Bulletin of the Iranian Mathematical Society https://doi.org/10.1007/s41980-018-0096-8 ORIGINAL PAPER
Existence of Optimal Mild Solutions for Multi-valued Impulsive Stochastic Partial Functional Integrodifferential Equations Zuomao Yan1 · Fangxia Lu1 Received: 14 July 2017 / Accepted: 25 December 2017 © Iranian Mathematical Society 2018
Abstract In this paper, we introduce a new class of multi-valued impulsive stochastic partial functional integrodifferential equations with infinite delay in the α-norm. Using stochastic analysis, analytic semigroup and fixed point strategy with the properties of fractional powers of closed operators, we establish the existence and uniqueness results of mild solutions for these equations with not instantaneous impulse. Then, the existence of optimal mild solutions is also proved. Particularly, the compactness of the operator semigroups is not needed. Finally, an example to illustrate the applications of main results is given. Keywords Multi-valued impulsive stochastic partial functional integrodifferential equations · Optimal mild solutions · Fractional powers of closed operators · Infinite delay · Fixed point theorem Mathematics Subject Classification Primary 34A37; Secondary 34A60 · 35R60 · 60H15
1 Introduction The theory of impulsive differential and integrodifferential systems has been an object of interest because of its wide applications in science and engineering. The reason for this applicability arises from the fact that impulsive differential and integrodifferential
Communicated by Asadollah Aghajani.
B
Zuomao Yan
[email protected] Fangxia Lu
[email protected]
1
Department of Mathematics, Hexi University, Zhangye 734000, Gansu, People’s Republic of China
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problems are an appropriate model for describing process which at certain moments change their state rapidly and which cannot be described using the classical differential problems. For some of these applications, we refer to [5,6,25] and references therein. On the other hand, stochastic differential systems play an important role in formulation and analysis of mechanical, electrical, control engineering, physical sciences and so on. There are many interesting results on the theory and applications of stochastic differential and integrodifferential systems, see [33,36,38,39]. However, besides impulse effects and delays, stochastic effects likewise exist in real systems. In recent years, several papers devoted to the existence and other quantitative and qualitative properties of solutions to partial impulsive stochastic differential and functional integrodifferential systems in abstract spaces using fixed point techniques (see [1–3,23,29,35,37,45]). As the generalization of classic impulsive differential equations, impulsive partial stochastic differential and integrodifferential inclusions have been extensively investigated in Hilbert spaces. For example, using the fixed point theorem for multi-valued mapping by Dhage, Lin and Hu [28] considered the existence of mild solutions for a class of impulsive neutral stochastic functional integrodifferential inclusions with nonlocal initial conditions in Hilbert space. Chadha and Pandey [10] investigated the existence of mild solution for impulsive neutral stochastic fractional integrodifferential inclusions with nonlocal conditions. Park and Jeong [30] derived the existence of mild solutions for a class of impulsive neutral stochastic functional integrodifferential inclusions with infinite delays in Hilbert spaces. Ren et al. [34] established the controllability of impulsive neutral stochastic functional differential inclusions with infinite delay in an abstract space. The approximate controllability for a class of impulsive partial neutral stochastic functional differential inclusions in Hilbert spaces is considered [41]. Yan and Yan [46] proved the existence results for a class of impulsive nonlocal stochastic functional integrodifferential inclusions in a real separable Hilbert space using Bohnenblust–Karlin’s fixed point theorem and fractional operators combined with approximation techniques. Yan and Zhang [47] discussed the existence of mild solutions for a class of impulsive fractional partial neutral stochastic integrodifferential inclusions with state-dependent delay by means of the nonlinear alternative of Leray–Schauder type for multi-valued maps by O’Regan. Boudaoui et al. [7] studied the existence of mild solutions for a first-order impulsive semilinear stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay by employing Leray–Schauder fixed point theorem for multi-valued mapping. In many cases, the models with instantaneous impulses cannot characterize many practical problems, for example, the dynamics of evolution processes in pharmacotherapy (see [4,40]). Recently, the existence results for abstract differential and integrodifferential equations with not instantaneous impulses are derived using fixed point theorems, such as in Hernández et al. [19,20], Pierri et al. [32], Kumar et al. [24], Yu and Wang [48], Gautam and Dabas [17], Colao et al. [11]. From a practical point of view, many physical phenomena in evolution processes are modeled as impulsive stochastic differential systems for which the impulses are not instantaneous. Using the Banach contraction mapping principle, Yan et al. [43] discussed the existence, uniqueness and continuous dependence of mild solutions for fractional impulsive partial functional stochastic integrodifferential equations with not instantaneous impulses. Yan and Jia [42] studied the existence of mild solutions for a new class of impulsive
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stochastic partial neutral functional integrodifferential equations with infinite delay in separable Hilbert spaces using the Hausdorff measure of noncompactness, the Darbo fixed point theorem and Darbo–Sadovskii fixed point theorem. In [44], the authors investigated the approximate controllability of a multi-valued fractional impulsive stochastic partial integrodifferential equation with infinite delay and not instantaneous impulses by means of the fixed point theorem for multi-valued operators by Dhage and analytic α-resolvent operator. In this paper, we consider the existence of optimal mild solutions to the following multi-valued impulsive stochastic partial functional integrodifferential equations with infinite delay and not instantaneous impulses of the form t h(t, s, xs )ds dw(t), d[x(t) − G(t, xt )] ∈ Ax(t)dt + F t, xt , 0
t ∈ (si , ti+1 ], i = 0, 1, . . . , N , x(0) = ϕ(t), t ∈ (− ∞, 0],
(1.1) (1.2)
x(t) = gi (t, xt ), t ∈ (ti , si ], i = 1, . . . , N ,
(1.3)
where the state x(·) takes values in a separable real Hilbert space H with inner product ·, · H and norm · H , A is the infinitesimal generator of an analytic semigroup of bounded linear operators {T (t)}t≥0 on H . The history xt : (−∞, 0] → H , xt (θ ) = x(t + θ ), belongs to some abstract phase space B defined axiomatically in Sect. 2. Let K be another separable Hilbert space with inner product ·, · K and norm · K . Suppose {w(t) : t ≥ 0} is a given K -valued Brownian motion or Wiener process with a finite trace nuclear covariance operator Q > 0 defined on a complete probability space (, F, P) equipped with a normal filtration {Ft }t≥0 , which is generated by the Wiener process w. h(t), t ∈ [0, b] is a bounded linear operator, and let 0 = t0 = s0 < t1 ≤ s1
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physics and mathematics. Motivated by the above consideration, we will study the interesting problem. This is one of our motivations. This paper has two main contributions: (i) We first introduce the new concept of optimal mild solutions for multi-valued impulsive stochastic systems, which is natural generalizations of the concept of optimality for differential systems well known in the theory of infinite dimensional systems. (ii) We investigated the existence and uniqueness of mild solutions of (1.1)–(1.3) using stochastic analysis, analytic semigroup, fractional powers of closed operators and the fixed point theorems for multi-valued operators by Covitz and Nadler, Dhage, respectively. Further, the existence of optimal mild solutions in the α-norm are obtained for the non-Lipschitz conditions cases. The known results appeared in [8,14,49] are generalized to the impulsive stochastic inclusions settings and the case of infinite delay without the assumptions of the compactness of the operator semigroups. Compared with the earlier results obtained in [8,14,49], there are at least three main advantages: (i) The impulsive stochastic models are more general and accurate than deterministic models in our daily life. (ii) When the nonlinear functions are nontrivial and convex, we get the existence results for optimal α-mild solutions under the mixed Lipschitz and Carathéodory conditions. (iii) The results are obtained using the analytic semigroup theory and fractional powers of closed operators. Furthermore, we see the additional way to improve the paper. For instance, using the properties of noncompact measure and weak sequentially closed graph operators, we establish sufficient conditions to guarantee optimality results. We do not assume that the operators are compact or we do not assume a compactness type condition on the multi-valued function. The rest of this paper is organized as follows. In Sect. 2, we introduce some notations and necessary preliminaries. In Sect. 3, we give the existence of mild solutions. In Sect. 4, the existence of optimal mild solutions is proved. In Sect. 5, an example is given to illustrate our results. Finally, concluding remarks are given in Sect. 6.
2 Preliminaries Let H , K be two real separable Hilbert spaces and we denote by ·, · H , ·, · K their inner products and by · H , · K their vector norms, respectively. L(K , H ) be the space of bounded linear operators mapping K into H equipped with the usual norm · H and L(H ) denotes the Hilbert space of bounded linear operators from H to H . Let {w(t) : t ≥ 0} denote an K -valued Wiener process defined on the probability space (, F, P) with covariance operator Q, that is, Ew(t), x K w(s), y K = (t ∧ s)Qx, y K , for all x, y ∈ K , where Q is a positive, self-adjoint, trace class operator on K . In particular, we denote w(t) an K -valued Q-Wiener process with respect to {Ft }t≥0 . To define stochastic integrals with respect to the Q-Wiener process w(t), we introduce the subspace K 0 = Q 1/2 (K ) of K which is endowed with the inner product ˜ Q −1/2 v ˜ K is a Hilbert space. We assume that there exists a u, ˜ v ˜ K 0 = Q −1/2 u, complete orthonormal system {en }∞ n=1 in K , a bounded sequence of nonnegative real
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numbers {λn }∞ n=1 such that Qen = λn en , and a sequence βn of independent Brownian motions such that ∞ λn en , e K βn (t), e ∈ K , t ∈ [0, b], w(t), e K = n=1
and Ft = Ftw , where Ftw is the σ -algebra generated by {w(s) : 0 ≤ s ≤ t}. Let L 02 = L 2 (K 0 , H ) be the space of all Hilbert–Schmidt operators from K 0 to H with the norm ψ 2L 0 = Tr((ψ Q 1/2 )(ψ Q 1/2 )∗ ) for any ψ ∈ L 02 . Clearly for any bounded operators 2
ψ ∈ L(K , H ) this norm reduces to ψ 2L 0 = Tr(ψ Qψ ∗ ). Let L Ft (, H ) be the 2 Banach space of all Ft -measurable pth power integrable random variables with values in the Hilbert space H . Let C([0, b]; L p (, H )) be the Banach space of continuous p maps from [0, b] to L p (, H ) satisfying the condition supt∈[0,b] E x(t) H < ∞. We use the notation P(H ) for the family of all subsets of H . Let us introduce the following notations: p
Pcl (H ) = {x ∈ P(H ) : x is closed}, Pbd (H ) = {x ∈ P(H ) : x is bounded}, Pcv (H ) = {x ∈ P(H ) : x is convex}, Pcp (H ) = {x ∈ P(H ) : x is compact}. Consider Hd : P(H ) × P(H ) → R+ ∪ {∞} given by ˜ Hd ( A, B) = max sup d(a, ˜ B), sup d( A, b) , a∈ ˜ A
˜ b∈ B
˜ = inf d(a, ˜ d(a, ˜ Then, (Pbd,cl (H ), Hd ) b) where d( A, ˜ b), ˜ B) = inf b∈ ˜ b). ˜ a∈ ˜ A B d(a, is a metric space and (Pcl (H ), Hd ) is a generalized metric space. In what follows, we briefly introduce some facts on multi-valued analysis. For more details, one can see [15,22]. A multi-valued map : H → P(H ) is convex (closed) valued if G(H ) is convex (closed) for all x ∈ H . is bounded on bounded sets if (D) = x∈D (x) is bounded in H for any bounded set D of H , that is, supx∈D {sup { y H : y ∈
(x)}} < ∞.
is called upper semicontinuous (u.s.c., in short) on H , if for any x ∈ H , the set
(x) is a nonempty, closed subset of H , and if for each open set B of H containing
(x), there exists an open neighborhood N of x such that (N ) ⊆ B.
is said to be completely continuous if (D) is relatively compact for every bounded subset D of H . If the multi-valued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph, i.e., xn → x∗ , yn → y∗ , yn ∈ (xn ) imply y∗ ∈ (x∗ ).
is said to be completely continuous if (D) is relatively compact, for every bounded subset D ⊆ H . A multi-valued map : J → Pcl (H ) is said to be measurable if for each x ∈ H , the function Y : J → R+ defined by Y (t) = d(x, (t)) = inf{d(x, z) : z ∈ (t)} is measurable.
