Controllability of Fractional Neutral Stochastic Integro-Differential Systems with Infinite Delay

Abstract

This paper is concerned with the controllability of a class of fractional neutral stochastic integro-differential systems with infinite delay in an abstract space. By employing fractional calculus and Sadovskii's fixed point principle without assuming severe compactness condition on the semigroup, a set of sufficient conditions are derived for achieving the controllability result.

1. Introduction

It is well known that the fractional calculus is a classical mathematical notion and is a generalization of ordinary differentiation and integration to arbitrary (noninteger) order. Nowadays, studying fractional-order calculus has become an active research field [1–7]. Much effort has been devoted to apply the fractional calculus to networks control. For example, Chen et al. [8], Delshad et al. [9], and Wang and Zhang [10] studied the synchronization for fractional-order complex dynamical networks; Zhang et al. [11] investigated a fractional order three-dimensional Hopfield neural network and pointed out that chaotic behaviors can emerge in a fractional network; Kaslik and Sivasundaram [12] discussed the local stability for fractional-order neural networks of Hopfield type by applying the linear stability theory of fractional-order system.

One of the emerging branches of this study is the theory of fractional evolution equations, say, evolution equations, where the integer derivative with respect to time is replaced by a derivative of fractional order. The increasing interest in this class of equations is motivated both by their application to problems from fluid dynamic traffic model, viscoelasticity, heat conduction in materials with memory, electrodynamics with memory, and also because they can be employed to approach nonlinear conservation laws (see [13] and references therein). In addition, neutral stochastic differential equations with infinite delay have become important in recent years as mathematical models of phenomena in both science and engineering, for instance, in the theory development in Gurtin and Pipkin [14] and Nunziato [15] for the description of heat conduction in materials with fading memory. It should be pointed out that 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. We mention here the recent papers [16, 17] concerning the existence of mild solutions of fractional stochastic systems.

As one of the fundamental concepts in mathematical control theory, controllability plays an important role both in deterministic and stochastic control problems such as stabilization of unstable systems by feedback control. Roughly speaking, controllability generally means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. Controllability problems for different nonlinear stochastic systems in infinite dimensional spaces have been extensively studied in many papers; see [18–22] and references therein. We would also like to mention that the controllability for stochastic systems with infinite delay has been investigated by Balasubramaniam et al. [23, 24] and Ren et al. [25] using some abstract spaces. Nevertheless, to the best of our knowledge, it seems that little is known about the controllability of fractional neutral stochastic differential equations with infinite delay, and the aim of this paper is to close this gap.

In this paper, we are interested in the controllability of a class of fractional neutral stochastic integro-differential systems with infinite delay of the followin form: ^𝑐𝐷^𝛼𝑑𝑡^𝛼𝑥(𝑡)+𝐺𝑡,𝑥_𝑡(𝑡)(𝑡)+=−𝐴𝑥+𝐵𝑢_𝑡0𝜎𝑡,𝑠,𝑥_𝑠(𝑠)𝑥(𝑡)(𝑡)(].𝑑𝑊,𝑡∈𝐽,=𝜙,𝑡∈−∞,0 (1) Here, 𝐽∶=[0,𝑏], 𝑏>0. 𝑥(⋅) takes value in a real separable Hilbert space ℍ with inner product (⋅,⋅) and norm |⋅|_ℍ. The fractional derivative ^𝑐𝐷^𝛼, 𝛼∈(1/2,1] is understood in the Caputo sense. −𝐴∶𝒟(−𝐴)⊂ℍ→ℍ is the infinitesimal generator of a analytic semigroup of a bounded linear operator 𝑆(𝑡), 𝑡≥0, on ℍ. Let 𝕂 be another separable Hilbert space with inner product (⋅,⋅)_𝕂 and norm ‖⋅‖_𝕂. 𝑊 is a given 𝕂-valued Wiener process with a finite trace nuclear covariance operator 𝑄≥0 defined on a filtered complete probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃). The control function 𝑢(⋅) takes value in 𝐿²(𝐽,𝕌) of admissible control functions for a separable Hilbert space 𝕌, and 𝐵 is a bounded linear operator from 𝕌 into ℍ. The histories 𝑥_𝑡∶Ω→𝐶_ℎ defined by 𝑥_𝑡={𝑥(𝑡+𝜃),𝜃∈(−∞,0]} belong to the phase space 𝐶_ℎ, which will be defined in Section 2. The initial data 𝜙={𝜙(𝑡),𝑡∈(−∞,0]} is an ℱ₀-measurable, 𝐶_ℎ-valued random variable independent of 𝑊 with finite second moments, and 𝐺∶𝐽×𝐶_ℎ→ℍ, 𝜎∶𝐽×𝐽×ℍ→ℒ⁰₂(𝕂,ℍ) are appropriate mappings specified later (here, ℒ⁰₂(𝕂,ℍ) denotes the space of all 𝑄-Hilbert-Schmidt operators from 𝕂 into ℍ, which is going to be defined later).

The structure of this paper is as follows. In Section 2, we briefly present some basic notations and preliminaries. The controllability result of system (1) is investigated by means of Sadovskii’s fixed point theorem and operator theory in Section 3. Conclusion is given in Section 4.

2. Preliminaries

For more details in this section, we refer the reader to Pazy [26], Da Prato and Zabczyk [27], and Samko et al. [28]. Throughout this paper, (ℍ,|⋅|_ℍ) and (𝕂,‖⋅‖_𝕂) denote two real separable Hilbert spaces. We denote by ℒ(𝕂,ℍ) the set of all linear bounded operators from 𝕂 into ℍ, equipped with the usual operator norm ‖⋅‖. In this paper, we use the symbol ‖⋅‖ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises.

