Approximate Controllability for Impulsive Riemann-Liouville Fractional Differential Inclusions

Abstract

We study the control systems governed by impulsive Riemann-Liouville fractional differential inclusions and their approximate controllability in Banach space. Firstly, we introduce the 𝑃𝐶_1−𝛼-mild solutions for the impulsive Riemann-Liouville fractional differential inclusions in Banach spaces. Secondly, by using the fractional power of operators and a fixed point theorem for multivalued maps, we establish sufficient conditions for the approximate controllability for a class of Riemann-Liouville fractional impulsive differential inclusions, which is a generalization and continuation of the recent results on this issue. At the end, we give an example to illustrate the application of the abstract results.

1. Introduction

The concept of controllability plays an important part in the analysis and design of control systems. Since Kalman [1] first introduced its definition in 1963, controllability of the deterministic and stochastic dynamical control systems in finite-dimensional and infinite-dimensional spaces is well developed in different classes of approaches, and more details can be found in papers [2–4]. Some authors [5–7] have studied the exact controllability for nonlinear evolution systems by using the fixed point theorems. In [5–7], to prove the controllability results for fractional-order semilinear systems, the authors made an assumption that the semigroup associated with the linear part is compact. But if 𝐶₀-semigroup 𝑇(𝑡) is compact or the operator 𝐵 is compact, then the controllability operator is also compact and hence the inverse of it does not exist if the state space 𝑉 is infinite dimensional [8]. Thus, it is shown that the concept of exact controllability is difficult to be satisfied in infinite-dimensional space. Therefore, it is important to study the weaker concept of controllability, namely, approximate controllability for differential equations. In these years, several researchers [9–17] have studied it for control systems.

In [13], Sakthivel et al. studied on the approximate controllability of semilinear fractional differential systems: ^𝑐𝐷^𝛼𝑥(𝑡)(𝑡)(𝑡)((𝑡[],𝑥(0)=𝐴𝑥+𝐵𝑢+𝑓𝑡,𝑥)),𝑡∈𝐽=0,𝑇=𝑥₀, (1) where ^𝑐𝐷^𝛼_𝑡 is Caputo’s fractional derivative of 0<𝛼<1 and 𝐴 is the infinitesimal generator of a 𝐶₀-semigroup 𝑇(𝑡) of bounded operators on the Hilbert space 𝑋; the control function 𝑢(⋅) is given in 𝐿²(𝐽,𝑈); 𝑈 is a Hilbert space; 𝐵 is a bounded linear operator from 𝑈 to 𝑋; 𝑓∶𝐽×𝑋→𝑋 is a given function satisfying some assumptions and 𝑥₀ is an element of the Hilbert space 𝑋.

In [16], Sukavanam and Kumar researched approximate controllability of fractional-order semilinear delay systems: 𝑑^𝛼𝑥(𝑡)𝑡^𝛼=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝑓𝑡,𝑥_𝑡(𝑡)[],𝑥,𝑢,𝑡∈0,𝜏₀(𝜃)=𝜙(𝜃),𝜃∈[−ℎ,0], (2) where 1/2<𝛼<1; 𝐴∶𝐷(𝐴)⊆𝑉→𝑉 is a closed linear operator with dense domain 𝐷(𝐴) generating a 𝐶₀-semigroup 𝑆(𝑡); the state 𝑥(⋅) takes values in the Banach space 𝑉; the control function 𝑢(⋅) takes values in 𝑉; 𝐵 is a bounded linear operator from 𝐿²([0,𝜏];𝑉) to 𝐿²([0,𝜏];𝑉); the operator 𝑓∶[0,𝜏]×𝐶([−ℎ,0];𝑉)×𝑉→𝑉 is nonlinear. If 𝑥∶[−ℎ,𝜏]→𝑉 is a continuous function, then 𝑥_𝑡∶[−ℎ,0]→𝑉 is defined as 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃) for 𝜃∈[−ℎ,0] and 𝜙∈𝐶([−ℎ,0];𝑉).