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has a fixed point if there is x ∈ H such that x ∈ (x). Definition 2.1 A multi-valued operator : H → Pcl (H ) is called: (a) γ -Lipschitz if there exists γ > 0 such that Hd ( (x), (y)) ≤ γ d(x, y), x, y ∈ H . (b) A contraction if it is γ -Lipschitz with γ < 1. Definition 2.2 Let A be the infinitesimal generator of an analytic semigroup T (t). For every β > 0, we define (−A)β = ((−A)−β )−1 . For β = 0, (−A)β = I . We note that D((−A)β ) is a Banach space equipped with the norm x β = (−A)β x , x ∈ D((−A)β ). By X β , we denote this Banach space. We collect some basic properties of fractional powers (−A)β appearing in Pazy [31]. Lemma 2.3 [31]. Let A be the infinitesimal generator of an analytic semigroup T (t). If 0 ∈ ρ(A), then (i) If β ≥ α > 0 then X β ⊂ X α . (ii) If β, α > 0 then (−A)β+α x = (−A)β (−A)α x for every x ∈ D((−A)η ) where η = max(β, α, β + α). (iii) For every x ∈ D((−A)β ), then T (t)(−A)β x = (−A)β T (t)x. (iv) For every β > 0, there exists a positive constant Mβ such that (−A)β T (t) H ≤
Mβ . tβ
We introduce the space PC(Hα ) formed by all Ft -adapted measurable, Hα -valued stochastic processes {x(t) : t ∈ [0, b]} such that x is continuous at t = ti , x(ti ) = x(ti− ) and x(ti+ ) exists for all i = 1, ..., N . In this paper, we always assume that p
1
PC(Hα ) is endowed with the norm x PC = (sup0≤t≤b E x(t) α ) p . Then (PC(Hα ), · PC ) is a Banach space. The notation Br (x, Hα ) stands for the closed ball with center at x and radius r > 0 in Hα . In this paper, we assume that the phase space (B, · B ) is a seminormed linear space of F0 -measurable functions mapping (−∞, 0] into Hα , and satisfying the following fundamental axioms by Hale and Kato (see, e.g., in [18]). (A) If x : (−∞, σ + b] → Hα , b > 0, is such that x|[σ,σ +b] ∈ C([σ, σ + b], Hα ) and xσ ∈ B, then for every t ∈ [σ, σ + b] the following conditions hold: (i) xt is in B; (ii) x(t) α ≤ H˜ xt B ; (iii) xt B ≤ K (t − σ ) sup{ x(s) α : σ ≤ s ≤ t} + M(t − σ ) xσ B , where H˜ ≥ 0 is a constant; K , M : [0, ∞) → [1, ∞), K is continuous and M is locally bounded, and H˜ , K , M are independent of x(·).
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(B) The space B is complete. Remark 2.4 In impulsive stochastic functional differential systems with infinite delay, the map [σ, σ + b] → B, t → xt , is in general discontinuous. For this reason, this property has been omitted from our description of the phase space B. The next result is a consequence of the phase space axioms. Lemma 2.5 Let x : (−∞, b] → Hα be an Ft -adapted measurable process such that p the F0 -adapted process x0 = ϕ(t) ∈ L F0 (, B) and x|[0,b] ∈ PC([0, b], Hα ), then xs B ≤ Mb E ϕ B +K b sup E x(s) α , 0≤s≤b
where K b = sup{K (t) : 0 ≤ t ≤ b}, Mb = sup{M(t) : 0 ≤ t ≤ b}. Definition 2.6 An Ft -adapted stochastic process x : (−∞, b] → Hα is called a mild solution of the system (1.1)–(1.3) if x0 = ϕ ∈ B, x(·)|[0,b] ∈ PC and (i) x(t) is measurable and adapted to Ft , t ≥ 0. (ii) x(t) ∈ Hα has càdlàg paths on t ∈ [0, b] a.s and for each t ∈ [0, b], x(t) satisfies x(t) = g j (t, x(t)) for all t ∈ (t j , s j ], j = 1, . . . , N , and for each s ∈ [0, t) the function AT (t − s) f (s, xs ) is integrable and x(t) = T (t)[ϕ(0) − G(0, ϕ)] + G(t, xt ) t t AT (t − s)G(s, xs )ds + T (t − s) f (s)dw(s) + 0
0
for all t ∈ [0, t1 ] and x(t) = T (t − si )[gi (si , x(si )) − G(si , x(si ))] + G(t, xt ) t t + AT (t − s)G(s, xs )ds + T (t − s) f (s)dw(s) si
si
for all t ∈ (si , ti+1 ], i = 1, . . . , N , where f ∈ S F,x = { f ∈ L p ([0, b], t L(K , H )) : f (t) ∈ F(t, xt , 0 h(t, s, xs )ds) a.e. t ∈ [0, b]}. Lemma 2.7 [13]. For any p ≥ 1 and for arbitrary L 02 (K , H )-valued predictable process φ(·) such that sup E
s∈[0,t]
s 0
2 p t p 2 p 1/ p p φ(v)dw(v) ≤ ( p(2 p − 1)) (E φ(s) ) ds , L 02 H
0
t ∈ [0, ∞). In the rest of this paper, we denote by C p = ( p( p − 1)/2) p/2 .
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Lemma 2.8 (Covitz and Nadler [12]) Let (H , d) be a complete metric space. If : H → Pcl (H ) is a contraction, then fix = ∅. Lemma 2.9 (Dhage [16]) Let H be a Hilbert space, 1 : H → Pcl,cv,bd (H ) and
2 : H → Pcp,cv (H ) be two multi-valued maps satisfying: (a) 1 is a contraction with a contraction constant k, and (b) 2 is u.s.c. and completely continuous. Then either (i) the operator inclusion λx ∈ 1 x + 1 x has a solution for λ = 1, or (ii) the set E = {x ∈ H : λx ∈ 1 x + 2 x, λ > 1} is unbounded.
3 Existence of Mild Solutions In this section, we shall present the existence and uniqueness of mild solutions of (1.1)–(1.3). For some α ∈ (0, 1), let us list the following hypotheses. (H1) The analytic semigroup T (t) generated by A and there exists a constant M ≥ 1 such that T (t) H ≤ M. (H2) For β ∈ (0, 1), the function G : [0, b] × B → Hβ+α , and there exists a constant L G > 0 such that (i) E (−A)β G(t, ψ1 ) − (−A)β G(t, ψ2 ) αp ≤ L G ψ1 − ψ2 B , p
t ∈ [0, b], ψ1 , ψ2 ∈ B,
(ii) E (−A)β G(t, ψ) αp ≤ L G ( ψ B +1), t ∈ [0, b], ψ ∈ B. p
(H3) There exists a ah > 0 such that p t ≤ ah ψ1 − ψ2 p [h(t, s, ψ ) − h(t, s, ψ )]ds E 1 2 B 0
α
for all t ∈ [0, b], ψ1 , ψ2 ∈ B. (H4) The multi-valued map F : [0, b] × B × Hα → Pcl (L(K , H )) has the property that F(t, φ, y) : [0, b] → Pcl (L(K , H )) is measurable for each (φ, y) ∈ B × Hα .
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(H5) There exists a function l(t) ∈ L 1 ([0, b], R+ ), such that p
E Hd (F(t, ψ1 , y1 ), F(t, ψ2 , y2 )) p
≤ l(t)[ ψ1 − ψ2 B + y1 − y2 αp ] for all t ∈ [0, b], ψi ∈ B, yi ∈ Hα , i = 1, 2, and d p (0, F(t, 0, 0)) ≤ l(t) for a.e. t ∈ [0, b]. (H6) The functions gi : (ti , si ] × B → Hα , i = 1, . . . , N , are continuous and there exists γi > 0, i = 1, . . . , N , such that (i) p
E gi (t, ψ1 ) − gi (t, ψ2 ) αp ≤ γi ψ1 − ψ2 B , t ∈ (ti , si ], ψ1 , ψ2 ∈ B,
(ii) p
E gi (t, ψ) αp ≤ γi ( ψ B +1), t ∈ (ti , si ], ψ ∈ B. p
Theorem 3.1 Let x0 ∈ L F0 (, Hα ). If the assumptions (H1), (H2)(i), (H3)–(H6)(i) are satisfied, then the system (1.1)–(1.3) has a unique mild solution on [0, b], provided that 4
p−1
p K b max 1≤i≤N
p p−1 p p−1 (1 + 8 (2 p−1 M p + 1) (−A)−β H M )γi + 4
p/2−1 p−2 b pβ L G + 4 p−1 C p Mα pβ − p + 1 p − 2 − pα ×b p/2−1− pα (1 + ah ) l L 1 < 1. p
+M1−β
(3.1)
Proof We introduce the space Bb of all functions x : (−∞, b] → Hα such that x0 ∈ B and the restriction x|[0,b] ∈ PC. Let · b be a seminorm in Bb defined by
x b = x0 B +
1
p
sup E x(t) αp
, x ∈ Bb .