Let (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) be a filtered complete probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and ℱ₀ contains all 𝑃-null sets. 𝑊=(𝑊_𝑡)_𝑡≥0 is a 𝑄-Wiener process defined on (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) with covariance operator 𝑄 such that Tr𝑄<∞. We assume that there exist a complete orthonormal system {𝑒_𝑘}_𝑘≥1 in 𝕂, a bounded sequence of nonnegative real numbers 𝜆_𝑘 such that 𝑄𝑒_𝑘=𝜆_𝑘𝑒_𝑘, 𝑘=1,2,…, and a sequence of independent Brownian motions {𝛽_𝑘}_𝑘≥1 such that (𝑊(𝑡),𝑒)_𝕂=^∞_𝑘=1𝜆_𝑘𝑒_𝑘,𝑒_𝕂𝛽_𝑘(𝑡),𝑒∈𝕂,𝑡≥0. (2) Let ℒ⁰₂=ℒ₂(𝑄^1/2𝕂,ℍ) be the space of all Hilbert-Schmidt operators from 𝑄^1/2𝕂 to ℍ with the inner product ⟨𝜑,𝜙⟩_ℒ⁰₂=Tr[𝜑𝑄𝜙∗].

Suppose that 0∈𝜌(−𝐴), where 𝜌(−𝐴) is the resolvent set of −𝐴, then the semigroup 𝑆(⋅) is uniformly bounded. That is to say, ‖𝑆(𝑡)‖≤𝑀, 𝑡≥0, for some constant 𝑀>0. Then, for 𝛼∈(0,1], it is possible to define the fractional power operator 𝐴^𝛼 as a closed linear operator on its domain 𝒟(𝐴^𝛼). Furthermore, the subspace 𝒟(𝐴^𝛼) is dense in ℍ, and the expression ‖𝑥‖_𝛼=𝐴^𝛼𝑥,𝑥∈𝒟𝐴^𝛼, (3) defines a norm on ℍ_𝛼∶=𝒟(𝐴^𝛼). The following properties are well known.

Lemma 1 (Pazy [26]). Suppose that the preceding conditions are satisfied.(a)If 0<𝛽<𝛼≤1, then ℍ_𝛼⊂ℍ_𝛽 and the embedding is compact whenever the resolvent operator of 𝐴 is compact.(b)For every 𝛼∈(0,1], there exists a positive constant 𝐶_𝛼 such that 𝐴^𝛼𝑆(𝑡≤𝐶)^𝛼𝑡^𝛼,𝑡>0. (4)

Let now us recall some basic definitions and results of fractional calculus.

Definition 2. The fractional integral of order 𝛼 with the lower limit 0 for a function 𝑓 is defined as 𝐼^𝛼1𝑓=Γ(𝛼)^𝑡₀𝑓(𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠,𝑡>0,𝛼>0, (5) provided the right-hand side is pointwise defined on [0,∞), where Γ(⋅) is the gamma function.

Definition 3. The Caputo derivative of order 𝛼 with the lower limit 0 for a function 𝑓 can be written as ^𝑐𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑓^𝑛(𝑠)(𝑡−𝑠)^{𝛼+1−𝑛}𝑑𝑠=𝐼^𝑛−𝛼𝑓^𝑛(𝑡),𝑡>0,0≤𝑛−1<𝛼<𝑛. (6)

If 𝑓 is an abstract function with values in ℍ, then the integrals that appear in the previous definitions are taken in Bochner’s sense.

Assume that ℎ∶(−∞,0]→(0,+∞) with 𝑙=∫⁰_−∞ℎ(𝑡)𝑑𝑡<+∞ is a continuous function. Recall that the abstract phase space 𝒞_ℎ is defined by 𝒞_ℎ=𝜑∶(−∞,0]⟶ℍ,forany𝑎>0,𝐸𝜑(𝜃)^21/2isaboundedandmeasurablefunctionon[−𝑎,0_0−∞],ℎ(𝑠)sup_{𝑠≤𝜃≤0}𝐸𝜑(𝜃)^21/2𝑑𝑠<+∞. (7) If 𝒞_ℎ is endowed with the norm 𝜑_𝒞ℎ=⁰_−∞ℎ(𝑠)sup_{𝑠≤𝜃≤0}𝐸𝜑(𝜃)^21/2𝑑𝑠,𝜑∈𝒞_ℎ, (8) then (𝒞_ℎ,‖⋅‖_𝒞ℎ) is a Banach space (see Liu et al. [29]).

At the end of this section, we recall the fixed point theorem of Sadovskii [30].

Lemma 4. Let Φ be a condensing operator on a Banach space ℍ; that is, Φ is continuous, and take bounded sets into bounded sets, and 𝜇(Φ(𝐵))≤𝜇(𝐵) for every bounded set 𝐵 of ℍ with 𝜇(𝐵)>0. If Φ(𝑁)⊂𝑁 for a convex, closed, and bounded set 𝑁 of ℍ, then Φ has a fixed point in ℍ (where 𝜇(⋅) denotes Kuratowski's measure of noncompactness.)

3. Main Results

In this section, we obtain controllability of system (1). We first present the definition of mild solutions.

Definition 5. An ℍ-valued stochastic process {𝑥(𝑡),𝑡∈(−∞,𝑏]} is said to be a mild solution of system (1) if (i)𝑥(𝑡) is ℱ_𝑡-adapted and measurable for each 𝑡≥0; (ii)𝑥(𝑡) is continuous on [0,𝑏] almost surely and for each 𝑠∈[0,𝑡), the function (𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑡−𝑠)𝐺(𝑠,𝑥_𝑠) is integrable such that the following stochastic integral equation is verified: 𝑥(𝑡)=𝑆_𝛼(𝑡)𝜙(0)+𝐺0,𝜙−𝐺𝑡,𝑥_𝑡−^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑡−𝑠)𝐺𝑠,𝑥_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()×𝑡−𝑠_𝑠0𝜎𝑠,𝜏,𝑥_𝜏𝑑𝑊(𝜏)𝑑𝑠; (9)(iii)𝑥(𝑡)=𝜙(𝑡) on (−∞,0] with ‖𝜙‖²_𝒞ℎ<∞, where 𝑆_𝛼(𝑡)𝑥=^∞₀𝜂_𝛼(𝜃)𝑆𝑡^𝛼𝜃𝑇𝑥𝑑𝜃,_𝛼(𝑡)𝑥=𝛼^∞₀𝜃𝜂_𝛼(𝜃)𝑆𝑡^𝛼𝜃𝑥𝑑𝜃 (10) with 𝜂_𝛼 a probability density function defined on (0,∞).

