  

Γ(t + τ, t − s + τ ) − Γ(t, t − s) Ψ3 (t − s) dW(s)

α

2 C(α) 2

ε sup E Ψ3 (t) . δ t∈R

As to L003 , it is straightforward. We obtain

Z ε 

2 

L003 = E Γ(t + τ, t − s + τ ) − Γ(t, t − s) Ψ3 (t − s) dW(s) α 0

2

≤ 4 C(α) M 2 ε sup E Ψ3 (t) . t∈R

Combining these evaluations, we conclude that Γ5 X ∈ AP (R; L2 (Ω; Hα )). The proof for Γ6 X(·) is similar to that of Γ5 X(·) except that Eq. (2.7) and Eq. (4.1) are used instead of Eq. (2.6) and Eq. (4.2), respectively.   

Consider the nonlinear operator Ξ on the space AP (R; L2 (Ω; Hα )), · ∞,α defined by ΞX = Ξ1 X + Ξ2 X for all X ∈ AP (R; L2 (Ω; Hα )),

14

PAUL BEZANDRY AND TOKA DIAGANA

where Z (Ξ1 X)(t)

t

= −F1 (t, Xt ) −

Z

−∞

Z (Ξ2 X)(t)

A(s)Γ(t, s) F1 (s, Xs )) ds t

t

Z

∞

Γ(t, s) F2 (s, Xs )) ds −

=

∞

A(s)Γ(t, s) F1 (s, Xs ) ds +

−∞ Z t

Γ(t, s) F2 (s, Xs ) ds t

Z

∞

Γ(t, s) F3 (s, Xs ) dW(s) −

+ −∞

Γ(t, s) F3 (s, Xs ) dW(s). t

In view of Lemma 4.4, Lemma 4.5, and Lemma 4.6, it follows that Ξ maps AP (R; L2 (Ω; Hα )) into itself. In order to apply Krasnoselskii’s fixed point theorem, we need to construct two mappings: a contraction map and a compact map. Lemma 4.7. The operator Ξ1 is a contraction provided K(α, β, δ, µ, K1 ) < 1. Proof. Let X, Y ∈ AP (R; L2 (Ω; Hα )). Using (H.1)-(H4), we obtain

2

E F (t, Xt ) − F (t, Yt )

α

2

≤ k(α)K1 E Xt − Yt τ,α

2

≤ k(α) · K1 X − Y , ∞,α

which yields



F (·, X· ) − F (·, Y· )

∞,α



≤ k 0 (α) · K10 X − Y

∞,α

.

Now for Γ1 and Γ2 , we have the following evaluations

2

E (Γ1 X)(t) − (Γ1 Y )(t) α

Z t 2

≤ E

A(s)Γ(t, s)[F1 (s, Xs ) − F1 (s, Ys )] ds α

−∞

≤ n(α, µ)2

Z

t

δ



(t − s)−α e− 4 (t−s) ds ×

−∞

Z ×

t

−∞

2  δ

(t − s)−α e− 4 (t−s) E F1 (s, Xs ) − F1 (s, Ys ) ds α

≤ n(α, µ)2 k(α) K1

Z

t

−∞

2 2 δ

(t − s)−α e− 4 (t−s) ds X − Y

∞,α

,

and hence



Γ1 X − Γ1 Y

∞,α

≤ n(α, µ) · k 0 (α) · K10

41−α Γ(1 − α))

− Y .

X

δ 1−α ∞,α

SQUARE-MEAN ALMOST PERIODIC SOLUTIONS

15

Similarly,

2

E (Γ2 X)(t) − (Γ2 Y )(t) α

Z ∞ 2

≤ E

A(s)Γ(t, s)[F1 (s, Xs ) − F1 (s, Ys )] ds α t Z ∞  ≤ m(α, β)2 e−δ(s−t) ds × t Z ∞  × e−δ(s−t) EkF1 (s, Xs ) − F1 (s, Ys )k2α ds t

2 Z ∞ 2

2 ≤ m(α, β) k(α) K1 e−δ(s−t) ds X − Y , ∞,α

t

and hence,



Γ2 X − Γ2 Y

∞,α

≤

m(α, β) · k 0 (α) · K10

.

