We study the existence of mild solutions for a class of impulsive fractional partial neutral stochastic integro-differential inclusions with state- dependent delay

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF SOLUTIONS TO IMPULSIVE FRACTIONAL PARTIAL NEUTRAL STOCHASTIC INTEGRO-DIFFERENTIAL

INCLUSIONS WITH STATE-DEPENDENT DELAY

ZUOMAO YAN, HONGWU ZHANG

Abstract. We study the existence of mild solutions for a class of impulsive fractional partial neutral stochastic integro-differential inclusions with state- dependent delay. We assume that the undelayed part generates a solution operator and transform it into an integral equation. Sufficient conditions for the existence of solutions are derived by using the nonlinear alternative of Leray-Schauder type for multivalued maps due to O’Regan and properties of the solution operator. An example is given to illustrate the theory.

1. Introduction

The study of impulsive functional differential or integro-differential systems is linked to their utility in simulating processes and phenomena subject to short-time perturbations during their evolution. The perturbations are performed discretely and their duration is negligible in comparison with the total duration of the processes and phenomena. Now impulsive partial neutral functional differential or integro-differential systems have become an important object of investigation in recent years stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. With regard to this matter, we refer the reader to [11, 12, 19, 20, 33]. Besides impulsive effects, stochastic effects likewise exist in real systems. Therefore, impulsive stochastic differential equations describing these dynamical systems subject to both impulse and stochastic changes have attracted considerable attention. Particularly, the papers [5, 22, 27] considered the existence of mild solutions for some impulsive neutral stochastic functional differential and integro-differential equations with infinite delay in Hilbert spaces. As the generalization of classic impulsive differential equations, impulsive stochastic differential inclusions in Hilbert spaces have attracted the re- searchers great interest. Among them, Ren et al [30] established the controllability of impulsive neutral stochastic functional differential inclusions with infinite delay in an abstract space by means of the fixed point theorem for discontinuous multi-valued operators due to Dhage.

2000Mathematics Subject Classification. 34A37, 60H10, 34K50, 34G25, 26A33.

Key words and phrases. Impulsive stochastic integro-differential inclusions;

state-dependent delay; multi-valued map; fractional neutral integro-differential inclusions.

c

2013 Texas State University - San Marcos.

Submitted September 25, 2012. Published March 29, 2013.

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On the other hand, fractional differential equations have gained considerable im- portance due to their application in various sciences, such as physics, mechanics, chemistry, engineering, etc.. In the recent years, there has been a significant de- velopment in ordinary and partial differential equations involving fractional deriva- tives; see the monograph of Kilbas et al [23] and the papers [1, 3, 7, 24, 25] and the references therein. The existence of solutions for fractional semilinear differential or integro-differential equations is one of the theoretical fields that investigated by many authors [2, 16, 32]. Several papers [4, 15] devoted to the existence of mild solutions for abstract fractional functional differential and integro-differential equations with state-dependent delay in Banach spaces by using fixed point tech- niques. Recently, the existence, uniqueness and other quantitative and qualitative properties of solutions to various impulsive semilinear fractional differential and integrodifferential systems have been extensively studied in Banach spaces. For example, Balachandran et al [6], Chauhan et al [8], Debbouche and Baleanu [14], Mophou [28], Shu et al [31]. However, the deterministic models often fluctuate due to noise, which is random or at least appears to be so. Therefore, we must move from deterministic problems to stochastic ones. In this paper, we consider the existence of a class of impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay of the form

dD(t, xt)∈ Z t

0

(t−s)^α−2

Γ(α−1) AD(s, xs)ds dt+F(t, x_ρ(t,x_t₎)dw(t), (1.1) t∈J = [0, b], t6=t_k, k= 1, . . . , m,

x0=ϕ∈ B, (1.2)

∆x(tk) =Ik(xt_k), k= 1, . . . , m, (1.3) where the state x(·) takes values in a separable real Hilbert space H with inner product (·,·) and norm k · k, 1< α <2,A : D(A)⊂H →H is a linear densely defined operator of sectorial type onH. The time historyxt: (−∞,0]→H given byxt(θ) =x(t+θ) belongs to some abstract phase space Bdefined axiomatically;

LetKbe another separable Hilbert space with inner product (·,·)Kand normk·kK. Suppose{w(t) :t≥0}is a givenK-valued Brownian motion or Wiener process with a finite trace nuclear covariance operatorQ >0 defined on a complete probability space (Ω,F, P) equipped with a normal filtration {Ft}t≥0, which is generated by the Wiener processw. We are also employing the same notationk · k for the norm L(K, H), where L(K, H) denotes the space of all bounded linear operators from K into H. The initial data {ϕ(t) : −∞ < t ≤ 0} is an F₀-adapted, B-valued random variable independent of the Wiener processwwith finite second moment.

F, G, D(t, ϕ) =ϕ(0) +G(t, ϕ), ϕ∈ B, ρ, Ik(k= 1, . . . , m), are given functions to be specified later. Moreover, let 0< t1 <· · · < tm < b, are prefixed points and the symbol ∆x(tk) =x(t⁺_k)−x(t⁻_k), wherex(t⁻_k) andx(t⁺_k) represent the right and left limits ofx(t) att=tk, respectively.

We notice that the convolution integral in (1.1) is known as the Riemann- Liouville fractional integral (see [9, 10]). In [10], the authors established the existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay. To the best of our knowledge, the existence of mild solutions for the impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay in Hilbert spaces has not been investigated yet. Motivated by this consideration, in this paper we will

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study this interesting problem, which are natural generalizations of the concept of mild solution for impulsive fractional evolution equations well known in the theory of infinite dimensional deterministic systems. Specifically, sufficient conditions for the existence are given by means of the nonlinear alternative of Leray-Schauder type for multivalued maps due to O’Regan combined with the solution operator.

The known results appeared in [6, 8, 14, 28, 31] are generalized to the fractional stochastic multi-valued settings and the case of infinite delay.

The rest of this paper is organized as follows. In Section 2, we introduce some notations and necessary preliminaries. In Section 3, we give our main results.

In Section 4, an example is given to illustrate our results. In the last section, concluding remarks are given.