0≤t≤b
We consider the multi-valued map : Bb → P(Bb ) defined by x, the set of ρ ∈ Bb such that
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ρ(t) =
⎧ ϕ(t), ⎪ ⎪ ⎪ ⎪ T (t)[ϕ(0) − G(0, ϕ)] + G(t, xt ) ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + 0 ASα (t − s)G(s, xs )ds ⎪ ⎪ ⎪ t ⎪ ⎪ ⎨ + 0 T (t − s) f (s)dw(s),
t ∈ (−∞, 0],
t ∈ [0, t1 ], i = 0,
gi (t, xt ), ⎪ ⎪ ⎪ ⎪ ⎪ T (t − s )[g (s , x ) − G(s , x )] ⎪ i i i si i si ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ +G(t, xt ) + si AT (t − s)G(s, xs )ds ⎪ ⎪ ⎪ t ⎩ + si T (t − s) f (s)dw(s),
t ∈ (ti , si ], i ≥ 1,
t ∈ (si , ti+1 ], i ≥ 1,
where f ∈ S F,x . For ϕ ∈ B, we define ϕ˜ by ϕ(t) ˜ =
ϕ(t), T (t)ϕ(0),
− ∞ < t ≤ 0, 0 ≤ t ≤ b,
˜ −∞ < t ≤ b. It is clear to see that x satisfies then ϕ˜ ∈ Bb . Set x(t) = y(t) + ϕ(t), Definition 2.6 if and only if y satisfies y0 = 0 and
y(t) =
⎧ −T (t)G(0, ϕ) + G(t, yt + ϕ˜t ) ⎪ ⎪ t ⎪ ⎪ ⎪ + 0 AT (t − s)G(s, ys + ϕ˜ s )ds ⎪ ⎪ ⎪ ⎪ + 0t T (t − s) f (s)dw(s), ⎪ ⎪ ⎨ g (t, y + ϕ˜ ), i
t
t ∈ [0, t1 ], i = 0, t ∈ (ti , si ], i ≥ 1,
t
T (t − si )[gi (si , ysi + ϕ˜si ) − G(si , ysi + ϕ˜ si )] ⎪ ⎪ ⎪ ⎪ +G(t, ⎪ ⎪ t yt + ϕ˜t ) ⎪ ⎪ + AT (t − s)G(s, ys + ϕ˜s )ds ⎪ ⎪ sti ⎪ ⎩ + si T (t − s) f (s)dw(s),
t ∈ [si , ti+1 ], i ≥ 1,
where f ∈ S F,y = { f ∈ L p ([0, b], L(K , H )) : f (t) ∈ F(t, yt + ϕ˜t , ϕ˜s )ds)) a.e. t ∈ [0, b]}. Let Bb0 = {y ∈ Bb : y0 = 0 ∈ B}. For any y ∈ Bb0 ,
y b = y0 B +
p
=
0≤t≤b
0
1 sup E y(t) αp
t
h(t, s, ys +
1
p
sup E y(t) αp
0≤t≤b
thus (Bb0 , · b ) is a Banach space. Moreover, from Lemma 2.5, we have p
p
p
yt + ϕ˜t B ≤ 2 p−1 ( yt B + ϕ˜t B ) p
p
p
≤ 4 p−1 K b sup E y(s) αp +Mb E y0 B 0≤s≤b
p +K b
123
sup E ϕ(s) ˜ 0≤s≤b
p αp +Mb
p
ϕ˜0 B
,
(3.2)
Bulletin of the Iranian Mathematical Society
≤4
p−1
p
K b sup E y(s) αp
0≤s≤b p p +K b M E ϕ(0)
p p αp +Mb ϕ B ,
t ∈ [0, b].
(3.3)
¯ : B 0 → P(B 0 ) defined by y, ¯ the set of ρ¯ ∈ B 0 Consider the multi-valued map
b b b such that ρ(t) ¯ = 0, t ∈ (−∞, 0] and
ρ(t) ¯ =
⎧ −T (t)G(0, ϕ) + G(t, yt + ϕ˜t ) ⎪ ⎪ t ⎪ ⎪ ⎪ + AS (t − s)G(s, ys + ϕ˜s )ds α ⎪ 0t ⎪ ⎪ ⎪ + T (t − s) f (s)dw(s), ⎪ ⎪ ⎨ g (t, y0 + ϕ˜ ), i
t
t
T (t − si )[gi (si , ysi + ϕ˜si ) − G(si , ysi + ϕ˜ si )] ⎪ ⎪ ⎪ ⎪ ⎪ +G(t, ⎪ t yt + ϕ˜t ) ⎪ ⎪ + ⎪ si AT (t − s)G(s, ys + ϕ˜ s )ds ⎪ ⎪ ⎩ + t T (t − s) f (s)dw(s), si
t ∈ [0, t1 ], i = 0, t ∈ (ti , si ], i ≥ 1,
t ∈ (si , ti+1 ], i ≥ 1,
¯ has a fixed point in B 0 , then has a fixed point in B 0 which is a where f ∈ S F,y . If
b b ¯ satisfies the assumptions of Lemma 2.8. solution of(1.1)–(1.3). We shall show that
The proof will be given in two steps. ¯ ∈ Pcl (B 0 ) for each y ∈ B 0 . Step 1. y b b ¯ such that ρ¯n → ρ¯∗ in B 0 . Then ρ¯∗ (t) ∈ B 0 Indeed, let y (n) → y ∗ , (ρ¯n )n≥0 ∈ y b b and there exists f n ∈ S F,y (n) such that ⎧ −T (t)G(0, ϕ) + G(t, yt(n) + ϕ˜t ) ⎪ ⎪ ⎪ ⎪ + t AT (t − s)G(s, y (n) + ϕ˜ )ds ⎪ s ⎪ s ⎪ 0t ⎪ ⎪ + T (t − s) f (s)dw(s), ⎪ n 0 ⎪ ⎪ (n) ⎨ gi (t, yt + ϕ˜ t ), ρ¯n (t) = (n) ⎪ ⎪ T (t − si )[gi (si , ysi + ϕ˜ si ) ⎪ (n) (n) ⎪ ⎪ ⎪ −G(s i , ysi + ϕ˜ si )] + G(t, yt + ϕ˜ t ) ⎪ ⎪ t ⎪ ⎪ + si AT (t − s)G(s, ys(n) + ϕ˜s )ds ⎪ ⎪ t ⎩ + si T (t − s) f n (s)dw(s),
t ∈ [0, t1 ], i = 0, t ∈ (ti , si ], i ≥ 1,
t ∈ (si , ti+1 ], i ≥ 1.
Using the closedness property of the values of F and (H1), (H2)(i)–(H5), we can t t (n) prove that si AT (t − s)G(s, ys + ϕ˜s )ds + si T (t − s) f n (s)dw(s) is closed for each t ∈ (si , ti+1 ], i = 0, . . . , N . Then
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ρ¯n (t) → ρ¯∗ (t) ⎧ ϕ) + G(t, yt∗ + ϕ˜t ) ⎪ ⎪ −T (t)G(0, t ⎪ ∗ ⎪ ⎪ ⎪ + 0t AT (t − s)G(s, ys + ϕ˜s )ds ⎪ ⎪ ⎪ ⎪ + 0 T (t − s) f ∗ (s)dw(s), ⎪ ∗ ⎪ ⎪ ⎨ gi (t, yt + ϕ˜t ), ∗ = T (t − si )[gi (si , ysi + ϕ˜si ) ⎪ (n) ⎪ ⎪ −G(si , ysi + ϕ˜si )] ⎪ ⎪ ∗ ⎪ ⎪ +G(t, ⎪ t yt + ϕ˜t ) ⎪ ⎪ ⎪ + si AT (t − s)G(s, ys∗ + ϕ˜s )ds ⎪ ⎪ t ⎩ + si T (t − s) f ∗ (s)dw(s),
t ∈ [0, t1 ], i = 0, t ∈ (ti , si ], i ≥ 1,
t ∈ (si , ti+1 ], i ≥ 1.
¯ ¯ and ( y)(t) ∈ Pcl (Bb0 ). Therefore, ρ¯∗ (t) ∈ ( y)(t) ¯ Step 2. is a contractive multi-valued map. ¯ Then there exists f ∈ S F,y such that Let t ∈ [0, t1 ] and y, yˆ ∈ Bb0 and let ρ¯ ∈ y. ρ(t) ¯ = − T (t)G(0, ϕ) + G(t, yt + ϕ˜t ) + t + T (t − s) f (s)dw(s).
t
AT (t − s)G(s, ys + ϕ˜s )ds
0
0
From (H5), there exists v(t) ∈ F(t, yˆt + ϕ˜t ) such that p
p
E f (t) − v(t) H ≤ l(t) yt − yˆt B . Consider : [0, t1 ] → P(L(K , H )), given by p
p
(t) = {v(t) ∈ H : E f (t) − v(t) H ≤ l(t) yt − yˆt B }. Since the multi-valued operator W (t) = (t) ∩ F(t, yˆt + ϕ˜t ) is measurable (see [9, Proposition III.4]), there exists a function fˆ(t), which is a measurable selection for W . So, fˆ(t) ∈ F(t, yˆt + ϕ˜ t ) and p p E f (t) − fˆ(t) H ≤ l(t) yt − yˆt B .
For t ∈ [0, t1 ], we define ρ(t) ˆ = − T (t)G(0, ϕ) + G(t, yˆt + ϕ˜t ) + t + T (t − s) fˆ(s)dw(s). 0
Then, we have for t ∈ [0, t1 ],
123
t 0
AT (t − s)G(s, yˆs + ϕ˜s )ds
Bulletin of the Iranian Mathematical Society
E ρ(t) ¯ − ρ(t) ˆ αp ≤ 3 p−1 E G(t, yt + ϕ˜t ) − G(t, yˆt + ϕ˜ t ) αp t p + 3 p−1 E AT (t − s)[G(s, y + ϕ ˜ ) − G(s, y ˆ + ϕ ˜ )]ds s s s s 0 α t p + 3 p−1 E T (t − s)[ f (s) − fˆ(s)]dw(s) α
0
≤ 3 p−1 L G (−A)−β H yt − yˆt B t p p−1 + 3 p−1 M1−β t1 (t − s)− p(1−β) E (−A)β G(s, ys + ϕ˜s ) p
p
0
− (−A)β G(s, yˆs + ϕ˜s ) αp ds t 2/ p p/2 p (t − s)− pα E f (s) − fˆ(s) H + 3 p−1 C p Mαp ds ≤3
p−1
L G (−A) p
0 −β
p−1
+ 3 p−1 M1−β t1
0
p
p
H yt − yˆt B t
(t − s)− p(1−β) L G ys − yˆs B ds p
t
p/2−1
pα − 2p−2
(t − s) ds +3 0 t p × l(s)(1 + ah ) ys − yˆs B ds p−1
C p Mαp
0
≤ 12 p−1 K b L G (−A)−β H y − yˆ b p
p
p
pβ
t1 p L G ds y − yˆ b pβ − p + 1 p/2−1 p−2 p + 12 p−1 K b C p Mαp p − 2 − pα t p/2−1− pα p × t1 (1 + ah ) l(s)ds y − yˆ b p
p
+ 12 p−1 K b M1−β
0
≤ 12 p−1 K b L G (−A)−β H +M1−β p
+C p Mαp
p
p−2 p − 2 − pα
p/2−1
p
p/2−1− pα
t1
pβ
t1 LG pβ − p + 1 (1 + ah ) l L 1
p
y − yˆ b .