X − Y δ ∞,α 

Lemma 4.8. The nonlinear operator Ξ2 is continuous. Moreover, its image is contained in a compact set.  Proof. Let us consider the set V = X ∈ AP (R; L2 (Ω, Hα )) : kXk2∞,α ≤ R0 for some fixed R0 > 0. For the continuity, let X n ∈ AP (R; L2 (Ω, Hα )) be a sequence which converges to some X ∈ AP (R; L2 (Ω, Hα )), that is, kX n − Xk∞,α → 0 as n → ∞. It follows from the estimates in Lemma 2.3 that

Z t

2

E Γ(t, s)[F2 (s, Xsn ) − F2 (s, Xs )] ds α

−∞

"Z

t



F2 (s, Xsn ) − F2 (s, Xs ) ds

−α − δ2 (t−s)

≤E

c(α)(t − s) −∞

e

#2 .

Now, using the continuity of F2 and the Lebesgue Dominated Convergence Theorem we obtain that

Z t

2

E Γ(t, s)[F2 (s, Xsn ) − F2 (s, Xs )] ds → 0 as n → ∞ . α

−∞

By similar arguments, we can also show that

Z ∞

2

Γ(t, s)[F2 (s, Xsn ) − F2 (s, Xs )] ds → 0 as n → ∞ . E α

t

For the term containing the Brownian motion W, we use the estimates (4.1) and (4.2) to obtain

Z t

2

Γ(t, s)[F3 (s, Xsn ) − F3 (s, Xs )] dW(s) E α

−∞

≤ C(α)2

Z

t

−∞

2

(t − s)−2α e−δ (t−s) E F2 (s, Xsn ) − F3 (s, Xs ) ds .

16

PAUL BEZANDRY AND TOKA DIAGANA

Now, using the continuity of F3 and the Lebesgue Dominated Convergence Theorem we obtain that

Z t

2

E Γ(t, s)[F3 (s, Xsn ) − F3 (s, Xs )] dW(s) → 0 as n → ∞ . α

−∞

By similar arguments, we can also show that

Z ∞

2

E Γ(t, s)[F3 (s, Xsn ) − F3 (s, Xs )] dW(s) → 0 as n → ∞ . α

t

Therefore,



Ξ2 X n − Ξ2 X

∞,α

→ 0 as n → ∞.

We now show that Ξ2 maps V into a compact set; in particular, we show that Ξ2 (V ) is an equicontinuous set. Indeed, let ε > 0, t1 < t2 , and X ∈ V be arbitrary. Now

2

E (Ξ2 X)(t2 ) − (Ξ2 X)(t1 ) α

2

2



≤ 4E (Γ3 X)(t2 ) − (Γ3 X)(t1 ) + 4E (Γ4 X)(t2 ) − (Γ4 X)(t1 ) α α

2

2



+4E (Γ5 X)(t2 ) − (Γ5 X)(t1 ) + 4E (Γ6 X)(t2 ) − (Γ6 X)(t1 ) . α

α

We have

2

E (Γ3 X)(t2 ) − (Γ3 X)(t1 ) α

Z t2

2

Z

U (t2 , s)P (s) Ψ2 (s) ds + 2E ≤ 2E t1 t2

Z

2E

α

t1

−∞ t1

2

Z

U (t2 , s)P (s) Ψ2 (s) ds + 2E

2

[U (t2 , s) − U (t1 , s)]P (s) Ψ2 (s) ds

α

Z

t2

2 ∂U (τ, s)

dτ P (s) Ψ2 (s) ds = ∂τ α α t1 t1 −∞

Z t2

2

Z t1  Z t2  2



A(τ )U (τ, s)P (s) Ψ2 (s) dτ ds = 2E U (t2 , s)P (s) Ψ2 (s) ds + 2E α

t1

−∞



α

t1

= N1 + N2 . Clearly, N1

nZ ≤ E

t2

o2

U (t2 , s)P (s) Ψ2 (s) ds α

t1

nZ ≤ c(α)2 E

t2

t1

o2 δ

(t2 − s)−α e− 2 (t2 −s) Ψ2 (s) ds .