2. Preliminaries

In this section, we introduce some basic definitions, notation and lemmas which are used throughout this paper.

Let (Ω,F, P) be a complete probability space equipped with some filtration {F_t}_t≥0satisfying the usual conditions (i.e., it is right continuous andF₀contains all P-null sets). Let {ei}^∞_i=1 be a complete orthonormal basis of K. Suppose that {w(t) : t ≥ 0} is a cylindrical K-valued Wiener process with a finite trace nuclear covariance operator Q ≥ 0, denote Tr(Q) = P∞

i=1λi = λ < ∞, which satisfies thatQei =λiei. So, actually, w(t) = P∞

i=1

√λiwi(t)ei, where{wi(t)}^∞_i=1 are mutually independent one-dimensional standard Wiener processes. We assume thatFt=σ{w(s) : 0≤s≤t}is theσ-algebra generated bywandFb=F.

Let L(K, H) denote the space of all bounded linear operators from K into H equipped with the usual operator normk · kL(K,H). Forψ∈L(K, H) we define

kψk²_Q = Tr(ψQψ^∗) =

∞

X

n=1

kp

λ_nψe_nk².

If kψk²_Q <∞, then ψ is called a Q-Hilbert-Schmidt operator. Let L_Q(K, H) denote the space of allQ-Hilbert-Schmidt operatorsψ. The completionL_Q(K, H) of L(K, H) with respect to the topology induced by the norm k · kQ where kψk²_Q = (ψ, ψ) is a Hilbert space with the above norm topology.

The collection of all strongly measurable, square integrable, H-valued random variables, denoted byL2(Ω, H) is a Banach space equipped with normkx(·)kL₂ = (Ekx(·, w)k²)¹², where the expectation, E is defined by Ex = R

Ωx(w)dP. Let C(J, L₂(Ω, H)) be the Banach space of all continuous maps from J intoL₂(Ω, H) satisfying the condition sup_0≤t≤bEkx(t)k² <∞. Let L⁰₂(Ω, H) denote the family of allF0-measurable, H-valued random variablesx(0).

Definition 2.1 ([13]). We call S ⊂ Ω a P-null set if there is B ∈ F such that S⊆B andP(B) = 0.

Definition 2.2 ([13]). A stochastic process {x(t) : t ≥ 0} in a real separable Hilbert spaceH is a Wiener process if for eacht≥0,

(i) x(t) has continuous sample paths and independent increments.

(ii) x(t)∈L²(Ω, H) andE(x(t)) = 0.

(iii) Cov(w(t)−w(s)) = (t−s)Q, whereQ∈L(K, H) is a nonnegative nuclear operator.

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Definition 2.3 ([13]). Brownian motion is a continuous adapted real-valued pro- cessx(t), t≥0 such that

(i) x(0) = 0.

(ii) x(t)−x(s) is independent ofF_s for all 0≤s < t.

(iii) x(t)−x(s) isN(0, t−s)-distributed for all 0≤s≤t.

Definition 2.4 ([13]). Normal filtration {Ft : 0 ≤t ≤ b} is a right-continuous, increasing family of subσ-algebras ofF.

Definition 2.5([13]). The processxisF0-adapted if eachx(0) is measurable with respect toF0.

We say that a function x : [µ, τ] → H is a normalized piecewise continuous function on [µ, τ] if x is piecewise continuous and left continuous on (µ, τ]. We denote byPC([µ, τ], H) the space formed by the normalized piecewise continuous, Ft-adapted measurable processes from [µ, τ] into H. In particular, we introduce the spacePC formed by allFt-adapted measurable,H-valued stochastic processes {x(t) :t∈[0, b]}such thatxis continuous att6=tk,x(tk) =x(t⁻_k) andx(t⁺_k) exists for k = 1,2..., m. In this paper, we always assume that PC is endowed with the norm

kxk_PC= ( sup

0≤t≤b

Ekx(t)k²)¹². Then, we have the following conclusion.

Lemma 2.6. The set(PC,k · k_PC)is a Banach space.

Proof. Let{xn} be a Cauchy sequence inPC, and fix any ε >0. There isn0 ∈N such that for alln > n0andp∈N

kxn+p−xnkPC= ( sup

0≤t≤b

Ekxn+p(t)−xn(t)k²)¹² < ε

for eacht ∈[0, b]. From the above inequality it follows that the sequence xn(t) is a Cauchy sequence inL²(Ω, H); moreover, by the completeness of L²(Ω, H) with respect tok · kL₂, for its limitx(t) := limxn(t), we obtain

Ekxn(t)−x(t)k²< ε²

for alln > n0. Consequently,kxn−xk_PC →0 asn→ ∞. Next, we need to show thatx∈ PC. In fact, we verify thatxis continuous. By

x(t+ ∆t)−x(t) =x(t+ ∆t)−xn(t+ ∆t) +xn(t+ ∆t)−xn(t) +xn(t)−x(t), it follows that

Ekx(t+ ∆t)−x(t)k²≤3Ekx(t+ ∆t)−xn(t+ ∆t)k²

+ 3Ekxn(t+ ∆t)−xn(t)k²+ 3Ekxn(t)−x(t)k². Using the uniform convergence ofxntoxwith respect tok · kL₂ and the continuity ofxn, the continuity ofxfollows. The proof is complete.

To simplify notation, we put t0 = 0, tm+1 =b and for x ∈ PC, we denote by ˆ

xk∈C([tk, tk+1];L2(Ω, H)),k= 0,1, . . . , m, the function given by ˆ

xk(t) :=

(x(t) fort∈(tk, tk+1], x(t⁺_k) fort=tk.

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Moreover, forB⊆ PCwe denote by ˆBk,k= 0,1, . . . , m, the set ˆBk ={ˆxk:x∈B}.

The notationBr(x, H) stands for the closed ball with center atxand radiusr >0 inH.

Lemma 2.7. A setB⊆ PC is relatively compact inPC if, and only if, the setBˆ_k is relatively compact in C([t_k, t_k+1];L₂(Ω, H)), for everyk= 0,1, . . . , m.