¯ ¯ ∈ ( y)(t), we For any t ∈ (ti , si ], i = 1, . . . , N . Let y(t), yˆ (t) ∈ Bb0 and let ρ(t) have ρ(t) ¯ = gi (t, yt + ϕ˜t ). For any t ∈ (ti , si ], i = 1, . . . , N , we define ρ(t) ˆ = gi (t, yˆt + ϕ˜t ).
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Then, for any t ∈ (ti , si ], i = 1, . . . , N , we have E ρ(t) ¯ − ρ(t) ˆ αp = E gi (t, yt + ϕ˜t ) − gi (t, yˆt + ϕ˜ t ) αp p p p ≤ γi yt − yˆt B ≤ 4 p−1 K b γi y − yˆ b . Similarly, for any t ∈ (si , ti+1 ], i = 1, . . . , N . Let y(t), yˆ (t) ∈ Bb0 and let ρ(t) ¯ ∈ ¯ ( y)(t). Then there exists f ∈ S F,y such that ρ(t) ¯ = T (t − si )[gi (si , ysi + ϕ˜si ) − G(si , ysi + ϕ˜si )] t + G(t, yt + ϕ˜t ) + AT (t − s)G(s, ys + ϕ˜s )ds si
t
+
T (t − s) f (s)dw(s).
si
From (H5), there exists v(t) ∈ F(t, yˆt + ϕ˜t ) such that p
p
E f (t) − v(t) H ≤ l(t) yt − yˆt B . Consider : (sk , tk+1 ] → P(L(K , H )), given by p p (t) = v(t) ∈ H : E f (t) − v(t) H ≤ l(t) yt − yˆt B . Since the multi-valued operator W (t) = (t) ∩ F(t, yˆt + ϕ˜t ) is measurable (see Proposition III.4, [9]), there exists a function fˆ(t), which is a measurable selection for W . So, fˆ(t) ∈ F(t, yˆt + ϕ˜t ) and p p E f (t) − fˆ(t) H ≤ l(t) yt − yˆt B .
For any t ∈ (si , ti+1 ], i = 1, . . . , N , we define ρ(t) ˆ = T (t − si )[gi (si , yˆsi + ϕ˜si ) − G(si , yˆsi + ϕ˜si )] t + G(t, yˆt + ϕ˜t ) + AT (t − s)G(s, yˆs + ϕ˜s )ds +
si t
T (t − s) fˆ(s)dw(s).
si
Then, for any t ∈ (si , ti+1 ], i = 1, . . . , N , we have E ρ(t) ¯ − ρ(t) ˆ αp ≤ 4 p−1 E T (t − si )[gi (si , ysi + ϕ˜si ) − gi (si , yˆsi + ϕ˜si ) + G(si , ysi + ϕ˜si ) − G(si , yˆsi + ϕ˜si )] αp + 4 p−1 E G(t, yt + ϕ˜ t ) − G(t, yˆt + ϕ˜t ) αp
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+ 4 p−1 E
t
si
p AT (t − s)[G(s, ys + ϕ˜s ) − G(s, yˆs + ϕ˜s )]ds α
t p p−1 ˆ +4 E T (t − s)[ f (s) − f (s)]dw(s) α
si
≤8
p−1
M [γi ysi − yˆsi B +L G (−A)−β H ysi − yˆsi B ] p
p
p
p
+ 4 p−1 L G (−A)−β H yt − yˆt B t p−1 p p−1 +4 M1−β (ti+1 − si ) (t − s)− p(1−β) p
p
si
× E (−A)β G(s, ys + ϕ˜s ) − (−A)β G(s, yˆs + ϕ˜s ) αp ds t 2/ p p/2 p (t − s)− pα E f (s) − fˆ(s) H + 4 p−1 C p Mαp ds si
≤8
p−1
M [γi ysi − yˆsi B +L G (−A)−β H ysi − yˆsi B ] p
p
p
p
+ 4 p−1 L G (−A)−β H yt − yˆt B t p p−1 p p−1 +4 M1−β (ti+1 − si ) (t − s)− p(1−β) L G ys − yˆs B ds p
si
+4
p−1
× ≤ 32
t
C p Mαp
p
t
(t − s)
pα − 2p−2
p/2−1 ds
si p
l(s)(1 + ah ) ys − yˆs B ds
si p−1
M p K b [γi + L G (−A)−β H ] y − yˆ b p
p
p
+ 16 p−1 K b L G (−A)−β H y − yˆ b p
p
p
(ti+1 − si ) pβ p L G y − yˆ b pβ − p + 1 p/2−1 p−2 p−1 p p + 16 K b C p Mα (ti+1 − si ) p/2−1− pα p − 2 − pα t p × (1 + ah ) l(s)ds y − yˆ b 0 p p−1 p ≤ 16 K b 2 p−1 M p [γi + L G (−A)−β H ] p
p
+ 16 p−1 K b M1−β
+ L G (−A)−β H +M1−β p
+ C p Mα
p−2 p − 2 − pα
p
(ti+1 − si ) pβ LG pβ − p + 1
p/2−1
(ti+1 − si )
p/2−1− pα
(1 + ah ) l L 1
p
× y − yˆ b .
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Thus, for all t ∈ [0, b], we have p p ρ¯ − ρˆ b ≤ L˜ y − yˆ b ,
and p ¯ ¯ yˆ ) ≤ L˜ y − yˆ p ,
Hd ( y, b
where p L˜ = 4 p−1 K b max
1≤i≤N
(1 + 8 p−1 M p )γi + 4 p−1 (2 p−1 M p + 1)
b pβ LG × (−A) pβ − p + 1 p/2−1 p−2 + 4 p−1 C p Mα b p/2−1− pα (1 + ah ) l L 1 < 1. p − 2 − pα −β
p H
p +M1−β
¯ is a contraction on B 0 . In view of Lemma 2.8, we conclude that
¯ has a Hence,
b 0 ∗ ∗ ˜ t ∈ (−∞, b]. Then, x is a fixed unique fixed point y ∈ Bb . Let x(t) = y (t) + ϕ(t), point of the operator , which implies that x is a unique mild solution of the system (1.1)–(1.3). We use the below conditions instead of (H3)–(H5) to avoid the Lipschitz continuity of h, f used in Theorem 3.1. Assume that (−A)−α is compact for α ∈ (0, 1). In addition, we make the following hypotheses. (H7) (i) For each (t, s) ∈ = {(t, s) ∈ [0, b] × [0, b] : s ≤ t}, the function h(t, s, ·) : B → Hα is continuous and for each x ∈ B, the function h(·, ·, x) : → Hα is strongly measurable. (ii) There exists a continuous function m h : → [0, ∞), such that p
h(t, s, ψ) αp ≤ m h (t, s)h ( ψ B ) for a.e. t, s ∈ [0, b], ψ ∈ B, where h : [0, ∞) → (0, ∞) is a continuous nondecreasing function. (H8) The multi-valued map F : [0, b] × B × Hα → Pbd,cl,cv (L(K , H )); for each t ∈ [0, b], the function F(t, ·, ·) : B × Hα → Pbd,cl,cv (L(K , H )) is u.s.c. and for each (ψ, y) ∈ B × H , the function F(·, ψ, y) is measurable; for each fixed (ψ, y) ∈ B × Hα , the set S F,ψ =
f ∈ L p ([0, b], (L(K , H ))) : t f (t) ∈ F t, ψ, h(t, s, ψ)ds for a.e t ∈ [0, b] 0
is nonempty.
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(H9) There exist a continuous function m f : [0, b] → [0, ∞) and a continuous nondecreasing function f : [0, ∞) → (0, ∞) such that p
p
F(t, ψ, y) H = sup{E f H : f ∈ F(t, ψ, y)} p
p
≤ m f (t) f ( ψ B + y H ) for a.e. t ∈ [0, b] and each ψ ∈ B, y ∈ H with
∞
1
1 ds = ∞. f (s) + h (s)
Lemma 3.2 [27]. Let [0, b] be a compact interval and H be a Hilbert space. Let F be a multi-valued map satisfying (H7) and let P˜ be a linear continuous operator from L p ([0, b], H ) to C([0, b], H ). Then, the operator ˜ F , x) P˜ ◦ S F : C([0, b], H ) → Pcp,cv (H ), x → ( P˜ ◦ S F )(x) := P(S is a closed graph in C([0, b], H ) × C([0, b], H ). p
Theorem 3.3 Let x0 ∈ L F0 (, Hα ). If the assumptions (H1), (H2) and (H6)–(H9) are satisfied, then the system (1.1)–(1.3) has at least one mild solution on [0, b], provided that 4
p−1
p K b max 1≤i≤N
p−1 p p−1 (1 + 12 (2 p−1 M p + 1) M )γi + 6
× (−A)−β H +M1−β p
p
b pβ L G < 1. pβ − p + 1
(3.4)
¯ be defined as in the proof of Theorem 3.1. For y ∈ Proof Let Bb , Bb0 , ,
p 0 0 Br (0, Bb ) = {y ∈ Bb : E y α ≤ r }, from Lemma 2.5 and (3.3), we have p p p p p yt + ϕ˜t B ≤ 4 p−1 [K b r + (K b M p H˜ p + Mb ) ϕ B ] = r ∗ , t ∈ [0, b]. (3.5)
¯ 1 : B 0 → B 0 is defined by Now, we define the two operators as follows. The map
b b
¯ 1 y)(t) = (
⎧ −T (t)G(0, ϕ) + G(t, yt + ϕ˜ t ) ⎪ t ⎪ ⎪ ⎪ + AT (t − s)G(s, ys + ϕ˜s )ds, ⎪ ⎪ ⎨ g (t, y0 + ϕ˜ ), i
t
t
T (t − si )[gi (si , ysi + ϕ˜si ) ⎪ ⎪ ⎪ ⎪ ⎪ −G(s ⎪ t i , ysi + ϕ˜si )] + G(t, yt + ϕ˜t ) ⎩ + si AT (t − s)G(s, ys + ϕ˜s )ds,
t ∈ [0, t1 ], i = 0, t ∈ (ti , si ], i ≥ 1, t ∈ [si , ti+1 ], i ≥ 1,
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¯ 2 : B 0 → P(B 0 ) is defined by and
b b ⎧ t ⎨ 0 T (t − s) f (s)dw(s), ¯ 2 y)(t) = 0, (
⎩ t si T (t − s) f (s)dw(s),
t ∈ [0, t1 ], i = 0, t ∈ (ti , si ], i ≥ 1, t ∈ [si , ti+1 ], i ≥ 1.