In [18], Rykaczewski studied the approximate controllability of an inclusion of the form 𝑥̇(𝑡)(𝑡)((𝑡(𝑡)[],𝑥(0)∈𝐴𝑥+𝐹𝑡,𝑥))+𝐵𝑢,𝑡∈𝐽=0,𝑏=𝑥₀, (3) where 𝐴 is a linear operator which generates a compact semigroup, 𝐹 is u.h.c. multivalued perturbation with weakly compact values, and the state 𝑥(⋅) takes values in the Hilbert space 𝐻. 𝑈 is a Hilbert space of all admissible controls. 𝐵∶𝑈→𝐻 is a continuous linear operator.

Fractional differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, and so forth; see [19–27] for example. As a consequence there was an intensive development of the theory of differential equations of fractional order. One can see the monographs of Kilbas et al. [28] and Podlubny [29] and the references therein. The definitions of Riemann-Liouville fractional derivatives or integrals with initial conditions are strong tools to resolve some fractional differential problems in the real world. Heymans and Podlubny [30] have verified that it was possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals, and such initial conditions are more appropriate than physically interpretable initial conditions. Furthermore, they have investigated that the impulse response with Riemann-Liouville fractional derivatives was seldom used in the fields of physics, such as viscoelasticity. In recent years, many authors [18, 27, 31] were devoted to mild solutions to fractional evolution equations with Caputo fractional derivative, and there have been a lot of interesting works. As for the study of the fractional differential systems with Caputo fractional derivative, we can refer to [27, 31, 32] for the existence results. Its approximate controllability was considered in [9, 13–16]. The approximate controllability of Caputo fractional inclusion systems has been investigated by [10]. We know that differential inclusions are strong tools to solve some problems in various fields of engineering, physics, and optimal control; see [10, 32–35]. However, the approximate controllability for the impulsive fractional differential evolution inclusion with Riemann-Liouville fractional derivatives is still an untreated topic in the literature.

Motivated by the above work, in this paper, we consider the following system: 𝐷^𝛼_𝑡𝑥(𝑡)∈𝐴𝑥(𝑡)+𝐹(𝑡,𝑥(𝑡))+𝐵𝑢(𝑡),𝑡∈𝐽^′=(0,𝑏]⧵𝑡₁,𝑡₂,…,𝑡_𝑚,Δ𝐼₀^1−𝛼+𝑥(𝑡)_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡⁻_𝑘𝐼,𝑘=1,2,…,𝑚,_𝑡1−𝛼𝑥(𝑡)_𝑡=0=𝑥₀∈𝑋, (4) where 1/2<𝛼≤1 and 𝐷^𝛼_𝑡 denotes the Riemann-Liouville fractional derivative of order 𝛼 with the lower limit zero. 𝐹∶𝐽×𝑋→𝒫(𝑋)∶=2^𝑋⧵{∅} is a nonempty, bounded, closed, and convex multivalued map. 𝐴∶𝐷(𝐴)⊆𝑋→𝑋 is the infinitesimal generator of a 𝐶₀-semigroup 𝑇(𝑡) (𝑡≥0) on a Banach space 𝑋. 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑏, 𝐼_𝑘∶𝑋↦𝑋, Δ𝐼₀^1−𝛼+𝑥(𝑡_𝑘)=𝐼₀^1−𝛼+𝑥(𝑡⁺_𝑘)−𝐼₀^1−𝛼+𝑥(𝑡⁻_𝑘), 𝐼₀^1−𝛼+𝑥(𝑡⁺_𝑘) and 𝐼₀^1−𝛼+𝑥(𝑡⁻_𝑘) denote the right and the left limits of 𝐼₀^1−𝛼+𝑥(𝑡) at 𝑡=𝑡_𝑘, 𝑘=1,2,…,𝑚. The control function 𝑢(𝑡) takes value in 𝑉=𝐿^𝑝([0,𝑏];𝑈), 𝑝>1/𝛼, and 𝑈 is a Banach space; 𝐵 is a linear operator from 𝑉 to 𝐿^𝑝([0,𝑏];𝑋).