Proof. LetB ⊆ PCbe a subset and{x⁽ⁱ⁾(·)} be any sequence of B. Since ˆB0 is a relatively compact subset ofC([0, t1];L2(Ω, H)). Then, there exists a subsequence ofx⁽ⁱ⁾, labeled{x⁽ⁱ⁾₁ } ⊂B, andx1∈C([0, t1];L2(Ω, H)), such that

x⁽ⁱ⁾₁ →x1 inC([0, t1];L2(Ω, H)) as i→ ∞.

Similarly, Bˆk is a relatively compact subset of C([tk, tk+1];L2(Ω, H)), for k = 1,2, . . . , m. Then, there exists a subsequence of x⁽ⁱ⁾, labeled {x⁽ⁱ⁾_k } ⊂ B, such thatxk∈C([tk, tk+1];L2(Ω, H)), and

x⁽ⁱ⁾_k →x^k inC([tk, tk+1];L2(Ω, H)) asi→ ∞.

Setting

x(t) =











x₁(t), t∈[0, t₁], x2(t), t∈(t1, t2], . . .

xm(t), t∈(tm, b], then

x⁽ⁱ⁾_m →x in PC as i→ ∞.

Thus, the setB is relatively compact.

If setB ⊆ PC is relatively compact in PC and {x⁽ⁱ⁾(·)} be any sequence of B.

Then, for eacht∈[0, t₁], there exists a subsequence ofx⁽ⁱ⁾, labeled{x⁽ⁱ⁾₁ } ⊂B,and x₁∈ PC, such thatx⁽ⁱ⁾₁ →x₁ inPC as i→ ∞. From the definition of the set ˆB₀, we can get

ˆ

x⁽ⁱ⁾₁ →xˆ₁ inC([0, t₁];L₂(Ω, H)) asi→ ∞.

Similarly, for eacht∈[tk, tk+1](k= 1,2, . . . , m), there exists a subsequence ofx⁽ⁱ⁾, labeled{x⁽ⁱ⁾_k } ⊂B andxk ∈ PC, such that x⁽ⁱ⁾_k →xk inPC as i→ ∞. From the definition of the set ˆBk, we can get

ˆ

x⁽ⁱ⁾_k →xˆ_k inC([t_k, t_k+1];L₂(Ω, H)) asi→ ∞.

Thus, the set ˆBk is relatively compact in C([tk, tk+1];L2(Ω, H)), for every k =

0,1, . . . , m. The proof is complete.

In this article, we assume that the phase space (B,k · kB) is a seminormed linear space ofF0-measurable functions mapping (−∞,0] into H, and satisfying the following fundamental axioms due to Hale and Kato (see e.g., in [18]).

(A) Ifx: (−∞, σ+b]→H,b >0, is such thatx|[σ,σ+b] ∈C([σ, σ+b], H) and xσ∈ B, then for everyt∈[σ, σ+b] the following conditions hold:

(i) xtis in B;

(ii) kx(t)k ≤H˜kxtkB;

(iii) kxtkB ≤K(t−σ) sup{kx(s)k :σ ≤s ≤t}+M(t−σ)kxσkB, where H˜ ≥0 is a constant;K, M : [0,∞)→[1,∞),K is continuous and M is locally bounded, and ˜H, K, M are independent ofx(·).

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(B) For the functionx(·) in (A), the functiont→xtis continuous from [σ, σ+b]

intoB.

(C) The spaceBis complete.

The next result is a consequence of the phase space axioms.

Lemma 2.8. Let x: (−∞, b]→H be anFt-adapted measurable process such that the F0-adapted process x0=ϕ(t)∈L⁰₂(Ω,B) andx|J ∈ PC(J, H), then

kxskB≤MbEkϕkB+Kb sup

0≤s≤b

Ekx(s)k,

whereKb= sup{K(t) : 0≤t≤b},Mb= sup{M(t) : 0≤t≤b}.

Proof. For each fixed x ∈ H, we consider the function ξ(t) defined by ξ(t) = sup{kx_sk_B: 0≤s≤t}, 0≤t≤b. Obviously, ξis increasing. This combined with the phase space axioms, we have

ξ(t)≤M(t)kϕkB+K(t) sup

0≤s≤t

kx(s)k

≤Mbkϕk_B+Kbkx(t)k.

SinceEkϕk_B<∞, Ekx(t)k<∞, the previous inequality holds. Consequently E(ξ(t))≤E(MbkϕkB+Kbkx(t)k)

≤MbEkϕk_B+Kb sup

0≤s≤b

Ekx(s)k

for eacht∈J. By the definition ofξ, we have

ξ(b) =E(ξ(b))≤MbEkϕk_B+Kb sup

0≤s≤b

Ekx(s)k,

andkxskB≤ξ(b) for eachs∈J; therefore,

kxsk_B≤MbEkϕk_B+Kb sup

0≤s≤b

Ekx(s)k.

The proof is complete.

Let P(H) denote all the nonempty subsets of H. Let Pbd,cl(H), Pcp,cv(H), Pbd,cl,cv(H), andPcd(H) denote respectively the family of all nonempty bounded- closed, compact-convex, bounded-closed-convex and compact-acyclic subsets ofH (see [17]). Forx∈H andY, Z ∈ P_bd,cl(H), we denote byD(x, Y) = inf{kx−yk: y ∈Y} and ˜ρ(Y, Z) = sup_a∈Y D(a, Z), and the Hausdorff metricH_d:P_bd,cl(H)× Pbd,cl(H)→R⁺ byHd(A, B) = max{ρ(A, B),˜ ρ(B, A)}.˜

A multi-valued mapGis called upper semicontinuous (u.s.c.) onH if, for each x0 ∈ H, the set G(x0) is a nonempty, closed subset of H and if, for each open setS ofH containingG(x0), there exists an open neighborhood S ofx0 such that G(S)⊆V. F is said to be completely continuous ifG(V) is relatively compact, for every bounded subsetV ⊆H.

If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only ifF has a closed graph, i.e. xn → x∗, yn → y∗, yn∈G(xn) implyy∗∈G(x∗).

A multi-valued map G: J → Pbd,cl,cv(H) is measurable if for each x∈H, the functiont7→D(x, G(t)) is a measurable function onJ.