Definition 6. System (1) is said to be controllable on the interval 𝐽, if for every initial stochastic process 𝜙∈𝒞_ℎ defined on (−∞,0], there exists a stochastic control 𝑢∈𝐿²(𝐽,𝕌), which is adapted to the filtration {ℱ_𝑡}_𝑡≥0 such that the mild solution 𝑥(𝑡) of (1) satisfies 𝑥(𝑏)=𝑥^∗, where 𝑥^∗ and 𝑏 are preassigned terminal state and time, respectively.

The following properties of 𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡) that appeared in Zhou and Jiao [7] are useful.

Lemma 7. Under previous assumptions on 𝑆(𝑡), 𝑡≥0, and 𝐴, then (i)for any fixed 𝑡≥0, 𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡) are linear and bounded operators such that for any 𝑥∈ℍ, |𝑆_𝛼(𝑡)𝑥|_ℍ≤𝑀|𝑥|_ℍ, |𝑇_𝛼(𝑡)𝑥|_ℍ≤(𝑀𝛼/Γ(1+𝛼))|𝑥|_ℍ; (ii)𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡) are strongly continuous; (iii)for any 𝑥∈ℍ, 𝛽∈(0,1), and 𝜃∈(0,1], one has 𝐴𝑇_𝛼(𝑡)𝑥=𝐴^1−𝛽𝑇_𝛼(𝑡)𝐴^𝛽𝐴𝑥,_𝜃𝑇_𝛼(𝑡)≤𝛼𝐶_𝜃𝑡^𝛼𝜃Γ(2−𝜃)Γ(([].1+𝛼1−𝜃)),𝑡∈0,𝑏 (11)

In this paper, we will work under the following assumptions. (A₀)−𝐴 is the infinitesimal generator of an analytic semigroup of bounded linear operators 𝑆(𝑡) in ℍ; 𝑆(𝑡), 𝑡>0, is continuous in the uniform operator topology and ‖𝑆(𝑡)‖≤𝑀 for some 𝑀>0.(A₁) The function 𝐺∶𝐽×𝒞_ℎ→ℍ is continuous, and there exist some constants 𝑀_𝐺>0, 𝛽∈(0,1) with 𝛼𝛽>1/2, such that 𝐺 is ℍ_𝛽-valued and 𝐸𝐴^𝛽𝐺(𝑡,𝑥)−𝐴^𝛽𝐺𝑡,𝑦²_ℍ≤𝑀_𝐺𝑥−𝑦²_𝒞ℎ,𝑥,𝑦∈𝒞_ℎ𝐸𝐴,𝑡∈𝐽,_𝛽𝐺(𝑡,𝑥)²_ℍ≤𝑀_𝐺‖𝑥‖²_𝒞ℎ+1. (12)(A₂)𝜎∶𝐽×𝐽×𝒞_ℎ→𝐿(𝕂,ℍ) satisfies the following: (2a) for each (𝑡,𝑠)∈𝐷∶=𝐽×𝐽, 𝜎(𝑡,𝑠,⋅)∶𝒞_ℎ→𝐿(𝕂,ℍ) is continuous and for each 𝑥∈𝒞_ℎ, 𝜎(⋅,⋅,𝑥)∶𝐷→𝐿(𝕂,ℍ) is strongly measurable; (2b) there is a positive integrable function 𝑚∈𝐿¹([0,𝑏]) and a continuous nondecreasing function Ξ∶[0,∞)→(0,∞) such that for every (𝑡,𝑠,𝑥)∈𝐽×𝐽×𝒞_ℎ, we have ^𝑡₀𝐸|𝜎(𝑡,𝑠,𝑥)|²_ℒ02𝑑𝑠≤𝑚(𝑡)Ξ‖𝑥‖²_𝒞ℎ,liminf_𝑟→∞Ξ(𝑟)𝑟𝑑𝑠=Λ<∞; (13)(2c) for any 𝑥,𝑦∈𝒞_ℎ, 𝑡≥0, there exists a positive constant 𝑀_𝜎 such that ^𝑡₀𝐸𝜎(𝑡,𝑠,𝑥)−𝜎𝑡,𝑠,𝑦²_ℒ02𝑑𝑠≤𝑀_𝜎𝑥−𝑦²_𝒞ℎ. (14)(A₃) The linear operator 𝑊 from 𝐿²(𝐽,𝕌) into ℍ defined by 𝑊𝑢=^𝑏₀(𝑏−𝑠)^𝛼−1𝑇_𝛼(𝑏−𝑠)𝐵𝑢(𝑠)𝑑𝑠 (15) has an invertible operator 𝑊⁻¹ defined on 𝐿²(𝐽,𝕌)/ker𝑊 (see [31]), and there exist a pair of constants 𝑁₁,𝑁₂>0 such that ‖𝐵‖²≤𝑁₁,𝑊⁻¹≤𝑁₂. (16)(A₄) There exists a compact set ℧⊆ℍ such that 𝑆(𝑡)𝜎(𝑠,𝜏,𝜉)∈℧ for all 𝑡∈𝐽, 0≤𝜏≤𝑠≤𝑏 and 𝜉∈𝒞_ℎ with finite second moment.(A₅) Assume that the following relationship holds: 201+5𝑁₁𝑁₂𝑁𝑙²×𝑀₀+𝑀_𝐺𝑃+𝑁𝑏⁻¹Tr𝑄Λ^𝑏₀𝑚(𝑠)𝑑𝑠<1, (17) where 𝑏𝑁=^2𝛼2𝛼−1𝑀𝛼Γ(1+𝛼)²,𝑀₀=𝐴^−𝛽2𝑀_𝐺,𝛼𝑃=²𝐶²_1−𝛽Γ²1+𝛽Γ²𝑏1+𝛼𝛽^2𝛼𝛽.2𝛼𝛽−1 (18)(19)