Similarly, N2

nZ ≤ E

t1

Z

−∞

nZ ≤ c20 E

t2

 o2

A(τ )U (τ, s)P (s) Ψ2 (s) dτ ds α

t1 t1

−∞

Z

t2

 o2

A(s)U (τ, s)P (s) Ψ2 (s) dτ ds α

t1

nZ ≤ c20 n(α, µ)2 E

t1

−∞

Z

t2

t1

(τ − s)−α e−

δ 2 (τ −s)

 o2

Ψ2 (s) dτ ds .

SQUARE-MEAN ALMOST PERIODIC SOLUTIONS

17

Therefore,

2

E (Γ3 X)(t2 ) − (Γ3 X)(t1 ) α

o2 n Z t2 δ

≤ 2 c(α)2 E (t2 − s)−α e− 2 (t2 −s) Ψ2 (s) ds t1

nZ +2 c20 n(α, µ)2 E

t1

Z

−∞

t2

(τ − s)−α e−

δ 2 (τ −s)

t1

 o2

Ψ2 (s) dτ ds

2

≤ K(α, δ, µ) (t2 − t1 ) sup E Ψ2 (t) , 2

t∈R

where K(α, δ, µ) is a positive constant. Similar computations show that

2

E (Γ4 X)(t2 ) − (Γ4 X)(t1 ) α

o2

n Z t2

2 ≤ 2 c(α) E e− δ(s−t1 ) Ψ2 (s) ds t1

+2

c20

nZ m(α, β) E

∞

2

t2

Z

t2

t1

 o2

e− δ(s−τ ) Ψ2 (s) dτ ds

2

≤ K(α, δ, β) (t2 − t1 )2 sup E Ψ2 (t) , t∈R

where K(α, δ, β) is a positive constant. Let us now evaluate Γ5 X. We have

2

E (Γ5 X)(t2 ) − (Γ5 X)(t1 ) α

Z t2

2

≤ 2E U (t2 , s)P (s) Ψ3 (s) dW(s)

α

t1

t1

Z

+2E =

N10

+

−∞ N20 .

2

[U (t2 , s) − U (t1 , s)]P (s) Ψ3 (s) dW(s)

α

Let us start with the first term. By Ito isometry identity, we have

N10

≤ c(α)2 ≤ c(α)2

nZ Z

t2

t1 t2

t1

2

(t2 − s)−2α e−2γ(t2 −s) E Ψ3 (s) ds

2 

(t2 − s)−2α e− 2γ (t2 −s) ds sup E Ψ3 (s) . s∈R

18

PAUL BEZANDRY AND TOKA DIAGANA

Similarly, N20

=

=

=

≤

Z

2

t1 h Z t2 ∂

i

E U (τ, s) dτ P (s) Ψ3 (s) dW(s)

−∞ t1 ∂τ

α

Z

2

t1 h Z t2

i

E A(τ )U (τ, s) dτ P (s) Ψ3 (s) dW(s)

−∞ t1

α

Z

t2 n Z t1 o 2

E A(τ )U (τ, t1 ) U (t1 , s)P (s) Ψ3 (s) dW(s) dτ

t1

−∞ α "Z # 2

t2 Z t1

E A(τ )U (τ, s) P (s) Ψ3 (s) dW(s) dτ

α

−∞

t1

≤ c20 (t2 − t1 )

Z

t2 Z

E

t1

≤ c20 n(α, µ)2 (t2 − t1 )

t1

2

A(s)U (τ, s) P (s) Ψ3 (s) dW(s) dτ

−∞ t2 n Z t1

Z

−∞

t1

≤ c20 n(α, µ)2 (t2 − t1 )2

Z

t1

−∞

≤

c20

α

2 o

(τ − s)−2α e−2γ(τ −s) E Ψ3 (s) ds dτ

2

(t1 − s)−2α e−2γ(t1 −s) E Ψ3 (s) ds

2


n(α, µ) (t2 − t1 ) Γ 1 − 2α (2γ)1−2α sup E Ψ3 (s) ds . 2

2

s∈R

Hence,

2

E (Γ5 X)(t2 ) − (Γ5 X)(t1 ) α h i ≤ K K(Γ, α, γ) (t2 − t1 ) + K1 (Γ, α, γ, µ) (t2 − t1 )2 , where K(Γ, α, γ, µ) is a positive constant depending on Γ, α, γ, and µ. Similar computations show