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Definition 2.9 ([17]). LetG:H → Pbd,cl(H) be a multi-valued map. ThenGis called a multi-valued contraction if there exists a constant κ∈(0,1) such that for eachx, y∈H we have

H_d(G(x)−G(y))≤κkx−yk.

The constantκis called a contraction constant ofG.

A closed and linear operator A is said to be sectorial of type ω if there exist 0 < θ < π/2, M > 0 and ω ∈R such that its resolvent exists outside the sector ω+S_θ:={ω+λ:λ∈C|arg(−λ)< θ}andk(λ−A)⁻¹k ≤ _|λ−ω|^M , λ /∈ω+S_θ. To give an operator theoretical approach we recall the following definition.

Definition 2.10 ([10]). LetAbe a closed and linear operator with domainD(A) defined on a Hilbert space H. We call A the generator of a solution operator if there exist ω ∈Rand a strongly continuous function Sα :R⁺ →L(H) such that {λ^α: Re(λ)> ω} ⊂ρ(A) andλ^α−1(λ^α−A)⁻¹x=R∞

0 e^−λtS_α(t)dt,Re(λ)> ω, x∈ H. In this case,S_α(·) is called the solution operator generated byA.

We note that, ifA is sectorial of type ω with 0< θ < π(1− ^α₂) then Ais the generator of a solution operator given by

Sα(t) = 1 2πi

Z

Σ

e^−λtλ^α−1(λ^α−A)⁻¹dλ, where Σ is a suitable path lying outside the sectorω+Sα.

Cuesta [10] proved that, if A is a sectorial operator of type ω < 0, for some M >0 and 0< θ < π(1−^α₂), there isC >0 such that

kSα(t)k ≤ CM

1 +|ω|t^α, t≥0. (2.1)

Moreover, we have the following results.

Lemma 2.11 ([10]). Let Sα(t) be a solution operator on H with generator A.

Then, we have

(a) Sα(t)D(A)⊂D(A)andASα(t)x=Sα(t)Axfor all x∈D(A), t≥0;

(b) Let x∈D(A) andt≥0. Then Sα(t)x=x+Rt 0

(t−s)^α−2

Γ(α−1) ASα(s)xds;

(c) Let x∈H andt >0. Then Rt 0

(t−s)^α−2

Γ(α−1) Sα(s)x ds∈D(A)and Sα(t)x=x+A

Z t 0

(t−s)^α−2

Γ(α−1) Sα(s)xds.

Note that the Laplace transform of the abstract function f ∈L²(R⁺, L(K, H)) is defined by

f˜(ς) = Z ∞

0

e^−ςtf(t)dw(t).

Now we consider the problem dx(t) =

Z t 0

(t−s)^α−2

Γ(α−1) Ax(s)ds dt+f(t)dw(t), t >0,1< α <2, (2.2)

x0=ϕ∈H. (2.3)

Formally applying the Laplace transform, we obtain λ˜x(ς)−ϕ=λ^1−αA˜x(ς) + ˜f(λ)dw(λ),

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which establishes the result

λ˜x(ς) =λ^α−1R(λ^α, A)ϕ+λ^α−1R(λ^α, A) ˜f(λ)dw(λ).

This implies that

x(t) =Sα(t)ϕ+ Z t

0

Sα(t−s)f(s)dw(s).

Let x: (−∞, b] →H be a function such thatx, x⁰ ∈ PC. If xis a solution of (1.1)-(1.3), from the partial neutral integro-differential inclusions theory, we obtain x(t)∈Sα(t)[ϕ(0)−G(0, ϕ)]+G(t, xt)+

Z t 0

Sα(t−s)F(s, x_ρ(s,x_s₎)dw(s), t∈[0, t1].

By using thatx(t⁺₁) =x(t⁻₁) +Ik(xt₁), fort∈(t1, t2] we have x(t)∈Sα(t−t1)[x(t⁺₁)−G(t1, x_t+

1)] +G(t, xt) + Z t

t1

Sα(t−s)F(s, x_ρ(s,x_s₎)dw(s)

=Sα(t−t1)[x(t⁻₁) +I1(xt₁)−G(t1, x_t+

1)] +G(t, xt) +

Z t t1

Sα(t−s)F(s, x_ρ(s,x_s₎)dw(s).

By repeating the same procedure, we can easily deduce that x(t)∈S_α(t−t_k)[x(t⁻_k) +I_k(x_t_k)−G(t₁, x_t+

k

)] +G(t, x_t) +

Z t t_k

Sα(t−s)F(s, xρ(s,x_s))dw(s)

holds for any t∈ (tk, tk+1], k = 2, . . . , m. This expression motivates the following definition.

Definition 2.12. AnFt-adapted stochastic process x: (−∞, b]→ H is called a mild solution of the system (1.1)-(1.3) if x0=ϕ, x_ρ(s,x_s₎∈ B for everys ∈J and

∆x(tk) =Ik(xt_k), k= 1, . . . , m, the restriction ofx(·) to the interval (tk, tk+1](k= 0,1, . . . , m) is continuous, and

x(t)∈











Sα(t)[ϕ(0)−G(0, ϕ)] +G(t, xt) +Rt

0Sα(t−s)F(s, x_ρ(s,x_s₎)dw(s), t∈[0, t1], Sα(t−t1)[x(t⁻₁) +I1(xt₁)−G(t1, x_t+

1)] +G(t, xt) +Rt

t₁Sα(t−s)F(s, x_ρ(s,x_s₎)dw(s), t∈(t1, t2], . . .

S_α(t−t_m)[x(t⁻_m) +I_m(x_t_m)−G(t_m, x_t+

m)] +G(t, x_t) +Rt

t_mSα(t−s)F(s, x_ρ(s,x_s₎)dw(s), t∈(tm, b].

Now we have a nonlinear alternative of Leray-Schauder type for multivalued maps due to O’Regan.

Lemma 2.13([29]). Let H be a Hilbert space withV an open,convex subset of H andy∈H. Suppose

(a) Φ :V → Pcd(H)has closed graph, and

(b) Φ :V → Pcd(H)is a condensing map withΦ(V)a subset of a bounded set inH hold.