Denote by 𝐶((−∞,𝑏],ℍ) the space of all continuous ℍ-valued stochastic processes {𝜉(𝑡),𝑡∈(−∞,𝑏]}. Let 𝒞_𝑏=𝑥∶𝑥∈𝐶((−∞,𝑏],ℍ),𝑥₀=𝜙∈𝒞_ℎ. (20) Set ‖⋅‖_𝑏 to be a seminorm defined by ‖𝑥‖_𝑏=𝑥_0𝒞ℎ+sup_{𝑠∈[0,𝑏]}𝐸|𝑥(𝑠)|^21/2,𝑥∈𝒞_𝑏. (21) We have the following useful lemma that appeared in Liu et al. [29].

Lemma 8. Assume that 𝑥∈𝒞_𝑏; then, for all 𝑡∈𝐽, 𝑥_𝑡∈𝒞_ℎ. Moreover, 𝑙𝐸|𝑥(𝑡)|^21/2≤𝑥_𝑡𝒞ℎ≤𝑙sup_{𝑠∈[0,𝑡]}𝐸|𝑥(𝑠)|^21/2+𝑥_0𝒞ℎ, (22) where 𝑙=∫⁰_−∞ℎ(𝑠)𝑑𝑠<∞.

The main object of this paper is to explain and prove the following theorem.

Theorem 9. Assume that assumptions (A₀)–(A₅) hold. Then, system (1) is controllable on 𝐽 provided that 𝐿∶=3𝑙²1+3𝑁₁𝑁₂𝑁𝑀₀+𝑀_𝐺𝑃+9𝑁₁𝑁₂𝑁²Tr𝑄𝑀_𝜎𝑙²<1. (23)

Proof. Using assumption (A₃), for an arbitrary process 𝑥(⋅), define the control process 𝑢_𝑥(𝑡)=𝑊⁻¹𝑥^∗−𝑆_𝛼(𝑏)𝜙(0)+𝐺0,𝜙+𝐺𝑏,𝑥_𝑏+^𝑏₀(𝑏−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑏−𝑠)𝐺𝑠,𝑥_𝑠−𝑑𝑠_𝑏0(𝑏−𝑠)^𝛼−1𝑇_𝛼()×𝑏−𝑠_𝑠0𝜎𝑠,𝜏,𝑥_𝜏𝑑𝑊(𝜏)𝑑𝑠(𝑡). (24) We transform (1) into a fixed point problem. Consider the operator 𝑄∶𝒞_𝑏→𝒞_𝑏 defined by (𝑡)=𝜙(𝑡)(];𝑆𝑄𝑥)(,𝑡∈−∞,0_𝛼(𝑡)𝜙(0)+𝐺0,𝜙−𝐺𝑡,𝑥_𝑡−∫^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑡−𝑠)𝐺𝑠,𝑥_𝑠+∫𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)𝑑𝑠+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()×∫𝑡−𝑠_𝑠0𝜎𝑠,𝜏,𝑥_𝜏𝑑𝑊(𝜏)𝑑𝑠,𝑡∈𝐽. (25) It follows from Hölder inequality, Lemma 7, and assumption (A₁) that 𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑡−𝑠)𝐺𝑠,𝑥_𝑠𝑑𝑠²_ℍ≤𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝐴^1−𝛽𝑇_𝛼×(𝑡−𝑠)𝐴^𝛽𝐺𝑠,𝑥_𝑠ℍ𝑑𝑠²≤𝑃𝐸^𝑡₀(𝑡−𝑠)^{𝛼(𝛽−1)}(𝑡−𝑠)^𝛼−1×𝐴^𝛽𝐺𝑠,𝑥_𝑠ℍ𝑑𝑠²≤𝑃^𝑡₀(𝑡−𝑠)^{2(𝛼𝛽−1)}𝑑𝑠×^𝑡₀𝑀_𝐺1+𝑥_𝑠²_𝒞ℎ𝑑𝑠, (26) which deduces that (𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑡−𝑠)𝐺(𝑠,𝑥_𝑠) is integrable on 𝐽 by Bochner's theorem (see [32]), where 𝑃 is defined in (19).
In what follows, we will show that using the control 𝑢_𝑥(⋅), the operator 𝑄 has a fixed point, which is then a mild solution for system (1).
For 𝜙∈𝒞_ℎ, define 𝜙(𝑡)=𝜙(𝑡)(];𝑆,𝑡∈−∞,0_𝛼(𝑡)𝜙(0),𝑡∈𝐽. (27) Then, 𝜙∈𝒞_𝑏. Set 𝑥(𝑡)=𝜙(𝑡)+𝑧(𝑡), 𝑡∈(−∞,𝑏]. It is easy to check that 𝑥 satisfies (1) if and only if 𝑧₀=0 and 𝑧(𝑡)=𝑆_𝛼(𝑡)𝐺0,𝜙−𝐺𝑡,𝜙_𝑡+𝑧_𝑡−^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑡−𝑠)𝐺𝑠,𝜙_𝑠+𝑧_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝜙+𝑧(𝑠)+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼()×𝑡−𝑠_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝑠, (28) where 𝑢_𝜙+𝑧 is obtained by replacing 𝑥_𝑡 by 𝜙_𝑡+𝑧_𝑡 in (24). Let 𝒞⁰_𝑏=𝑧∈𝒞_𝑏,𝑧₀=0∈𝒞_ℎ. (29) For each 𝑧∈𝒞⁰_𝑏, we have ‖𝑧‖_𝑏=𝑧_0𝒞ℎ+sup_{𝑠∈[0,𝑏]}𝐸|𝑧(𝑠)|^21/2=sup_{𝑠∈[0,𝑏]}𝐸|𝑧(𝑠)|^21/2. (30) Thus, (𝒞⁰_𝑏,‖⋅‖_𝑏) is a Banach space. For 𝑞>0, set 𝐵_𝑞=𝑦∈𝒞⁰_𝑏,𝑦²_𝑏≤𝑞; (31) then, for each 𝑞, 𝐵_𝑞 is clearly a bounded closed convex set in 𝒞⁰_𝑏. For 𝑧∈𝐵_𝑞, from Lemma 8, we see that 𝑧_𝑡+𝜙_𝑡²_𝒞ℎ≤2𝑧_𝑡²_𝒞ℎ+𝜙_𝑡²_𝒞ℎ≤4𝑙²sup_{𝑠∈[0,𝑡]}𝐸|𝑧(𝑠)|²+𝑧₀²_𝒞ℎ+𝑙²sup_{𝑠∈[0,𝑡]}𝐸𝜙(𝑠)²+𝜙₀²_𝒞ℎ≤4𝑙²𝑞+𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ. (32)
Consider the map Π∶𝒞⁰_𝑏→𝒞⁰_𝑏 defined by (𝑡)=(];𝑆Π𝑧)(0,𝑡∈−∞,0_𝛼(𝑡)𝐺0,𝜙−𝐺𝑡,𝜙_𝑡+𝑧_𝑡−∫^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼()𝜙𝑡−𝑠×𝐺𝑠,_𝑠+𝑧_𝑠𝑑𝑠+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝜙+𝑧(𝑠)𝑑𝑠+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()×∫𝑡−𝑠_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝑠,𝑡∈𝐽. (33) A similar argument as (26), we can show that Π is well defined on 𝐵_𝑞 for each 𝑞>0. Note that the operator 𝑄 with a fixed point is equivalent to show that the operator Π has fixed point. To this end, we decompose Π as Π=Π₁+Π₂, where the operators Π₁ and Π₂ are defined on 𝐵_𝑞, respectively, by Π₁𝑧(𝑡)=𝑆_𝛼(𝑡)𝐺0,𝜙−𝐺𝑡,𝜙_𝑡+𝑧_𝑡−^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑡−𝑠)𝐺𝑠,𝜙_𝑠+𝑧_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝜙+𝑧(𝑠)Π𝑑𝑠,₂𝑧(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()×𝑡−𝑠_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝑠. (34) Thus, Theorem 9 follows from the next theorem.