¯ =
¯1+
¯ 2 . In what follows, we show that the operators
¯ 1 and
¯ 2 satisfy Then
the conditions of Lemma 2.9. For better readability, we divide the proof into several steps. ¯ 1 has closed-convex values. Next, we shall show Step 1. Obviously, the operator
¯ 1 maps bounded sets into bounded sets in B 0 . that
b Indeed, it is enough to show that there exists a positive constant μ˜ such that for ¯ 1 y αp ≤ μ. ¯ 1 y, y ∈ Br (0, B 0 ) one has E
˜ Then, by (H1), (H2) and (H6), each
b we have for t ∈ [0, t1 ], ¯ 1 y)(t) αp E (
≤ 3 p−1 E T (t)G(0, ϕ) αp +3 p−1 E G(t, yt + ϕ˜t ) αp t p + 3 p−1 E AT (t − s)G(s, y + ϕ ˜ )ds s s 0
≤3
p−1
M (−A)−β H L G ( ϕ B +1) p
p
α
p
+ 3 p−1 (−A)−β H L G ( yt + ϕ˜t B +1) t p−1 p−1 p +3 M1−β t1 (t − s)− p(1−β) E (−A)β G(s, ys + ϕ˜ s ) αp ds p
p
0
≤ 3 p−1 M p (−A)−β H L G ( ϕ B +1) p
p
+ 3 p−1 (−A)−β H L G ( yt + ϕ˜t B +1) t p p−1 p + 3 p−1 M1−β t1 (t − s)− p(1−β) L G ( ys + ϕ˜s B +1)ds p
≤3
p−1
M (−A) p
0 −β
p
p
p
H L G ( ϕ B +1)
+ 3 p−1 (−A)−β H L G (r ∗ + 1) p
p(β+1)
p
p
+ 3 p−1 M1−β M1
t1 L G (r ∗ + 1) := μ0 . pβ − p + 1
For any t ∈ (ti , si ], i = 1, . . . , N , we have ¯ 1 y)(t) αp ≤ γi ( yt + ϕ˜ t +1) ≤ γi (r ∗ + 1) := μi . E (
B p
Similarly, for any t ∈ (si , ti+1 ], i = 1, . . . , N , we have ¯ 1 y)(t) αp E (
≤ 3 p−1 E T (t − si )[gi (si , ysi + ϕ˜ si ) − G(si , ysi + ϕ˜si )] αp + 3 p−1 E G(t, yt + ϕ˜t ) αp
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+ 3 p−1 E
t si
p AT (t − s)G(s, ys + ϕ˜s )ds α
p
≤ 6 p−1 M p [γi ( ysi + ϕ˜si B +1)
+ (−A)−β H L G ( ysi + ϕ˜ si B +1)] p
p
+ 3 p−1 (−A)−β H L G ( yt + ϕ˜t B +1) p
p
p
+ 3 p−1 M1−β (ti+1 − si ) p−1 t × (t − s)− p(1−β) E (−A)β G(s, ys + ϕ˜s ) αp ds ≤6
si p−1
p
M p [γi ( ysi + ϕ˜si B +1)
+ (−A)−β H L G ( ysi + ϕ˜ si B +1)] p
p
+ 3 p−1 (−A)−β H L G ( yt + ϕ˜t B +1) p
p
p
+ 3 p−1 M1−β (ti+1 − si ) p−1 t p × (t − s)− p(1−β) L G ( ys + ϕ˜s B +1)ds ≤6
si p−1
M p [γi (r ∗ + 1)+ (−A)−β H L G (r ∗ + 1)] p
+ 3 p−1 (−A)−β H L G (r ∗ + 1) p
p
+ 3 p−1 M1−β
(ti+1 − si ) pβ L G (r ∗ + 1) := νi . pβ − p + 1
¯ 1 y α ≤ μ. Take μ˜ = max0≤i≤N {μi + νi }, we have E
˜ ¯ 1 is a contraction operator. Step 2.
Let t ∈ [0, t1 ] and u, v ∈ Bb0 . From (H2) and Lemmas 2.3, 2.5, we have p
¯ 1 v)(t) αp ¯ 1 u)(t) − (
E (
≤ 2 p−1 E G(t, u t + ϕ˜t ) − G(t, vt + ϕ˜t ) αp t p p−1 +2 E AT (t − s)[G(s, u s + ϕ˜s ) − G(s, vs + ϕ˜s )]ds 0
≤ 2 p−1 (−A)−β H L G u t − vt B t p−1 p p−1 p +2 M1−β t1 (t − s)− p(1−β) L G u s − vs B ds p
p
≤ 8 p−1 K b
α
p
0
(−A)−β H +M1−β p
p
pβ t1 p L G u − v b . pβ − p + 1
For any t ∈ (ti , si ], i = 1, . . . , N , we have ¯ 1 u)(t) − (
¯ 1 v)(t) αp = E gi (t, u t + ϕ˜t ) − gi (t, vt + ϕ˜t ) αp E (
p
≤ γi u t − vt B p
p
≤ 4 p−1 K b γi u − v b .
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Similarly, for any t ∈ (ti , ti+1 ], i = 1, . . . , N , we have ¯ 1 u)(t) − (
¯ 1 v)(t) αp E (
≤ 3 p−1 E R(t − si )[gi (si , u si + ϕ˜ si ) − gi (si , u si + ϕ˜si ) + G(si , u si + ϕ˜si ) − G(si , u si + ϕ˜si )] αp + 3 p−1 E G(t, u t + ϕ˜t ) − G(t, u t + ϕ˜t ) αp t p p−1 +3 E AT (t − s)[G(s, u s + ϕ˜s ) − G(s, vs + ϕ˜ s )]ds
α p
si
≤6
p−1
M [γi u si − vsi B + (−A)−β H L G u si − vsi B ] p
p
p
+ 3 p−1 (−A)−β H L G u t − vt B t p p−1 p p−1 +3 M1−β (ti+1 − si ) (t − s)− p(1−β) L G u s − vs B ds si p p−1 p p−1 −β p Kb 2 (γi + (−A) H ) + (−A)−β H ≤ 24 p
p
+ M1−β
p
(ti+1 − si ) pβ p L G u − v b . pβ − p + 1
Thus, for all t ∈ [0, b], we have ¯ 1 u)(t) − (
¯ 1 v)(t) αp ≤ L 0 u − v . E (
b p
Taking supremum over t, ¯ 1u −
¯ 1v p ≤ L 0 u − v p ,
b b p
where L 0 = 4 p−1 K b max1≤i≤N {(1 + 8 p−1 M p )γi + 4 p−1 ((2 p−1 M p + 1) pβ
p(β+1)
b b (−A)−β H +M1−β pβ− p+1 + M1−β M1 pβ− p+1 )L}. By(3.4), we see that L 0 < 1. ¯ 1 is a contraction operator. Hence,
Step 3. 2 y is convex for each y ∈ Bb0 . ¯ 2 y, then there exist f 1 , f 2 ∈ S F,y , for any t ∈ In fact, if ρ˜1 , ρ˜2 belong to
[si , ti+1 ], i = 0, 1, . . . , N , such that p
p
ρ˜k (t) =
p
t si
p
T (t − s) f k (s)dw(s), k = 1, 2.
Let 0 ≤ λ ≤ 1. For any t ∈ [si , ti+1 ], i = 0, 1, . . . , N , we have
t
(λρ˜1 + (1 − λ)ρ˜2 )(t) =
T (t − s)[λ f 1 (s) + (1 − λ) f 2 (s)]dw(s).
si
¯ 2 y. Since S F,y is convex (because F has convex values) we have (λρ˜1 +(1−λ)ρ˜2 ) ∈
¯ 2 maps bounded sets into bounded sets in B 0 Step 4.
b
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Bulletin of the Iranian Mathematical Society
Indeed, it is enough to show that there exists a positive L such that each ρ(t) ˜ ∈ ¯ 2 y, y ∈ Br (0, B 0 ), one has E ρ˜ αp ≤ L.
b ¯ 2 y, then there exists f ∈ S F,y , for any t ∈ [si , ti+1 ], i = 0, 1, . . . , N , If ρ(t) ˜ ∈
such that t ρ(t) ˜ = T (t − s) f (s)dw(s). si
By (H7)–(H9), we get that for any t ∈ [si , ti+1 ], i = 0, 1, . . . , N , E ρ(t) ˜
αp
t p ≤ E T (t − s) f (s)dw(s) si α t 2/ p p/2 p p − pα ≤ C p Mα E f (s) H ds (t − s) si
≤ C p α Mαp
t
(t − s)
pα − 2p−2
p/2−1
t
m f (s) f
p
ys + ϕ˜s B si si s p p−1 + (ti+1 − si ) m h (s, τ )h ( yτ + ϕ˜τ B )dτ ds ds
0
p/2−1 p−2 ≤ C p Mαp (ti+1 − si ) p/2−1− pα p − 2 − pα ti+1 × f (r ∗∗ ) m f (s)ds := Li , si
t where r ∗∗ = r ∗ + (ti+1 − si ) p−1 h (r ∗ ) sii+1 m h (s, s)ds. Take L = max0≤i≤N {Li }, ¯ 2 y, we have E ρ˜ αp ≤ L. then for each ρ(t) ˜ ∈
¯ 2 is completely continuous. Step 5.