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Then either

(i) Φhas a fixed point inV; or

(ii) There exist y∈∂V andλ∈(0,1) withy∈λΦ(y) + (1−λ){y0}.

3. Main results

In this section we shall present and prove our main result. Assume that ρ : J× B →(−∞, b] is continuous. In addition, we make the following hypotheses:

(H1) The function t → ϕt is continuous from R(ρ⁻) = {ρ(s, ψ) ≤ 0,(s, ψ) ∈ J × B} into B and there exists a continuous and bounded function J^ϕ : R(ρ⁻)→(0,∞) such thatkϕtk_B≤J^ϕ(t)kϕk_B for eacht∈ R(ρ⁻).

(H2) The multi-valued map F :J× B → Pbd,cl,cv(L(K, H)); for eacht∈J, the functionF(t,·) :B → Pbd,cl,cv(L(K, H)) is u.s.c. and for eachψ∈ B, the functionF(·, ψ) is measurable; for each fixedψ∈ B, the set

S_F,ψ={f ∈L²(J, L(K, H)) :f(t)∈F(t, ψ) for a.et∈J} is nonempty.

(H3) There exists a positive function l : J → R⁺ such that the function s 7→

(_{1+|ω|(t−s)}¹ α)²l(s) belongs toL¹([0, t],R⁺), t∈J, and lim sup

kψk²_B→∞

kF(t, ψ)k² l(t)kψk²_B =γ

uniformly int∈J for a nonnegative constantγ, where kF(t, ψ)k²= sup{Ekfk²:f ∈F(t, ψ)}.

(H4) The function G:J× B →H is continuous and there existL, L1>0 such that

EkG(t, ψ1)−G(t, ψ2)k²≤Lkψ1−ψ2k²_B, t∈J, ψ1, ψ2∈ B, EkG(t, ψ)k²≤L1(kψk²_B+ 1), t∈J, ψ∈ B,

with 4[(CM)²+ 1]LK_b²<1.

(H5) The functions I_k :B →H are completely continuous and there exist con- stantsc_k such that

lim sup

kψk²_B→∞

EkIk(ψ)k² kψk²_B =ck

for everyψ∈ B,k= 1, . . . , m.

Remark 3.1. Let ϕ ∈ B and t ≤ 0. The notation ϕt represents the function defined by ϕt(τ) = ϕ(t+θ). Consequently, if the function x(·) in axiom (A) is such thatx0=ϕ, thenxt=ϕt. We observe thatϕt is well-defined fort <0 since the domain of ϕ is (−∞,0]. We also note that, in general, ϕt ∈ B; consider, for/ instance, a discontinuous function inCr×L^p(h, H) forr >0 (see [21]).

Remark 3.2. The condition (H1) is frequently verified by continuous and bounded functions. In fact, ifB verifies axiom (C2) in the nomenclature of [21], then there exists ˜L > 0 such that kϕk_B ≤ L˜sup_τ≤0kϕ(τ)k for every ϕ ∈ B continuous and bounded, see [21, Proposition 7.1.1] for details. Consequently,

kϕtkB≤L˜sup_τ≤0ϕ(τ) kϕkB

,

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for every continuous and bounded function ϕ∈ B \ {0} and everyt ≤0. We also observe that the spaceCr×L^p(h, H) verifies axiom (C2) see [21, p. 10] for details.

Lemma 3.3. Let x: (−∞, b] → H such that x0 = ϕ and x|_[0,b] ∈ PC(J, H). If (H1) be hold, then

kxskB≤(Mb+J₀^ϕ)kϕkB+Kbsup{kx(θ)k;θ∈[0,max{0, s}]}, s∈ R(ρ⁻)∪J, whereJ₀^ϕ= sup_t∈R(ρ−)J^ϕ(t).

Proof. For anys∈ R(ρ⁻), by (H1), we have

kx_sk_B≤ kϕ_sk_B≤J^ϕ(s)kϕk_B≤J₀^ϕkϕk_B.

For anys∈[0, b],x∈ PC(J, H). Using the phase spaces axioms, we have kxsk_B≤M(s)kϕk_B+K(s) sup{kx(s)k: 0≤s≤t}

≤M_bkϕk_B+K_bsup{kx(s)k: 0≤s≤t}.

Then, fors∈(−∞, b], we have

kxsk_B≤(Mb+J₀^ϕ)kϕk_B+Kbsup{kx(θ)k;θ∈[0,max{0, s}]}, s∈ R(ρ⁻)∪J.

The proof is complete.

Lemma 3.4 ([26]). Let J be a compact interval and H be a Hilbert space. Let F be a multivalued map satisfying (H2) andΓ be a linear continuous operator from L²(J, H) toC(J, H). Then the operator Γ◦S_F :C(J, H)→ P_cp,cv(C(J, H)) is a closed graph in C(J, H)×C(J, H).

Theorem 3.5. Let (H1)–(H5) be satisfied andx0∈L⁰₂(Ω, H), with ρ(t, ψ)≤tfor every(t, ψ)∈J× B. Then problem (1.1)-(1.3)has at least one mild solution on J, provided that

1≤k≤mmax {9(CM)²[1 + 2K_b²c_k+ 2K_b²L₁] + 6K_b²L₁}<1. (3.1) Proof. Consider the space BPC ={x: (−∞, b]→ H;x₀ = 0, x|J ∈ PC}endowed with the uniform convergence topology and define the multi-valued map Φ :BPC → P(BPC) by Φxthe set ofh∈ BPC such that

h(t) =











0, t∈(−∞,0],

S_α(t)[ϕ(0)−G(0, ϕ)] +G(t,x¯_t) +Rt

0S_α(t−s)f(s)dw(s), t∈[0, t₁], S_α(t−t₁)[¯x(t⁻₁) +I₁(¯x_t₁)−G(t₁,x¯_t+

1

)] +G(t,x¯_t) +Rt

t1S_α(t−s)f(s)dw(s), t∈(t₁, t₂], . . .