2

E (Γ6 X)(t2 ) − (Γ6 X)(t1 ) α h i ≤ K K(α, δ) (t2 − t1 ) + K(α, β, δ) (t2 − t1 )2 . where K(α, δ) and K(α, β, δ) are positive constants. From the theorem of Ascoli-Arzela, it follows that Ξ2 (V ) is contained in a compact set. The proof is complete.  Theorem 4.9. Suppose assumptions (H.1)-(H.2)-(H.3)-(H.4)-(H.5)-(H.6) hold and that K(α, β, δ, µ, K1 ) < 1, the evolution equation Eq. (1.2) has a square-mean almost periodic solution X satisfying X = Ξ1 X + Ξ2 X. Proof. Fix ε > 0 and let i= 2, 3. It follows from assumption (H.6) that there exists r > 0 such that

2

2

2

E Fi (t, Y ) ≤ εE Y τ,α for all t ∈ R and Y ∈ L2 (Ω, Hα ) with E Y τ,α > r .

SQUARE-MEAN ALMOST PERIODIC SOLUTIONS

19

Setting

2 n o

2

Mi = sup E Fi (t, Y ) : t ∈ R, E Y τ,α ≤ r . Therefore,

2

2

(4.4) E Fi (t, Y ) ≤ Mi + εE Y τ,α for all (t, Y ) ∈ R × L2 (Ω, Cτ,α ) . Also, using Lemma 2.3 , Eq. (4.1), Eq. (4.2), Eq. (4.4), and assumption (H.4) we can show that

2

E (Ξ1 X)(t) + (Ξ2 Y )(t) α

2  h  Γ(1 − α) 21−α 2  m(α, β) 2 i



X + K + sup (t, 0) ≤ 12 1 + n(α, µ)2

F

1 1 ∞,α δ 1−α δ t∈R "  h Γ(1 − α) 21−α 2  m(α) 2 i

+ M2 + ε Y ∞,α + 8c(α)2 1−α δ δ # h 

Γ(1 − 2α) K(α, β) i 

+ K(α) 1−2α + M3 + ε Y ∞,α δ 2δ

2 

  





= c1 (α, β, δ, µ, Γ) K1 X + sup F1 (t, 0) + c2 (α, β, δ, Γ) M2 + ε Y ∞,α



∞,α

t∈R



+ c3 (α, β, δ, Γ) M3 + ε Y ∞,α . Now, for ε, K1 small enough, choose R such that       c1 (α, β, δ, µ, Γ) K1 R +a +c2 (α, β, δ, Γ) M2 +εR +c3 (α, β, δ, Γ) M3 +εR ≤ R

2

where a = sup F1 (t, 0) , c1 (α, β, δ, µ, Γ) and ci (α, β, δ, Γ) (i = 2, 3) are constants t∈R

depending on(α, β , δ, and the classical gamma function Γ(·). )

Let W = Z ∈ AP (R; L2 (Ω; Hα ) : Z ∞,α ≤ R . For X, Y ∈ W , we have

2

E (Ξ1 X)(t) + (Ξ2 Y )(t) ≤ R . α

Thus (Ξ1 X)(t) + (Ξ2 Y )(t) ∈ W . In view of Lemma 4.4, Lemma 4.5, Lemma 4.6, Lemma 4.7, and Lemma 4.8, the proof can be completed by using the Krasnoselskii’s fixed point theorem (Theorem 4.2).  5. Example Throughout the rest of this paper, we suppose 0 < α < 0.5 < β < 1. Example 5.1. Fix T > 0. Let H= L2 [0, π] equipped with its  natural topology. 2 1 2 Here, for µ ∈ (0, 1), we take Hµ = L [0, π], H0 [0, π] ∩ H [0, π] equipped with µ,∞

its µ-norm k · kµ .