¯ 2 (Br (0, B 0 )) is equicontinuous. Claim 1
b Since T (·) is analytic, the function s → (−A)α T (s) is continuous in the uniform operator topology on (0, b]. Let si < ε < t ≤ ti+1 , i = 0, 1, . . . , N , and δ > 0 p such that (−A)α T (s1 ) − (−A)α T (s2 ) H < ε for every s1 , s2 ∈ [si , ti+1 ] with |s1 − s2 | < δ. For each x ∈ Br (0, Bb0 ), 0 < |η| < δ, t + η ∈ [si , ti+1 ]. Then, we have for each x ∈ Br (0, Bb0 ), and ρ˜ ∈ 2 x, there exists f ∈ S F,y such that, for each t ∈ [si , ti+1 ], i = 0, 1, . . . , N , ρ(t) ˜ =
t
T (t − s) f (s)dw(s).
si
By (H7)–(H9), we get E ρ(t ˜ + η) − ρ(t) ˜ αp t p p−1 ≤2 E [T (t + η − s) − T (t − s)] f (s)dw(s) si
α
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t+η p +2 p−1 E T (t + η − s) f (s)dw(s) t α t (−A)α T (t + η − s) ≤ 2 p−1 C p si
− (−A)α T (t − s) H E f (s) H p
+2
p
t+η
2/ p
p/2 ds
− pα
p H
2/ p
(t + η − s) E f (s) t ≤ 2 p−1 C p (ti+1 − si ) p/2−1 (−A)α T (t + η − s) p−1
Mαp C p
t
− (−A)α T (t − s) H p
si t
p/2 ds
p
ys + ϕ˜s B s p + (ti+1 − si ) p−1 m h (s, τ )h ( yτ + ϕ˜ τ B )dτ ds +2 ×
p−1
C p Mαp
si
0 t+η
t
m f (s) f
pα − 2p−2
(t + η − s)
p/2−1 ds
t+η
p
ys + ϕ˜s B
m f (s) f p−1 + (ti+1 − si ) t
s
p
m h (s, τ )h ( yτ + ϕ˜ τ B )dτ ds t ≤ 2 p−1 C p (ti+1 − si ) p/2−1 f (r ∗∗ )ε m f (s)ds 0
p/2−1
si
p−2 p − 2 − pα t+η p/2−1− pα ∗∗ ×η f (r ) m f (s)ds. + 2 p−1 C p Mαp
t
The right-hand side tends to zero as η → 0, and sufficiently small positive number ε. Hence, 2 maps Br (0, Bb0 ) into an equicontinuous family of functions. ¯ 2 (Br (0, B 0 ))} is precompact for every ¯ 2 (Br (0, B 0 ))(t) = {ρ(t) ˜ : ρ(t) ˜ ∈
Claim 2
b b t ∈ [0, b]. ¯ 2 y, there ˜ ∈
Let t ∈ (si , ti+1 ], i = 0, 1, . . . , N . For each x ∈ Br (0, Bb0 ), and ρ(t) exists f ∈ S F,y , such that ρ(t) ˜ =
t
T (t − s) f (s)dw(s).
si
Then, there exists 0 < δ < 1 such that 0 ≤ α < δ < 1, and
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E (−A)δ ρ(t) ˜ H t p δ = E (−A) T (t − s) f (s)dw(s) p
si
p
≤ C p Mδ
t si
p
t
H
(t − s)− pδ E f (s) H p
2 pδ − p−2
p/2−1
p/2
2/ p
t
ds p
ys + ϕ˜s B s p + (ti+1 − si ) p−1 m h (s, τ )h ( yτ + ϕ˜τ B )dτ ds
≤ C p Mδ
(t − s)
ds
si
m f (s) f
si
0
p/2−1 p−2 ≤ (ti+1 − si ) p/2−1− pδ p − 2 − pδ ti+1 ∗∗ × f (r ) m f (s)ds < ∞, p C p Mδ
si
which implies (−A)δ ρ(t) ˜ is bounded in H . By the compactness of (−A)−δ , we see ¯ 2 y)(t) is precompact in Hα for each t ∈ [0, b]. Together with the Arzela–Ascoli that (
¯ 2 is completely continuous. theorem, we conclude that
¯ 2 has a closed graph. Step 7.
¯ 2 y (n) , y (n) ∈ Br (0, B 0 )) and ρ˜n → ρ˜∗ . From Axiom (A), Let y (n) → y ∗ , ρ˜n ∈
b it is easy to see that yt(n) − yt B p
≤ 2 p−1 (K (t)) p sup{ y (n) (s) − y(s) αp : 0 ≤ s ≤ t} + 2 p−1 (M(t)) p y0(n) − y0 B p
= 2 p−1 (K (t)) p sup{ y (n) (s) − y(s) αp : 0 ≤ s ≤ t} ≤ 2 p−1 K b y (n) − y b → 0(n → ∞), t ∈ [0, b]. p
p
¯ 2 y. Now ρ˜n ∈
¯ 2 y (n) means that there exists f n ∈ S F,y (n) such We prove that ρ˜∗ ∈
that, for any t ∈ [si , ti+1 ], i = 0, 1, . . . , N , ρ˜n (t) =
t
T (t − s) f n (s)dw(s).
si
We must prove that there exists f ∗ ∈ S F,y ∗ such that, for any t ∈ [si , ti+1 ], i = 0, 1, . . . , N , ρ˜∗ (t) =
si
t
T (t − s) f ∗ (s)dw(s).
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Now, for any t ∈ [si , ti+1 ], i = 0, 1, . . . , N , we have p
ρ˜n (t) − ρ˜∗ (t) b → 0 as n → ∞. Consider the linear continuous operator : L p ([si , ti+1 ], Hα ) → C([si , ti+1 ], Hα ), i = 0, 1, . . . , N , ( f )(t) =
t
T (t − s) f (s)dw(s).
si
Then
p
p/2−1 p−2 (ti+1 − si ) p/2−1− pδ p − 2 − pδ ti+1 × f (r ∗∗ ) m f (s)ds,
( f ) b ≤ C p Mαp
si
and this implies that is continuous. From Lemma 3.2, it follows that ◦ S F is a closed graph operator. Moreover, from the definition of , we have that, for any t ∈ [si , ti+1 ], i = 0, 1, . . . , N , ρ˜n (t) ∈ (S F,y (n) ). Since y (n) → y ∗ , for some f ∗ ∈ S F,y ∗ it follows that, for any t ∈ [si , ti+1 ], i = 0, 1, . . . , N , we have t ρ˜∗ (t) = T (t − s) f ∗ (s)dw(s). si
¯ 2 is a completely continuous multi-valued map, u.s.c. with convex closed, Therefore,
compact values. ¯ has a solution in B 0 . Step 8. The operator
b ¯ =
¯ 1y +
¯ 2 y for some λ > 1, we get Let y ∈ Bb0 be any solution of λy ∈ y ⎧ 1 − T (t)G(0, ϕ) + λ1 G(t, yt + ϕ˜ t ) ⎪ ⎪ ⎪ λ ⎪ t ⎪ ⎪ + λ1 0 ASα (t − s)G(s, ys + ϕ˜s )ds ⎪ ⎪ ⎪ ⎪ 1 t ⎪ ⎪ ⎪ + λ 0 T (t − s) f (s)dw(s), ⎪ ⎪ ⎪ 1 ⎪ ⎪ g (t, yt + ϕ˜t ), ⎪ ⎨λ i y(t) = 1 T (t − si )[gi (si , ysi + ϕ˜ si ) λ ⎪ ⎪ ⎪ ⎪ ⎪ −G(si , ysi + ϕ˜ si )] ⎪ ⎪ ⎪ ⎪ + 1 G(t, y + ϕ˜ ) ⎪ t t ⎪ λ ⎪ ⎪ ⎪ 1 t ⎪ ⎪ + λ si AT (t − s)G(s, ys + ϕ˜s )ds ⎪ ⎪ t ⎪ ⎩ + λ1 si T (t − s) f (s)dw(s),
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t ∈ [0, t1 ], i = 0, t ∈ (ti , si ], i ≥ 1,
t ∈ [si , ti+1 ], i ≥ 1,
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where f ∈ S F,y . Then, by (H1), (H2) and (H6)–(H9), from the above equation, we have for t ∈ [0, t1 ], E y(t) αp ≤ 4 p−1 E T (t)G(0, ϕ) αp +4 p−1 E G(t, yt + ϕ˜t ) αp t p p−1 +4 E AT (t − s)G(s, ys + ϕ˜s )ds 0 α t p p−1 +4 E T (t − s) f (s)dw(s) α p
0
≤ 4 p−1 M p (−A)−β H L G ( ϕ B +1) p
+4 p−1 (−A)−β H L G ( yt + ϕ˜t B +1) p
p
p(β+1)
t1 p L G ( yt + ϕ˜t B +1) pβ − p + 1 t p/2−1 − 2 pα +4 p−1 C p Mαp (t − s) p−2 ds 0 t p × m f (s) f ys + ϕ˜s B 0 s p−1 p +t1 m h (s, τ )h ( yτ + ϕ˜ τ B )dτ ds. p
p
+4 p−1 M1−β M1
0
For any t ∈ (ti , si ], i = 1, . . . , N , we have p
E yλ (t) αp ≤ γi ( yt + ϕ˜t B +1). Similarly, for any t ∈ (si , ti+1 ], i = 1, . . . , N , we have E y(t) αp ≤ 4 p−1 E T (t − si )[gi (si , ysi + ϕ˜ si ) − G(si , ysi + ϕ˜si )] αp +4 p−1 E G(t, yt + ϕ˜t ) αp t p p−1 +4 E AT (t − s)G(s, ys + ϕ˜ s )ds si α t p +4 p−1 E T (t − s) f (s)dw(s) α
si
p
≤ 8 p−1 M p [γi ( yλsi + ϕ˜si B +1)
+ (−A)−β H L G ( yλsi + ϕ˜si B +1)] p
p
+4 p−1 (−A)−β H L( yλt + ϕ˜t B +1) p
p
+4 p−1 M1−β
p
(ti+1 − si ) pβ p L G ( yλt + ϕ˜ t B +1) pβ − p + 1
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+4
p−1
C p Mαp
t
(t − s)
pα − 2p−2
p/2−1 ds
s
i t p × m f (s) f ys + ϕ˜s B si
s
+(ti+1 − si ) p−1 0
p m h (s, τ )h ( yτ + ϕ˜τ B )dτ ds.