Sα(t−tm)[¯x(t⁻_m) +Im(¯xt_m)−G(tm,x¯_t+

m)] +G(t,x¯t) +Rt

tmS_α(t−s)f(s)dw(s), t∈(t_m, b],

where f ∈ SF,¯x_ρ = {f ∈ L²(L(K, H)) : f(t) ∈ F(t,x¯_ρ(s,¯_x_t₎) a.e. t ∈ J} and

¯

x: (−∞,0]→ H is such that ¯x0 =ϕ and ¯x= xon J. In what follows, we aim to show that the operator Φ has a fixed point, which is a solution of the problem (1.1)-(1.3).

Let ¯ϕ: (−∞,0) → H be the extension of (−∞,0] such that ¯ϕ(θ) = ϕ(0) = 0 on J and J₀^ϕ = sup{J^ϕ(s) : s ∈ R(ρ⁻)}. We now show that Φ satisfies all the conditions of Lemma 2.13. The proof will be given in several steps.

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Step 1. We shall show there exists an open set V ⊆ BPC with x ∈ λΦx for λ ∈ (0,1) and x /∈ ∂V. Let λ ∈ (0,1) and let x ∈ λΦx, then there exists an f ∈SF,x¯ρ such that we have

x(t) =











λS_α(t)[ϕ(0)−G(0, ϕ)] +λG(t,x¯_t) +λRt

0S_α(t−s)f(s)dw(s), t∈[0, t₁], λS_α(t−t₁)[¯x(t⁻₁) +I₁(¯x_t₁)−G(t₁,x¯_t+

1)] +λG(t,x¯_t) +λRt

t1S_α(t−s)f(s)dw(s), t∈(t₁, t₂],

. . .

λSα(t−tm)[¯x(t⁻_m) +Im(¯xtm)−G(tm,x¯_t+

m)] +λG(t,x¯t) +λRt

t_mSα(t−s)f(s)dw(s), t∈(tm, b],

for someλ∈(0,1). It follows from assumption (H3) that there exist two nonnegative real numbersa₁ anda₂ such that for anyψ∈ B andt∈J,

kF(t, ψ)k²≤a₁l(t) +a₂l(t)kψk²_B. (3.2) On the other hand, from condition (H5), we conclude that there exist positive constantsk(k= 1, . . . , m), γ1such that, for allkψk²_B> γ1,

EkIk(ψ)k²≤(c_k+_k)kψk²_B, max

1≤k≤m{9(CM)²[1 + 2K_b²(c_k+_k) + 2K_b²L₁] + 6K_b²L₁}<1. (3.3) Let

F1={ψ:kψk²_B≤γ1}, F2={ψ:kψk²_B> γ1}, C1= max{EkIk(ψ)k², x∈F1}.

Therefore,

EkIk(ψ)k²≤C1+ (ck+k)kψk²_B. (3.4) Then, by (H4), (3.2) and (3.4), from the above equation, fort∈[0, t₁], we have

Ekx(t)k²≤3EkSα(t)[ϕ(0)−G(0, ϕ)]k²+ 3EkG(t,x¯_t)k² + 3E

Z t

0

Sα(t−s)f(s)dw(s)

2

≤6(CM)²[Ekϕ(0)k²+L1(kϕk²_B+ 1)] + 3L1(kx¯tk²_B+ 1) + 3(CM)²Tr(Q)

Z t 0

1

1 +|ω|(t−s)^α 2

[a₁l(s) +a₂l(s)kx¯_ρ(s,¯_x_s₎k²_B]ds

≤6(CM)²[ ˜H²Ekϕk²_B+L1(kϕk²_B+ 1)] + 3L1(k¯xtk²_B+ 1) + 3(CM)²Tr(Q)a1

Z t1

0

1

1 +|ω|(t1−s)^α 2

l(s)ds

+ 3(CM)²Tr(Q)a₂ Z t

0

1

1 +|ω|(t−s)^α 2

l(s)k¯x_ρ(s,¯_x_s₎k²_Bds.

Similarly, for anyt∈(tk, tk+1], k= 1, . . . , m, we have Ekx(t)k²

≤3EkSα(t−tk)[¯x(t⁻_k) +Ik(¯xt_k)−G(tk,x¯_t+

k)]k²+ 3EkG(t,¯xt)k² + 3E

Z t

t_k

S_α(t−s)f(s)dw(s)

2

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≤9(CM)²[Ekx(t¯ ⁻_k)k²+C1+ (ck+k)k¯xt_kk²_B+L1(k¯x_t+

kk²_B+ 1)]

+ 3L1(kx¯tk²_B+ 1) + 3(CM)²a1Tr(Q) Z t_k+1

tk

1

1 +|ω|(t_k+1−s)^α 2

l(s)ds

+ 3(CM)²a₂Tr(Q) Z t

t_k

1

1 +|ω|(t−s)^α 2

l(s)k¯x_ρ(s,¯_x_s₎k²_Bds.

Then, for allt∈[0, b], we have Ekx(t)k²

≤Mf+ 9(CM)²[Ekx(t¯ ⁻_k)k²+ (ck+k)k¯xt_kk²_B+L1kx¯_t+ k

k²_B] + 3L₁k¯x_tk²_B+ 3(CM)²a₂Tr(Q)

Z t 0

1

1 +|ω|(t−s)^α ²

l(s)k¯x_ρ(s,¯_x_s₎k²_Bds, where

Mf= maxn

6(CM)²[ ˜H²Ekϕk²_B+L1(kϕk²_B+ 1)] + 3L1

+ 3(CM)²Tr(Q)a₁ Z t1

0

1

1 +|ω|(b−s)^α ²

l(s)ds, 9(CM)²(C₁+L₁) + 3L₁+ 3(CM)²a₁Tr(Q)

Z t_k+1 t_k

1

1 +|ω|(tk+1−s)^α ²

l(s)dso . By Lemmas 2.8 and 3.3, it follows thatρ(s, xs)≤s, s∈[0, t], t∈[0, b] and

kx_ρ(s,x_s₎k²_B≤2[(M_b+J₀^ϕ)Ekϕk_B]²+ 2K_b² sup

0≤s≤b

Ekx(s)k². (3.5) For eacht∈[0, b], we have

Ekx(t)k²≤M_∗+{9(CM)²[1 + 2K_b²(ck+k) + 2K_b²L1] + 6K_b²L1} sup

t∈[0,b]

Ekx(t)k²

+ 6(CM)²a₂K_b²Tr(Q) Z t

0

1

1 +|ω|(t−s)^α 2

l(s) sup

τ∈[0,s]

Ekx(τ)k²ds,

where

M∗=Mf+ 9(CM)²[C1+ (ck+k)C^∗+L1(C^∗+ 1)] + 3L1(C^∗+ 1) + 3(CM)²Tr(Q)a2C^∗

Z b 0

1

1 +|ω|(b−s)^α ²

l(s)ds, C^∗= 2[(M_b+J₀^ϕ)kϕkB]².