The purpose of this paper is to provide some suitable sufficient conditions for the existence of mild solutions and approximate controllability results for the impulsive fractional abstract Cauchy problems with Riemann-Liouville fractional derivatives. The main tools used in our study are fixed point theorem, semigroup theory for multivalued maps, and the theory from fractional differential equations. The rest of this paper is organized as follows. In Section 2, we present some preliminaries to prove our main results. In Section 3, by applying some standard fixed point principles, we prove the existence of the mild solutions for semilinear fractional differential equations, and the approximate controllability of the system (4) is proved. In Section 4, we give an example to illustrate our main results.

2. Preliminaries

In this section, we introduce some basic definitions and preliminaries which are used throughout this paper. The norm of a Banach space 𝑋 will be denoted by ‖⋅‖_𝑋. 𝐿_𝑏(𝑋,𝑌) denotes the space of bounded linear operators from 𝑋 to 𝑌. For the uniformly bounded 𝐶₀-semigroup 𝑇(𝑡) (𝑡≥0), we set 𝑀∶=sup_{𝑡∈[0,∞)}‖𝑇(𝑡)‖_{𝐿𝑏(𝑋)}<∞. Let 𝐶(𝐽,𝑋) denote the Banach space of all 𝑋 value continuous functions from 𝐽=[0,𝑏] to 𝑋 with the norm ‖𝑥‖_𝐶=sup_𝑡∈𝐽‖𝑥(𝑡)‖_𝑋. Let 𝐶_1−𝛼(𝐽,𝑋)={𝑥∶𝑡^1−𝛼𝑥(𝑡)∈𝐶(𝐽,𝑋)} with the norm ‖𝑥‖_𝐶1−𝛼=sup𝑡^1−𝛼‖𝑥(𝑡)‖_𝑋∶𝑡∈𝐽. (5) Obviously, the space 𝐶_1−𝛼(𝐽,𝑋) is a Banach space.

To define the mild solutions of (4), we also consider the Banach space 𝑃𝐶_1−𝛼(𝐽,𝑋)={𝑥∶(𝑡−𝑡_𝑘)^1−𝛼𝑥(𝑡)∈𝐶((𝑡_𝑘,𝑡_𝑘+1],𝑋), lim_𝑡→𝑡⁺_𝑘(𝑡−𝑡_𝑘)^1−𝛼𝑥(𝑡) exist, and 𝑥(𝑡⁻_𝑘) with 𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘), 𝑘=0,1,2,…,𝑚} with the norm ‖𝑥‖_{𝑃𝐶1−𝛼}=maxsup𝑡−𝑡_𝑘^1−𝛼‖𝑥(𝑡)‖_𝑋∶𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,…,𝑚. (6) It is easily known that the space 𝑃𝐶_1−𝛼(𝐽,𝑋) is a Banach space.

Let (𝑋,𝑑) be a metric space. We use the notations 𝑃_𝑐𝑙(𝑋)={(𝑋)},𝑃𝑌∈𝒫∶𝑌isclosed_𝑏(𝑋)={(𝑋)},𝑃𝑌∈𝒫∶𝑌isbounded_𝑐𝑣(𝑋)={(𝑋)},𝑃𝑌∈𝒫∶𝑌isconvex_𝑐𝑝(𝑋)=𝑌∈𝒫(𝑋)∶𝑌iscompact. (7)

Firstly, let us recall the following basic definitions from fractional calculus. For more details, one can see [28, 29].

Definition 1. The integral 𝐼^𝛼_𝑡𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠,𝛼>0, (8) is called Riemann-Liouville fractional integral of order 𝛼, where Γ is the gamma function.

Definition 2. For a function 𝑓(𝑡) given in the interval [0,∞), the expression 𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑓(𝑠)𝑑𝑡, (9) where 𝑛=[𝛼]+1 and [𝛼] denotes the integer part of number 𝛼, is called the Riemann-Liouville fractional derivative of order 𝛼.