Let ξ(t) = 4 p−1 [K b E y α,t +(K b M p H˜ p + Mb ) ϕ B ], 0 ≤ t ≤ b, p
p
p
p
p
p
p
where y α,t = sup0≤s≤t y(s) α . Then, for all t ∈ [0, b], we have E y(t) αp ≤ M ∗ + γi ξ(t) + 8 p−1 M p [γi ξ(t)+ (−A)−β H L G ξ(t)] p
+4 p−1 (−A)−β H L G ξ(t) p
b pβ L G ξ(t) pβ − p + 1 p/2−1 p−2 +4 p−1 C p Mαp b p/2−1− pα p − 2 − pα t s × m f (s) f ξ(s) + b p−1 m h (s, τ )h (ξ(τ ))dτ ds, p
+4 p−1 M1−β
0
0
where p p 4 p−1 M p (−A)−β H L G ( ϕ B +1) 1≤i
M ∗ = max
+ (−A)−β H +M1−β p
p
b pβ LG . pβ − p + 1
It is easy to see that p p p p ξ(t) ≤ 4 p−1 [K b M ∗ + (K b M p H˜ p + Mb ) ϕ B ] p +4 p−1 K b γi (1 + 8 p−1 M p )γi p +4 p−1 (2 p−1 M p + 1) (−A)−β H p
+M1−β
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b pβ L G ξ(t) pβ − p + 1
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p/2−1 p−2 +4 b p/2−1− pα p − 2 − pα s t p−1 × m f (s) f ξ(s) + b m h (s, τ )h (ξ(τ ))dτ ds. p−1
C p Mαp
0
0
Since L ∗ = 4 p−1 K b max1≤i≤N {(1 + 8 p−1 M p )γi + 4 p−1 ((2 p−1 M p + 1) (−A)−β p
p
p
H +M1−β
b pβ pβ− p+1 )L G }
< 1, we have
1 p−1 p ∗ p p p p ˜p 4 [K M + (K M + M ) ϕ ] H b b b B 1 − L∗ t s ∗ p−1 m f (s) f ξ(s) + b m h (s, τ )h (ξ(τ ))dτ ds, +d
ξ(t) ≤
0
0
1 p−1 C M ( p−2 ) p/2−1 b p/2−1− pα . Denoting by ζ (t) the rightwhere d ∗ = 1−L ∗4 p α p−2− pα hand side of the above inequality, we have ξ(t) ≤ ζ (t) for all t ∈ [0, b], and p
1 p−1 p ∗ p p p p ˜p 4 [K M + (K M + M ) ϕ ] H b b b B , 1 − L∗ t ζ (t) ≤ d ∗ m f (t) f ξ(t) + b p−1 m h (t, s)h (ξ(s))ds 0 t ∗ p−1 m h (t, s)h (ζ (s))ds . ≤ d m f (t) f ζ (t) + b ζ (0) =
0
t Let ϑ(t) = ζ (t) + b p−1 0 m h (t, s)h (ζ (s))ds then ϑ(0) = ζ (0), ζ (t) ≤ ϑ(t), and for each t ∈ [0, b] we have (ϑ(t)) = ζ (t) + b p−1 m h (t, t)h (ζ (t)) ≤ d ∗ m f (t) f (ϑ(t)) + b p−1 m h (t, t)h (ϑ(t)) ≤ max{d ∗ m f (t), b p−1 m h (t, t)}[ f (ϑ(t)) + h (ϑ(t))], t ∈ [0, b]. This implies that
ϑ(t)
ϑ(0)
ds ≤ f (s) + h (s)
b
max{d ∗ m f (t), b p−1 m h (t, t)}dt < ∞.
0
such that ϑ(t) ≤ K , t ∈ [0, b], and This inequality shows that there is a constant K p hence E y α ≤ ξ(t) ≤ ζ (t) ≤ ϑ(t) ≤ K , where K depends only on d ∗ , p, b and on the functions m f (·), m h (·, ·), h (·) and f (·). This shows that the set E is bounded in Bb0 . As a result the second assertion of Lemma 2.9 does not hold. Hence, the first assertion holds and the multi-valued map has a solution x on [0, b]. In ¯ has at least one fixed point y ∗ ∈ B 0 . Let view of Lemma 2.9, we conclude that
b ˜ t ∈ (−∞, b]. Then, x is a fixed point of the operator , which x(t) = y ∗ (t) + ϕ(t), implies that x is a mild solution of the system (1.1)–(1.3).
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4 Existence of Optimal Mild Solutions In this section, we give the existence of optimal mild solution for the system (1.1)– (1.3). Next, denote by f the set of all mild solutions x(t) to the system (1.1)–(1.3) p which are bounded over [0, b], that is, μ(x) = supt∈[0,b] E x(t) α . Assume here that f = ∅, and recall the following definition. Definition 4.1 A bounded mild solution x ∗ (t) to the system (1.1)–(1.3) is called an optimal mild solution to the systems (1.1)–(1.3) if μ(x ∗ ) ≡ μ∗ = inf x∈ f μ(x). Our proof is based on the following lemma. Lemma 4.2 [26]. If D˜ is a nonempty convex and closed subset of a uniformly convex ˜ then there exists a unique k0 ∈ D˜ such that v − k0 = Banach space X and v ∈ / D, inf k∈ D˜ v − k . To study the optimal mild solutions to the system (1.1)–(1.3), we require the following assumption. (S1) The functions G : [0, b] × B → Hα+β , F : [0, b] × B × Hα → Pbd,cl,cv (L(K , H )), gi : [0, b] × B → Hα , i = 1, . . . , N , are nontrivial. Moreover, G, gi are convex in ψ ∈ B for all i = 1, . . . , N . Theorem 4.3 If the assumption (S1) and the assumptions of Theorem 3.3 hold, then the system (1.1)–(1.3) has an optimal mild solution. Proof It suffices to prove that f is a convex and closed set because the trivial solution 0 ∈ / f , then we use Lemma 4.2 to deduce the uniqueness of the optimal mild solution. For the convexity of f , we consider two distinct bounded mild solutions x1 (t), x2 (t) ∈ f and a real number 0 ≤ λ ≤ 1, let x(t) = λx1 (t) + (1 − λ)x2 (t), t ∈ [0, b]. By (H1),(H2),(H6)–(H9) and (S1), we get that x(t) is continuous for every t ∈ [0, b]. For each f ∈ S F,x , and t ∈ [0, t1 ], we have x(t) = T (t)[ϕ(0) − G(0, ϕ)] + [λG(t, x1,t ) + (1 − λ)G(t, x2,t )] t + AT (t − s)[λG(s, x1,s ) + (1 − λ)G(s, x2,s )]ds 0 t + T (t − s)[λ f (s) + (1 − λ) f (s)]dw(s) 0
= T (t)[ϕ(0) − G(0, ϕ)] + G(t, λx1,t + (1 − λ)x2,t ) t + AT (t − s)(t − s)G(s, λx1,s + (1 − λ)x2,s )ds 0 t + T (t − s) f (s)dw(s) 0
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= T (t)[ϕ(0) − G(0, ϕ)] + G(t, xt ) t AT (t − s)(t − s)G(s, xs )ds + 0 t + T (t − s) f (s)dw(s). 0
For any t ∈ (ti , si ], i = 1, . . . , N , we have x(t) = λgi (t, x1,t ) + (1 − λ)gi (t, x2,t ) = gi (t, xt ). Similarly, for each f ∈ S F,x , and any t ∈ (si , ti+1 ], i = 1, . . . , N , we have x(t) = T (t − si ){[λgi (si , x1,si ) + (1 − λ)gi (si , x1,si )] −[λG(si , x1,si ) + (1 − λ)G(si , x2,si )]} +[λG(t, x1,t ) + (1 − λ)G(t, x2,t )] t + AT (t − s)[λG(s, x1,s ) + (1 − λ)G(s, x2,s )]ds
si t
+
T (t − s)[λ f (s) + (1 − λ) f (s)]dw(s)
si
= T (t − si )[gi (si , xsi ) − G(si , xsi )] + G(t, xt ) t t + AT (t − s)G(s, xs )ds + T (t − s) f (s)dw(s). si
si
Then x(t) is a mild solution to the system (1.1)–(1.3). Note that x(t) is bounded over [0, b] since μ(x) = sup E x(t) αp ≤ μ(x1 ) + (1 − λ)μ(x2 ) < ∞. t∈[0,b]
Thus, x(t) ∈ f . Now we show that f is closed. Let a sequence xn ∈ f such that limn→∞ xn (t) = x(t), t ∈ [0, b], and ⎧ T (t)[ϕ(0) − G(0, ϕ)] + G(t, xn,t ) ⎪ ⎪ t ⎪ ⎪ ⎪ + 0 AT (t − s)G(s, xn,s )ds ⎪ ⎪ t ⎪ ⎪ + T (t − s) f n (s)dw(s), ⎪ ⎪ ⎨ g (t, x0 ), i n,t xn (t) = T (t − s )[g (s , x ) − G(s , x )] i i i n,si i n,si ⎪ ⎪ ⎪ ⎪ ) +G(t, x n,t ⎪ ⎪ t ⎪ ⎪ + si AT (t − s)G(s, xn,s )ds ⎪ ⎪ ⎪ ⎩ + t T (t − s) f (s)dw(s), n si
t ∈ [0, t1 ], i = 0, t ∈ (ti , si ], i ≥ 1,
(4.1)
t ∈ (si , ti+1 ], i ≥ 1,
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where f ∈ S F,xn . By (H1), (H2), (H6)–(H9), taking limits in (4.1), we have for all t ∈ [0, b], ⎧ T (t)[ϕ(0) − G(0, ϕ)] + G(t, xt ) ⎪ ⎪ t ⎪ ⎪ ⎪ + AT (t − s)G(s, xs )ds ⎪ 0t ⎪ ⎪ ⎪ ⎪ ⎪ + 0 T (t − s) f (s)dw(s), ⎨ gi (t, xt ), x(t) = T (t − s )[g (s , x ) − G(s , x )] i i i si i si ⎪ ⎪ ⎪ ⎪ ) +G(t, x t ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ + si AT (t − s)G(s, xs )ds ⎪ ⎩ + t T (t − s) f (s)dw(s), si
t ∈ [0, t1 ], i = 0, t ∈ (ti , si ], i ≥ 1,
(4.2)
t ∈ (si , ti+1 ], i ≥ 1,
where f ∈ S F,x , which means that x(t) is a mild solution to the system (1.1)– (1.3). Finally, we show that x(t) is bounded over [0, b]. Indeed, we can write (4.2) as x(t) = [x(t)−xn (t)]+xn (t) for all t ∈ [0, b]. Then, we have for f n ∈ S F,xn , f ∈ S F,x and t ∈ [0, t1 ], E x(t) αp p ≤ 6 p−1 (−A)−β H E (−A)β G(t, xn,t ) − (−A)β G(t, xt ) αp p − 1 p−1 pβ−1 p−1 p −β p +6 M1−β (−A) H t1 pβ − 1 t × E (−A)β G(s, xn,s ) − (−A)β G(s, xs ) αp ds 0
p/2−1 p−2 +6 p−1 C p Mαp p − 2 − pα t p/2−1− pα p ×t1 E f n (s) − f (s) H ds 0
+2 p−1 E xn (t) αp . For any t ∈ (ti , si ], i = 1, . . . , N , we have E x(t) αp ≤ 2 p−1 E gi (t, xn,t ) − gi (t, xt ) αp +2 p−1 E xn (t) αp . Similarly, for f n ∈ S F,xn , f ∈ S F,x and any t ∈ (si , ti+1 ], i = 1, . . . , N , we have E x(t) αp ≤ 16 p−1 M p [E gi (si , xsn,i ) − gi (si , xsi ) αp + (−A)−β H E (−A)β G(si , xn,si ) − (−A)β G(si , xsi ) αp ] p
+8 p−1 (−A)−β H E (−A)β G(t, xn,t ) − (−A)β G(t, xt ) αp p − 1 p−1 p p +8 p−1 M1−β (−A)−β H (ti+1 − si ) pβ−1 pβ − 1 p
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t
× si
E (−A)β G(s, xn,s ) − (−A)β G(s, xs ) αp ds
p/2−1 p−2 +8 p − 2 − pα t p ×(ti+1 − si ) p/2−1− pα E f n (s) − f (s) H ds p−1
C p Mαp
si
+2
p−1
E xn (t)
αp
.