SinceL∗= max1≤k≤m{9(CM)²[1 + 2K_b²(ck+k) + 2K_b²L1] + 6K_b²L1}<1, we have sup

t∈[0,b]

Ekx(t)k²

≤ M_∗

1−L_∗ +6(CM)²a2K_b²Tr(Q) 1−L_∗

Z b 0

1

1 +|ω|(b−s)^α ²

l(s) sup

τ∈[0,s]

Ekx(τ)k²ds.

Applying Gronwall’s inequality in the above expression, we obtain sup

t∈[0,b]

Ekx(s)k²≤ M_∗

1−L_∗ expn6(CM)²a2K_b²Tr(Q) 1−L_∗

Z b 0

1

1 +|ω|(b−s)^α 2

l(s)dso

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and, therefore, kxk²_PC≤ M_∗

1−L∗

expn6(CM)²a₂K_b²Tr(Q) 1−L∗

Z b 0

1

1 +|ω|(b−s)^α 2

l(s)dso

<∞.

Then, there existsr^∗ such thatkxk²_PC 6=r^∗. Set

V ={x∈ BPC:kxk²_PC< r^∗}.

From the choice ofV, there is no x∈∂V such thatx∈λΦxforλ∈(0,1).

Step 2. Φ has a closed graph. Letx⁽ⁿ⁾→x^∗, hn ∈Φx⁽ⁿ⁾, x⁽ⁿ⁾∈V =Br^∗(0,BPC) andhn →h_∗. From Axiom (A), it is easy to see that (x⁽ⁿ⁾)s→x^∗s uniformly for s ∈ (−∞, b] as n → ∞. We prove that h_∗ ∈ Φx^∗. Now h_n ∈ Φx⁽ⁿ⁾ means that there existsf_n∈S_F,x_(n)

ρ such that, for eacht∈[0, t₁], h_n(t) =S_α(t)[ϕ(0)−G(0, ϕ)] +G(t,(x⁽ⁿ⁾)_t) +

Z t 0

S_α(t−s)f_n(s)dw(s), t∈[0, t₁].

We must prove that there existsf_∗∈S_F,x∗

ρ such that, for eacht∈[0, t1], h∗(t) =Sα(t)[ϕ(0)−G(0, ϕ)] +G(t,(x^∗)t) +

Z t 0

Sα(t−s)f∗(s)dw(s), t∈[0, t1].

Now, for everyt∈[0, t₁], we have

hn(t)−Sα(t)[ϕ(0)−G(0, ϕ)]−G(t,(x⁽ⁿ⁾)t)− Z t

0

Sα(t−s)fn(s)dw(s)

−

h∗(t)−Sα(t)[ϕ(0)−G(0, ϕ)]−G(t,(x^∗)t)

− Z t

0

S_α(t−s)f_∗(s)dw(s)

2

PC→0 asn→ ∞.

Consider the linear continuous operator Ψ :L([0, t1], H)→C([0, t1], H), Ψ(f)(t) =

Z t 0

Sα(t−s)f(s)dw(s).

From Lemma 3.4, it follows that Ψ◦SF is a closed graph operator. Also, from the definition of Ψ, we have that, for everyt∈[0, t1],

h_n(t)−S_α(t)[ϕ(0)−G(0, ϕ)]−G(t,(x⁽ⁿ⁾)_t)− Z t

0

S_α(t−s)f_n(s)dw(s)

∈Γ(S_F,x_(n)).

Sincex⁽ⁿ⁾→x^∗, for somef_∗∈S_F,x∗

ρ it follows that, for everyt∈[0, t1], h∗(t)−Sα(t)[ϕ(0)−G(0, ϕ)]−G(t,(x^∗)t) =

Z t 0

Sα(t−s)f∗dw(s).

Similarly, for anyt∈(t_k, t_k+1], k = 1, . . . , m, we have hn(t) =Sα(t−tk)[x⁽ⁿ⁾(t⁻_k) +Ik(x⁽ⁿ⁾t_k)−G(tk,(x⁽ⁿ⁾)_t+

k)] +G(t,(x⁽ⁿ⁾)t) +

Z t t_k

S_α(t−s)f_n(s)dw(s), t∈(t_k, t_k+1].

We must prove that there existsf∗∈S_F,x∗

ρ such that, for eacht∈(tk, tk+1], h_∗(t) =Sα(t−tk)[x^∗(t⁻_k) +Ik(x^∗t_k)−G(tk,(x^∗)_t+

k

)] +G(t,(x^∗)t)

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+ Z t

tk

Sα(t−s)f_∗(s)dw(s), t∈(tk, tk+1].

Now, for everyt∈(t_k, t_k+1], k= 1, . . . , m, we have

hn(t)−Sα(t−tk)[x⁽ⁿ⁾(t⁻_k) +Ik(x⁽ⁿ⁾t_k)−G(tk,(x⁽ⁿ⁾)_t+

k)]−G(t,(x⁽ⁿ⁾)t)

− Z t

t_k

S_α(t−s)f_n(s)dw(s)

−

h_∗(t)−S_α(t−t_k)

x^∗(t⁻_k) +I_k(x^∗_t_k)

−G(tk,(x^∗)_t+ k)

−G(t,(x^∗)t)− Z t

t_k

Sα(t−s)f∗(s)dw(s)

2

PC→0 asn→ ∞.

Consider the linear continuous operator Ψ : L²((t_k, t_k+1], H) → C((t_k, t_k+1], H), k= 1, . . . , m,

Ψ(f)(t) = Z t

t_k

S_α(t−s)f(s)dw(s).