Theorem 10. Assume that the assumptions of Theorem 9 hold. Then, Π=Π₁+Π₂ satisfies all the conditions in Lemma 4.

Proof. The proof is followed by the several steps.
Step  1. There exists a positive number 𝑞 such that Π(𝐵_𝑞)⊂𝐵_𝑞. If it is not true, then for each positive number 𝑞, there exists a function 𝑧^𝑞(⋅)∈𝐵_𝑞, but Π(𝑧^𝑞)∉𝐵_𝑞; that is, 𝐸|(Π𝑧^𝑞)(𝑡)|²_ℍ>𝑞 for some 𝑡=𝑡(𝑞)∈𝐽. An elementary calculation can show that 𝑞≤𝐸Π𝑧^𝑞(𝑡)²_ℍ≤5𝐸𝑆_𝛼(𝑡)𝐺0,𝜙²_ℍ+5𝐸𝐺𝑡,𝜙_𝑡+𝑧^𝑞_𝑡²_ℍ+5𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑡−𝑠)𝐺𝑠,𝜙_𝑠+𝑧^𝑞_𝑠𝑑𝑠²_ℍ+5𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()×𝑡−𝑠_𝑠0𝜎𝑠,𝜙_𝜏+𝑧^𝑞_𝜏𝑑𝑊(𝜏)𝑑𝑠²_ℍ+5𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢_{𝜙+𝑧𝑞}(𝑠)𝑑𝑠²_ℍ=5⁵_𝑖=1𝐼_𝑖. (35) We estimate each term on the right hand side of (35). By virtue of Lemma 8, assumptions (A₀)-(A₁), and (32), we obtain 𝐼₁+𝐼₂≤𝑀²𝐴^−𝛽2𝐸𝐴^𝛽𝐺0,𝜙²_ℍ+𝐴^−𝛽2𝐸𝐴^𝛽𝐺𝑡,𝜙_𝑡+𝑧^𝑞_𝑡²_ℍ≤𝑀₀𝑀²1+𝜙²_𝒞ℎ+1+𝜙_𝑡+𝑧^𝑞_𝑡²_𝒞ℎ≤𝑀₀𝑀²1+𝜙²_𝒞ℎ+1+4𝑙²𝑞+𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ; (36) in view of Lemma 7 and Hölder inequality, we have 𝐼₃≤𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝐴^1−𝛽𝑇_𝛼(𝑡−𝑠)×𝐴^𝛽𝐺𝑠,𝜙_𝑠+𝑧^𝑞_𝑠ℍ𝑑𝑠²≤𝛼²𝐶²_1−𝛽Γ²1+𝛽Γ²𝐸×1+𝛼𝛽_𝑡0(𝑡−𝑠)^𝛼𝛽−1𝐴^𝛽𝐺𝑠,𝜙_𝑠+𝑧^𝑞_𝑠ℍ𝑑𝑠²≤𝛼²𝐶²_1−𝛽Γ²1+𝛽Γ²1+𝛼𝛽^𝑡₀(𝑡−𝑠)^{2(𝛼𝛽−1)}×𝑑𝑠_𝑡0𝑀_𝐺1+𝜙_𝑠+𝑧^𝑞_𝑠²_𝒞ℎ≤𝑀𝑑𝑠_𝐺𝛼²𝐶²_1−𝛽Γ²1+𝛽Γ²𝑏1+𝛼𝛽^{2𝛼𝛽−1}×2𝛼𝛽−1_𝑡01+𝜙_𝑠+𝑧^𝑞_𝑠²_𝒞ℎ𝑑𝑠≤𝑀_𝐺𝑃1+4𝑙²𝑞+𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ, (37) where 𝑀₀ and 𝑃 are defined in (18) and (19), respectively.
Applying Burkhölder-Davis-Gundy's inequality and assumptions (A₂), we get 𝐼₄≤𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()×𝑡−𝑠_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧^𝑞_𝜏𝑑𝑊(𝜏)_ℍ𝑑𝑠²≤𝑀𝛼Γ(1+𝛼)^2𝑡₀(𝑡−𝑠)^2(𝛼−1)×𝑑𝑠_𝑡0𝐸^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧^𝑞_𝜏𝑑𝑊(𝜏)²_ℍ≤𝑑𝑠𝑀𝛼Γ(1+𝛼)²𝑏^2𝛼−1×2𝛼−1_𝑡0Tr𝑄^𝑠₀𝐸𝜎𝑠,𝜙_𝜏+𝑧^𝑞_𝜏²_ℒ02≤𝑑𝜏𝑑𝑠𝑀𝛼Γ(1+𝛼)²𝑏^2𝛼−1×2𝛼−1Tr𝑄_𝑡0𝑚(𝑠)Ξ𝜙_𝜏+𝑧^𝑞_𝜏²_𝒞ℎ≤𝑑𝑠𝑀𝛼Γ(1+𝛼)²Tr𝑄𝑏^2𝛼−1Ξ×2𝛼−14𝑙²𝑞+𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ^𝑡₀𝑚(𝑠)𝑑𝑠. (38)
On the other hand, in view of (24) and (A₃), we have 𝐸𝐵𝑢_𝑧𝑞+𝜙(𝑡)²_ℍ≤𝑁₁𝑁₂𝐸𝑥^∗−𝑆_𝛼(𝑏𝜙(0)𝜙)+𝐺0,𝜙+𝐺𝑏,_𝑏+𝑧^𝑞_𝑏+^𝑏₀(𝑏−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑏−𝑠)𝐺𝑠,𝜙_𝑠+𝑧^𝑞_𝑠−𝑑𝑠_𝑏0(𝑏−𝑠)^𝛼−1𝑇_𝛼()×𝑏−𝑠_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧^𝑞_𝜏𝑑𝑊(𝜏)𝑑𝑠(𝑡)²_ℍ; (39) thus, by the same procedure as (36)–(38), it follows that 