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To illustrate our main result, we study the existence of square-mean almost periodic solutions to the one-dimensional stochastic heat equation with with infinite delay given by (5.1)  " # " # 2  ∂ Φ   + a(t, x)Φ + F2 (t, Φt ) ∂t + F3 (t, Φt )dW(t), in R × [0, π]  ∂ ϕ + F1 (t, Φt ) = ∂x2     Φ(t, 0) = Φ(t, π) = 0, for all t ∈ R where a : R × [0, π] 7→ R is T -periodic in t ∈ R uniformly in x ∈ [0, π], and F1 : R×L2 (Ω; Bα ) 7→ L2 (Ω, Hβ ) and Fi (i = 2, 3) : R×L2 (Ω, Bα ) → L2 (Ω, L2 [0, π]) are square-mean almost periodic processes. Define the corresponding linear operator A(t) on L2 (Ω, L2 [0, π]) as follows: ∂2Φ + a(t, x)Φ for all Φ ∈ D(A(t)) = L2 (Ω, H10 [0, π] ∩ H2 [0, π]), ∂x2 where a : R × [0, π] 7→ R, in addition of being T -periodic satisfies: there exist δ0 ∈ (0, 1) such that a(t, x) ≤ δ0 for all t ∈ R and x ∈ [0, π]. Clearly, A(t + T ) = A(t) for all t ∈ R. Moreover, it is then easy to check that the evolution family U (t, s) associated with A(t) is exponentially stable, which yields dichotomy with dichotomy projections P (t) = I and Q(t) = 0. Indeed, Z t a(τ, x)dτ U (t, s) = T (t − s)e s for all t ≥ s, A(t)Φ =

where T (t) is the analytic semigroup associated with the second-order differential operator ∂2Φ AΦ = with D(A) = L2 (Ω, H10 [0, π] ∩ H2 [0, π]). ∂x2 Now kU (t, s)Φk ≤ e−(1−δ0 )(t−s) kΦk, t ≥ s. Moreover, U (t + T, s + T ) = U (t, s) for all t ≥ s. Now since Q(t) = 0, it follows that Γ(t, s) = U (t, s) and hence A(t + T )Γ(t + T, s + T ) = A(t)Γ(t, s). We have Theorem 5.2. Under previous assumptions, then the heat equation Eq. (5.1) has a solution Φ ∈ AP (R, L2 ([0, π], Hα )). References 1. P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations 1 (1988), 433–457. 2. P. Acquistapace, F. Flandoli, B. Terreni, Initial boundary value problems and optimal control for nonautonomous parabolic systems. SIAM J. Control Optim. 29 (1991), 89–118. 3. P. Acquistapace, B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova 78 (1987), 47–107. 4. H. Amann, Linear and Quasilinear Parabolic Problems, Birkh¨ auser, Berlin 1995.

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31. R. Schnaubelt, Asymptotic behavior of parabolic nonautonomous evolution equations, in: M. Iannelli, R. Nagel, S. Piazzera (Eds.), Functional Analytic Methods for Evolution Equations, in: Lecture Notes in Math., 1855, Springer-Verlag, Berlin, 2004, 401–472. 32. R. J. Swift, Almost Periodic Harmonizable Processes, Georgian Math. J. 3 (1996), no. 3, 275–292. 33. C. Tudor, Almost Periodic Solutions of Affine Stochastic Evolutions Equations, Stochastics and Stochastics Reports 38 (1992), 251-266. 34. A. Yagi, Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II, Funkcial. Ekvac. 33 (1990), 139–150. 35. A. Yagi, Abstract quasilinear evolution equations of parabolic type in Banach spaces, Boll. Un. Mat. Ital. B (7) 5 (1991), 341–368. 36. S. Zaidman, Topics in Abstract Differential Equations, Pitman Research Notes in Mathematics Ser. II John Wiley and Sons, New York, 1994–1995. Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, D.C. 20059 - USA E-mail address: [email protected] Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, D.C. 20059 - USA E-mail address: [email protected]