From Lemma 3.4, it follows that Ψ◦SF is a closed graph operator. Also, from the definition of Ψ, we have that, for everyt∈(tk, tk+1], k= 1, . . . , m,

h_n(t)−S_α(t−t_k)[x⁽ⁿ⁾(t⁻_k)+I_k(x⁽ⁿ⁾_t_k)−G(t_k,(x⁽ⁿ⁾)_t+ k

)]−G(t,(x⁽ⁿ⁾)_t)∈Γ(S_F,x_(n)

ρ).

Since x⁽ⁿ⁾ → x^∗, for some f∗ ∈S_F,x∗

ρ it follows that, for every t ∈ (tk, tk+1], we have

h_∗(t)−Sα(t−tk)[x^∗(t⁻_k) +Ik(x^∗t_k)−G(tk,(x^∗)_t+

k)]−G(t,(x^∗)t)

= Z t

t_k

S_α(t−s)f_∗(s)dw(s).

Therefore, Φ has a closed graph.

Step 3. We show that the operator Φ condensing. For this purpose, we decompose Φ as Φ₁ + Φ₂, where the map Φ₁ : V → P(BPC) be defined by Φ1x, the set h₁∈ BPC such that

h₁(t) =











0, t∈(−∞,0],

−Sα(t)G(0, ϕ) +G(t,x¯t), t∈[0, t1],

−Sα(t−t1)G(t1,x¯_t+

1) +G(t,x¯t), t∈(t1, t2], . . .

−Sα(t−tm)G(tm,x¯_t+

m) +G(t,x¯t), t∈(tm, b],

and the map Φ₂:V → P(BPC) be defined by Φ2x, the seth₂∈ BPC such that

h₂(t) =











0, t∈(−∞,0],

Sα(t)ϕ(0) +Rt

0Sα(t−s)f(s)ds, t∈[0, t1], Sα(t−t1)[¯x(t⁻₁) +I1(¯xt₁)] +Rt

t₁Sα(t−s)f(s)dw(s), t∈(t1, t2], . . .

Sα(t−tm)[¯x(t⁻_m) +Im(¯xt_m)] +Rt

t_mSα(t−s)f(s)dw(s), t∈(tm, b].

We first show that Φ₁is a contraction while Φ₂is a completely continuous operator.

Claim 1. Φ₁ is a contraction on BPC. Let t ∈ [0, t₁] and v^∗, v^∗∗ ∈ BPC. From (H4), Lemmas 2.8 and 3.3, we have

Ek(Φ₁v^∗)(t)−(Φ₁v^∗∗)(t)k²≤EkG(t, v^∗_t)−G(t, v^∗∗_t)k²

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≤Lkv^∗t−v^∗∗tk²_B

≤2LK_b²sup{kv^∗(τ)−v^∗∗(τ)k²,0≤τ≤t}

≤2LK_b² sup

s∈[0,b]

kv^∗(s)−v^∗∗(s)k²

= 2LK_b² sup

s∈[0,b]

kv^∗(s)−v^∗∗(s)k² (since ¯v=v onJ)

= 2LK_b²kv^∗−v^∗∗k²_PC. Similarly, for anyt∈(tk, tk+1], k= 1, . . . , m, we have

Ek(Φ₁v^∗)(t)−(Φ₁v^∗∗)(t)k²

≤2EkS_α(t−t_k)[−G(t_k, v^∗_t+ k

) +G(t_k, v^∗∗_t+ k

)]k²+ 2EkG(t, v^∗_t)−G(t, v^∗∗_t)k²

≤2(CM)²Lkv^∗_t+

k −v^∗∗_t+

kk²_B+ 2Lkv^∗t−v^∗∗tk²_B

≤4((CM)²+ 1)LK_b² sup

s∈[0,b]

kv^∗(s)−v^∗∗(s)k²

= 4((CM)²+ 1)LK_b² sup

s∈[0,b]

kv^∗(s)−v^∗∗(s)k² (since ¯v=vonJ)

= 4[(CM)²+ 1)]LK_b²kv^∗−v^∗∗k²_PC, Thus, for allt∈[0, b], we have

Ek(Φ1v^∗)(t)−(Φ₁v^∗∗)(t)k²≤L₀kv^∗−v^∗∗k²_PC. Taking supremum overt,

kΦ1v^∗−Φ₁v^∗∗k²_PC≤L₀kv^∗−v^∗∗k²_PC,

whereL₀= 4[(CM)²+ 1]LK_b²<1. Hence, Φ₁ is a contraction onBPC.

Claim 2. Φ₂ is convex for eachx∈V. In fact, ifh¹₂, h²₂ belong to Φ₂x, then there existf₁, f₂∈S_F,x_ρ such that

hⁱ₂(t) =Sα(t)ϕ(0) + Z t

0

Sα(t−s)fi(s)dw(s), t∈[0, t1], i= 1,2.

Let 0≤λ≤1. For eacht∈[0, t1] we have (λh¹₂+ (1−λ)h²₂)(t) =Sα(t)ϕ(0) +

Z t 0

Sα(t−s)[λf1(s) + (1−λ)f2(s)]dw(s).

Similarly, for anyt∈(t_k, t_k+1], k= 1, . . . , m, we have hⁱ₂(t) =Sα(t−tk)[¯x(t⁻_k) +Ik(¯xt_k)] +

Z t t_k

Sα(t−s)fi(s)dw(s), i= 1,2.

Let 0≤λ≤1. For eacht∈(tk, tk+1], k= 1, . . . , m, we have (λh¹₂+ (1−λ)h²₂)(t) =Sα(t−tk)[¯x(t⁻_k) +Ik(¯xt_k)]

+ Z t

t_k

Sα(t−s)[λf1(s) + (1−λ)f2(s)]dw(s).

SinceSF,¯x_ρis convex (becauseFhas convex values) we have (λh¹₂+(1−λ)h²₂)∈Φ2x.

Claim 3. Φ2(V) is completely continuous. We begin by showing Φ2(V) is equicon- tinuous. Ifx∈V, from Lemmas 2.8 and 3.3, it follows that

k¯x_ρ(s,¯_x_s₎k²_B≤2[(M_b+J₀^ϕ)kϕkB]²+ 2K_b²r^∗:=r⁰.