𝐼₅≤𝑀𝛼Γ(1+𝛼)^2𝑡₀(𝑡−𝑠)^2(𝛼−1)𝑑𝑠^𝑡₀𝐸𝐵𝑢_𝑧𝑞+𝜙(𝑠)²_ℍ𝐶𝑑𝑠≤𝑁₀+5𝑁₁𝑁₂×4𝑀_𝐺𝐴^−𝛽2𝑙²𝑞+4𝑙²𝑀_𝐺𝑃𝑞+2𝑁𝑏⁻¹×Tr𝑄Ξ4𝑙²𝑞+𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ×^𝑏₀𝑚(𝑠)𝑑𝑠, (40) where 𝐶₀=5𝑁₁𝑁₂𝐸𝑥^∗2_ℍ+2𝑀²𝐸𝜙(0)²_ℍ+2𝑀²𝑀₀1+𝜙²_𝒞ℎ+𝑀₀1+4𝑙²𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ+𝑀_𝐺𝑃1+4𝑙²𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ. (41)𝑀₀ and 𝑃 are defined in (18) and (19), respectively.
Combining these estimates (35) to (40) yields 𝑞≤𝐸Π𝑧^𝑞(𝑡)²_ℍ≤𝐿₀+20𝑀₀𝑙²1+5𝑁₁𝑁₂𝑁𝑞+20𝑙²𝑀_𝐺𝑃1+5𝑁₁𝑁₂𝑁𝑞+5𝑀𝛼Γ(1+𝛼)²Tr𝑄𝑏^2𝛼−12𝛼−1×𝑁Ξ4𝑙²𝑞+𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ^𝑏₀𝑚(𝑠)𝑑𝑠, (42) where 𝐿₀=5𝑁+5𝑀²𝑀₀1+𝜙²_𝒞ℎ𝐶₀+5𝑀₀1+4𝑙²𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ+5𝑀_𝐺𝑃1+4𝑙²𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ. (43) Dividing both sides of (42) by 𝑞 and taking 𝑞→∞, we obtain that 201+5𝑁₁𝑁₂𝑁𝑙²×𝑀₀+𝑀_𝐺𝑃+𝑁𝑏⁻¹Tr𝑄Λ^𝑏₀𝑚(𝑠)𝑑𝑠≥1, (44) which is a contradiction by assumption (A₅). Thus, for some positive number 𝑞, Π(𝐵_𝑞)⊂𝐵_𝑞.
Step  2. Π₁ is a contractive mapping. Let 𝑥,𝑣∈𝐵_𝑞. From the assumptions on 𝐺 and 𝜎, it is easy to verify that the following inequality holds: 𝐸𝑢_𝑥+𝜙(𝑠)−𝑢_𝑣+𝜙(𝑠)²_ℍ≤3𝑁₂𝐴^−𝛽2𝑀_𝐺𝑥_𝑏−𝑣_𝑏²_𝒞ℎ+𝛼²𝐶²_1−𝛽Γ²1+𝛽Γ²𝑏1+𝛼𝛽2𝛼𝛽−^{2𝛼𝛽−1}1^𝑏₀𝑀_𝐺𝑥_𝑠−𝑣_𝑠²_𝒞ℎ+𝑑𝑠𝑀𝛼Γ(1+𝛼)²𝑏^2𝛼−12𝛼−1^𝑏₀𝑀_𝜎Tr𝑄𝑥_𝑠−𝑣_𝑠²_𝒞ℎ𝑑𝑠. (45) Thus, by the assumptions, we have 𝐸Π₁𝑥(𝑡)−Π₁𝑣(𝑡)²_ℍ≤3𝐸𝐺𝑡,𝜙_𝑡+𝑥_𝑡−𝐺𝑡,𝜙_𝑡+𝑣_𝑡²_ℍ+3𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼()×𝐺𝜙𝑡−𝑠𝑠,_𝑠+𝑥_𝑠−𝐺𝑠,𝜙_𝑠+𝑣_𝑠𝑑𝑠²_ℍ+3𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝜙+𝑥−𝑢_𝜙+𝑣𝑑𝑠²_ℍ≤3𝑀₀𝑙²1+3𝑁₁𝑁₂𝑁sup_{0≤𝑠≤𝑏}𝐸|𝑥(𝑠)−𝑣(𝑠)|²_ℍ+3𝑃𝑀_𝐺𝑙²1+3𝑁₁𝑁₂𝑁sup_{0≤𝑠≤𝑏}𝐸|𝑥(𝑠)−𝑣(𝑠)|²_ℍ+9𝑁₁𝑁₂𝑁²Tr𝑄𝑀_𝜎𝑙²sup_{0≤𝑠≤𝑏}𝐸|𝑥(𝑠)−𝑣(𝑠)|²_ℍ, (46) where we have used the fact that 𝑢₀=𝑣₀=0. Hence, Π₁𝑥−Π₁𝑣²_𝑏≤𝐿‖𝑢−𝑣‖²_𝑏; (47) so, Π₁ is a contraction by (23).
Step  3. We show that the operator Π₂ is compact. Let 𝑞>0 be such that Π₂(𝐵_𝑞)⊂𝐵_𝑞. The proof will be divided into the following claims.
Claim  1. Π₂ maps bounded sets into bounded sets in 𝐵_𝑞.
Indeed, it is enough to show that there exists a positive constant Δ such that for each 𝑡∈𝐽, 𝑧∈𝐵_𝑞, we have ‖Π₂𝑧‖²_𝑏≤Δ. If 𝑧∈𝐵_𝑞, from (32), it follows that 𝑧_𝑡+𝜙_𝑡²_𝒞ℎ≤4𝑙²𝑞+𝑀²𝐸𝜙(0)²_ℍ+4𝜙²_𝒞ℎ∶=𝑞^′. (48) By the similar argument as before, we get 𝐸Π₂𝑧(𝑡)²_ℍ≤𝑀𝛼Γ(1+𝛼)^2𝑡₀(𝑡−𝑠)^2(𝛼−1)×𝑑𝑠_𝑡0𝐸^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)²_ℍ≤𝑑𝑠𝑀𝛼Γ(1+𝛼)²Tr𝑄𝑏^2𝛼−1Ξ𝑞2𝛼−1^′𝑡₀𝑚(𝑠)≤𝑑𝑠𝑀𝛼Γ(1+𝛼)²Tr𝑄𝑏^2𝛼−1Ξ𝑞2𝛼−1^′𝑏₀𝑚(𝑠)𝑑𝑠∶=Δ. (49) Therefore, for each 𝑧∈𝐵_𝑞, it holds that ‖Π₂𝑧‖²_𝑏≤Δ.