Choose n large enough and combine Theorem 3.3, for every ε > 0 we get E x(t) αp ≤ ε + 2 p−1 E xn (t) αp for all t ∈ [0, b]. Then one has μ(x) ≤ ε + 2 p−1 μ(xn ) < ∞. Thus, x ∈ f .
5 Application Consider the following multi-valued impulsive stochastic partial functional neutral integrodifferential equations of the form d z(t, x) − ∈
t
−∞ 0
∂2
π
b˜1 (s − t, η, x)z(s, η)dηds
t
z(t, x)dt + b˜2 (t, s − t, x, z(s, x))ds ∂x2 −∞ t s ˜ ˜ b3 (t)b4 (s, τ − s, x, z(τ, x))dτ ds dw(t), + 0
−∞
N (t, x) ∈ ∪i=1 [si , ti+1 ] × [0, π ],
z(t, 0) = z(t, π ) = 0, t ∈ [0, π ], z(τ, x) = ϕ(τ, x), (τ, x) ∈ (−∞, 0] × [0, π ], t ηi (s − t, z(s, x))ds, (t, x) ∈ (ti , si ] × [0, π ], z(t, x) = −∞
(5.1) (5.2) (5.3) (5.4)
where ϕ is continuous, w(t) denotes a one-dimensional standard Wiener process in H defined on a stochastic space (, F, P) and take H = L 2 ([0, π ]) with the norm · and define the operators A : D(A) ⊂ H → H by Aω = ω with the domain D(A) := {ω ∈ H : ω, ω are absolutely continuous, ω ∈ H , ω(0) = ω(π ) = 0}.
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Then Aω =
∞
n 2 ω, ωn ωn , ω ∈ D(A),
n=1
where ωn (x) = π2 sin(nx), n = 1, 2, . . . is the orthogonal set of eigenvectors of A. It is well known that A is the infinitesimal generator of an analytic semigroup T (t), t ≥ 0 −n 2 t (ω, ω )ω , ω ∈ H , and T (t) ≤ e−t for in H and is given by T (t)ω = ∞ n n n=1 e 1 1 − 21 = 1. all t ≥ 0. For each ω ∈ H , (−A)− 2 ω = ∞ n=1 n ω, ωn ωn and (−A) 1
The operator (−A) 2 is given by 1
(−A) 2 ω =
∞
nω, ωn ωn
n=1
1 on the space D((−A) 2 ) = {ω(·) ∈ H , ∞ n=1 nω, ωn ωn ∈ H }. ˜ Let r ≥ 0, 1 ≤ p < ∞ and let h : (−∞, −r ] → R be a nonnegative measurable function which satisfies the conditions (h-5), (h-6) in the terminology of Hino et al. [21]. Briefly, this means that h˜ is locally integrable and there is a non-negative, locally ˜ + τ ) ≤ γ (ξ )h(τ ˜ ) for all ξ ≤ 0 and bounded function γ on (−∞, 0] such that h(ξ θ ∈ (−∞, −r ) \ Nξ , where Nξ ⊆ (−∞, −r ) is a set whose Lebesgue measure ˜ H 1 ) the set consists of all classes of functions is zero. We denote by PC r × L 2 (h, 2 ϕ : (−∞, 0] → H 1 such that ϕ|[−r ,0] ∈ C([−r , 0], H 1 ), ϕ(·) is Lebesgue measurable 2 2 p on (−∞, −r ), and h˜ ϕ 1 is Lebesgue integrable on (−∞, −r ). The seminorm is 2
given by ϕ B =
sup
−r ≤θ≤0
ϕ(θ ) 1 + 2
−r −∞
˜ ) ϕ(θ ) p1 dθ h(θ
1
p
.
2
˜ H 1 ) satisfies axioms (A)–(C). Moreover, when r = 0 The space B = PC r × L p (h, 2 0 1 ˜ )dθ ) 21 , for and p = 2, we can take H˜ = 1, M(t) = γ (−t) 2 and K (t) = 1 + ( −t h(θ t ≥ 0 (see Theorem 1.3.8, [21] for details). Additionally, we choose β = 21 and assume that the following conditions hold: (i) The functions ∂ 0 and l˜1 = max
0
i b˜
π
1 (s,η,x) ∂xi
0
, i = 0, 1, 2, are measurable, b˜1 (s, η, π ) = b˜1 (s, η, 0) =
−∞ 0
π
∂ i b˜1 (s, η, x) ˜ ∂ xi h(s) 1
2
1 2
dηdsdx
: i = 0, 1, 2 < ∞.
(ii) The function b˜3 : R → R is continuous, and the functions b˜2 , b˜4 : R4 → R are continuous and there exists continuous functions a j : R×R → R, j = 1, 2, 3, 4,
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such that |b˜2 (t, s, x, y)| ≤ a1 (t)a2 (s)|y|, (t, s, x, y) ∈ R4 , |b˜4 (t, s, x, y)| ≤ a3 (t)a4 (s)|y|, (t, s, x, y) ∈ R4 with l˜2 = (
0
−∞
0 (a4 (s))2 1 1 (a2 (s))2 ˜ 2 ds) 2 < ∞. ˜h(s) ds) < ∞, l3 = ( −∞ h(s) ˜ : R2 → R, i = 1, . . . , N , is continuous and there
(iii) The functions ηi uous functions c˜i : R → R such that
exist contin-
|ηi (s, y)| ≤ c˜i (s)|y|, (s, y) ∈ R2 with L˜ i = (
0
1 (c˜i (s))2 ds) 2 ˜ h(s)
−∞
< ∞ for every i = 1, 2, . . . , N .
˜ H 1 ) with ϕ(θ )(x) = ϕ(θ, x), (θ, x) ∈ (−∞, 0] × B. Take ϕ ∈ B = PC 0 × L 2 (h, 2 Let z(s)(x) = z(s, x). G : [0, b]×B → H , F : [0, b]×B × H 1 → P(L(K , H )), gi : 2 [0, b] × B → H 1 be the operators defined by 2
G(t, ψ)(x) = ˜ F(t, ψ, Bψ)(x) = ˜ Bψ(x) =
0
π
b˜1 (s, v, x)ψ(s, x)dvds,
−∞ 0 0
b˜2 (t, s, x, ψ(s, x))ds −∞ t 0 −∞
0
gi (t, ψ)(x) =
0
−∞
˜ + Bψ(x),
b˜3 (t)b˜4 (s, τ, x, ψ(τ, x))dτ ds.
ηi (s, ψ(s, x))ds.
Under the above assumptions, the problem (5.1)–(5.4) can be written as (1.1)–(1.3). Moreover, using (i) we can prove that G is D(A)-valued and E AG(t, ψ) p 0 1 p 2 2 ˜ ˜ ≤ l1 ψ(0) + h(θ ) ψ(θ ) dθ −∞ 1 1 ≤ l˜1 (−A)− 2 (−A) 2 ψ(0) +
0
−∞
˜ ) (−A)− 2 2 (−A) 2 ψ(θ ) 2 dθ h(θ 1
˜ = l1 ψ(0) 1 + 2
0
−∞
1
˜ ) ψ(θ ) 21 dθ h(θ
1 p 2
1 p 2
2
p
= L G ψ B
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and E AG(t, ψ) − AG(t, ψ1 ) p ≤ L G ψ − ψ B for all (t, ψ), (t, ψ1 ) ∈ p [0, b] × B, where L G = l˜1 . Similarly, it follows by assumption (ii) that p ˜ E F(t, ψ, Bψ) p ≤ L F (t) ψ B
for all (t, ψ) ∈ [0, b] × B, where L F (t) = [a1 (t)l˜2 + b˜3 (t) a3 L 1 l˜3 ] p . By assumption (iii), we have 1
E (−A) 2 gi (t, ψ) p 1 ≤ L˜ i (−A) 2 ψ(0) + ≤
L˜ i
ψ(0) 1 + 2
0
−∞
0 −∞
˜ ) (−A) 2 ψ(θ ) 2 dθ h(θ 1
˜ ) ψ(θ ) 21 dθ h(θ
1 p 2
1 p 2
2
p
= γi ψ B 1
1
p
and E (−A) 2 gi (t, ψ) − (−A) 2 gi (t, ψ1 ) p ≤ γi ψ − ψ1 B for all (t, ψ), p (t, ψ1 ) ∈ (ti , si ] × B, where γi = L˜ i , i = 1, . . . , N ,. Therefore, (H1), (H2) and ∞ 1 (H6)–(H9) are all satisfied and h (s) = f (s) = s, 1 h (s)+ ds = ∞. If f (s) also the associated condition (3.4) holds, then, according to Theorem 3.3, problems (5.1)–(5.4) admit a mild solution on [0, b] under the above assumptions. Further, we can impose some suitable conditions on the above-defined functions to verify the assumptions on Theorem 4.3. Hence, by Theorems 4.3, the system (5.1)–(5.4) has an optimal mild solution on [0, b].
6 Conclusion This paper contains the existence of mild solutions and optimal mild solutions for a new class of multi-valued impulsive stochastic partial integrodifferential equations with infinite delay in Hilbert spaces. More precisely, using stochastic analysis, the properties of analytic semigroup and a fixed point theorem for contraction multi-valued maps, we investigated the uniqueness of mild solutions of the impulsive stochastic system. Then, the existence of mild solutions is studied when T (·) is analytic and F satisfies the Carathéodory condition by utilizing the fixed point theorem for multivalued operators by Dhage. Also, we discussed the existence results of optimal mild solutions. The conditions are formulated and proved under which the fractional power arguments. Finally, an application is provided to illustrate the applicability of the new results. There are two direct issues which require further study. First, we will investigate the existence of optimal mild solutions for a new class of multi-valued impulsive stochastic partial integrodifferential equations with state-dependent delay both in the case of a noncompact operator and a normal topological space. Second, we will be devoted to study the existence of optimal mild solutions of systems governed by impulsive
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partial stochastic differential inclusions with not instantaneous impulses in an infinite interval. Acknowledgements The authors would like to thank the editor and the reviewers for their constructive comments and suggestions. This work is supported by the National Natural Science Foundation of China (11461019), the President Fund of Scientific Research Innovation and Application of Hexi University (xz2013-10), and the Scientific Research Fund of Young Teacher of Hexi University (QN2015-01).
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