Claim  2. The set {Π₂𝑧,𝑧∈𝐵_𝑞} is an equicontinuous family of functions on 𝐽. Let 0<𝜖<𝑡<𝑏 and 𝛿>0 such that ‖𝑇_𝛼(𝑠₁)−𝑇_𝛼(𝑠₂)‖<𝜖, for every 𝑠₁,𝑠₂∈𝐽 with |𝑠₁−𝑠₂|<𝛿. For 𝑧∈𝐵_𝑞, 0<|ℎ|<𝛿, 𝑡+ℎ∈𝐽, we have 𝐸Π₂𝑧(𝑡+ℎ)−Π₂𝑧(𝑡)²_ℍ=𝐸₀^𝑡+ℎ(𝑡+ℎ−𝑠)^𝛼−1𝑇_𝛼()×𝑡+ℎ−𝑠_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏(𝜏−𝑑𝑊)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼()×𝑡−𝑠_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝑠²_ℍ≤3𝐸_𝑡^𝑡+ℎ(𝑡+ℎ−𝑠)^𝛼−1𝑇_𝛼()×𝑡+ℎ−𝑠_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝑠²_ℍ+3𝐸^𝑡₀(𝑡+ℎ−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−1𝑇_𝛼()×𝑡+ℎ−𝑠_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝑠²_ℍ+3𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡+ℎ−𝑠)−𝑇_𝛼(×𝑡−𝑠)_𝑠0𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝑠²_ℍ. (50) Applying assumption (A₂), Hölder inequality, together with Lemma 7, we get 𝐸Π₂𝑧(𝑡+ℎ)−Π₂𝑧(𝑡)²_ℍ≤3𝑍(𝛼)_𝑡^𝑡+ℎ(𝑡+ℎ−𝑠)^2(𝛼−1)×𝑑𝑠_𝑡𝑡+ℎ𝐸^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)²_ℍ(𝛼)𝑑𝑠+3𝑍^𝑡₀(𝑡+ℎ−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−12×𝑑𝑠_𝑡0𝐸^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)²_ℍ𝑑𝑠+3𝜖^𝑡₀(𝑡−𝑠)^2(𝛼−1)×𝑑𝑠_𝑡0𝐸^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)²_ℍ(𝛼)ℎ𝑑𝑠≤3𝑍^2𝛼−12𝛼−1_𝑡^𝑡+ℎTr𝑄𝑀_𝜎𝑚(𝑠)Ξ𝑞^′(𝛼)𝑑𝑠+3𝑍^𝑡₀(𝑡+ℎ−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−12𝑑𝑠×^𝑡₀Tr𝑄𝑀_𝜎𝑚(𝑠)Ξ𝑞^′𝑑𝑠+3𝜖^𝑡₀(𝑡−𝑠)^2(𝛼−1)𝑑𝑠^𝑡₀Tr𝑄𝑀_𝜎𝑚(𝑠)Ξ𝑞^′𝑑𝑠, (51) with 𝑍(𝛼)=(𝑀𝛼/Γ(1+𝛼))². Therefore, for 𝜖 sufficiently small, the right hand side of (51) tends to zero as ℎ→0. Thus, the set {Π₂𝑧,𝑧∈𝐵_𝑞} is equicontinuous by assumption (A₁).
Claim  3. Π₂ maps 𝐵_𝑞 into a precompact set in 𝐵_𝑞. Let 0<𝑡≤𝑏 be fixed and 𝜖 a real number satisfying 0<𝜖<𝑡. For 𝛿>0, define an operator Π₂^𝜖,𝛿 on 𝐵_𝑞 by Π₂^𝜖,𝛿𝑧(𝑡)=𝛼₀^{𝑡−𝜖∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜂_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃×^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏(𝜏𝜖𝑑𝑊)𝑑𝜃𝑑𝑠=𝑆^𝛼𝛿𝛼₀^{𝑡−𝜖∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜂_𝛼(𝜃)𝑆×(𝑡−𝑠)^𝛼𝜃−𝜖^𝛼𝛿×^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝜃𝑑𝑠. (52) Since 𝑆(𝑡)𝜎(𝑠,𝜏,𝜙_𝜏+𝑧_𝜏) belongs to the compact set ℧, the set {Π₂^𝜖,𝛿𝑧)(𝑡),𝑧∈𝐵_𝑞} is precompact in ℍ for every 𝜖∈(0,𝑡), 𝛿>0. Moreover, for each 𝑧∈𝐵_𝑞, we have 𝐸Π₂𝑧(𝑡)−Π₂^𝜖,𝛿𝑧(𝑡)²_ℍ≤2𝛼²𝐸^𝑡₀^𝛿₀𝜃(𝑡−𝑠)^𝛼−1𝜂_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃×^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝜃𝑑𝑠²_ℍ+2𝛼²𝐸^𝑡_𝑡−𝜖^∞_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜂_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃×^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)𝑑𝜃𝑑𝑠²_ℍ=𝐽₁+𝐽₂. (53) An application of Hölder inequality together with Burkhölder-Davis-Gundy's inequality, it follows that 𝐽₁≤2𝛼²𝑀^2𝑡₀(𝑡−𝑠)^2(𝛼−1)𝐸×^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)²_ℍ𝑑𝑠×^𝑡₀^𝛿₀𝜃𝜂_𝛼(𝜃)𝑑𝜃²𝑑𝑠≤2𝛼²𝑀²𝑏^𝑡₀(𝑡−𝑠)^2(𝛼−1)×Tr𝑄_𝑠0𝐸𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏²_ℒ02𝑑𝜏𝑑𝑠^𝛿₀𝜃𝜂_𝛼(𝜃)𝑑𝜃²≤2𝛼²𝑀²𝑏^𝑡₀Tr𝑄Ξ𝑞^′𝑚(𝑠)(𝑡−𝑠)^2(𝛼−1)×𝑑𝑠_𝛿0𝜃𝜂_𝛼(𝜃)𝑑𝜃²,𝐽₂≤2𝛼²𝑀^2𝑡_𝑡−𝜖(𝑡−𝑠)^2(𝛼−1)𝐸×^𝑠₀𝜎𝑠,𝜏,𝜙_𝜏+𝑧_𝜏𝑑𝑊(𝜏)²_ℍ×𝑑𝑠_{𝑡𝑡−𝜖∞0}𝜃𝜂_𝛼(𝜃)𝑑𝜃²≤𝑑𝑠2𝛼²𝑀²𝜖Γ²(1+𝛼)^𝑡_𝑡−𝜖Tr𝑄Ξ𝑞^′𝑚(𝑠)(𝑡−𝑠)^2(𝛼−1)𝑑𝑠. (54) From (54), we deduce that for each 𝑧∈𝐵_𝑞, 𝐸Π₂𝑧(𝑡)−Π₂^𝜖,𝛿𝑧(𝑡)²_ℍ⟶0as𝜖⟶0⁺,𝛿⟶0⁺. (55) Therefore, there are relatively compact sets arbitrary close to the set {Π₂(𝑡),𝑧∈𝐵_𝑞}; hence, the set {(Π₂𝑧)(𝑡),𝑧∈𝐵_𝑞} is also precompact in 𝐵_𝑞.
Thus, from Arzelá-Ascoli theorem together with assumptions on 𝑓 and 𝜎, we conclude that Π₂ is a compact operator. Therefore, Π is a condensing map on 𝐵_𝑞. This completes the proof.

Remark 11. In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, impulsive differential equations have been used to model the system. The technique used here can be extended to establish the controllability of neutral fractional stochastic integro-differential systems with impulsive effect and infinite delay. The controllability result can be obtained by suitably introducing the impulsive effects defined in [19].

4. Conclusions

In this paper, we have studied the controllability of fractional neutral stochastic integro-differential systems with infinite delay in an abstract space. Through fractional calculus and Sadovskii's fixed point principle, we have investigated the sufficient conditions for the controllability of the system considered.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11171062), Innovation Program of Shanghai Municipal Education Commission (no. 12ZZ063), Natural Science Foundation of Anhui Province (no. 1308085QA14), and Key Natural Science Foundation of Anhui Education Commission (no. KJ2013A133).