Lemma 3 (see [28]). Let 𝛼>0, 𝑚=[𝛼]+1, and let 𝑥_𝑚−𝛼(𝑡)=𝐼_𝑡^𝑚−𝛼𝑥(𝑡) be the fractional integral of order 𝑚−𝛼. If 𝑥(𝑡)∈𝐿¹(𝐽,𝑋) and 𝑥_𝑚−𝛼(𝑡)∈𝐴𝐶^𝑚(𝐽,𝑋), then one has the following equality: 𝐼^𝛼_𝑡𝐷^𝛼_𝑡𝑥(𝑡)=𝑥(𝑡)−^𝑚_𝑘=1𝑥^{(𝑚−𝑘)}_𝑚−𝛼(0)Γ()𝑡𝛼−𝑘+1^𝛼−𝑘. (10)

In order to study the 𝑃𝐶-mild solutions of (4) in Banach space 𝑃𝐶_1−𝛼(𝐽,𝑋), we give the following results which will be used throughout this paper.

Lemma 4. Let 0<𝛼≤1, and let 𝑥_1−𝛼(𝑡)=𝐼_𝑡^1−𝛼𝑥(𝑡) be the fractional integral of order 1−𝛼. If 𝑥(𝑡)∈𝑃𝐶_1−𝛼(𝐽,𝑋) and 𝑥_1−𝛼(𝑡)∈𝑃𝐶(𝐽,𝑋), then one has the following equality: 𝐼^𝛼_𝑡𝐷^𝛼_𝑡𝑥(𝑡)=𝑥(𝑡)−𝑥_1−𝛼(𝑡)_𝑡=0𝑡^𝛼−1Γ(𝛼),𝑡∈0,𝑡₁,𝑥(𝑡)−∑^𝑚_𝑘=1Δ𝑥_1−𝛼𝑡_𝑖Γ(𝛼)𝑡−𝑡_𝑖^𝛼−1−𝑥_1−𝛼(𝑡)_𝑡=0𝑡^𝛼−1Γ(𝛼),𝑡∈𝑡_𝑘,𝑡_𝑘+1, (11) where Δ𝑥_1−𝛼(𝑡_𝑘)=𝑥_1−𝛼(𝑡⁺_𝑘)−𝑥_1−𝛼(𝑡⁻_𝑘), 𝑘=1,2,…,𝑚.

Proof. If 𝑡∈[0,𝑡₁], then, by Lemma 3, we easily get 𝐼^𝛼₀⁺𝐷^𝛼₀⁺𝑥(𝑡)=𝑥(𝑡)−𝑥_1−𝛼(𝑡)_𝑡=0𝑡^𝛼−1Γ(𝛼). (12) If 𝑡∈(𝑡₁,𝑡₂], since 𝐼^𝛼_𝑡𝐷^𝛼_𝑡𝑥(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐿𝐷^𝛼_𝑡𝑥(𝑠)=𝑑𝑑𝑠1𝑑𝑡Γ(𝛼+1)^𝑡₀(𝑡−𝑠)^𝛼𝐿𝐷^𝛼_𝑡𝑥(𝑠)𝑑𝑠, (13) then, by (13), we have 1Γ(𝛼+1)^𝑡₀(𝑡−𝑠)^𝛼𝐿𝐷^𝛼_𝑡𝑥(𝑠)=1𝑑𝑠Γ(𝛼+1)^𝑡₀(𝑡−𝑠)^𝛼𝑑𝐼𝑑𝑠^01−𝛼+𝑥(𝑠=1)𝑑𝑠Γ()Γ()×𝛼+11−𝛼_𝑡0(𝑡−𝑠)^𝛼𝑑𝑑𝑠^𝑠₀(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)=1𝑑𝜏𝑑𝑠Γ()Γ()×𝛼+11−𝛼_𝑡10(𝑡−𝑠)^𝛼𝑑𝑑𝑠^𝑠₀(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)+1𝑑𝜏𝑑𝑠Γ()Γ()×𝛼+11−𝛼_𝑡𝑡1(𝑡−𝑠)^𝛼𝑑𝑑𝑠^𝑡1₀(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)+1𝑑𝜏𝑑𝑠Γ()Γ()×𝛼+11−𝛼_𝑡𝑡1(𝑡−𝑠)^𝛼𝑑𝑑𝑠^𝑠_𝑡1(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)=1𝑑𝜏𝑑𝑠Γ(𝛼)Γ()×1−𝛼_𝑡10(𝑡−𝑠)^𝛼−1𝑠₀(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)+1𝑑𝜏𝑑𝑠Γ()Γ()×)𝛼+11−𝛼(𝑡−𝑠^𝛼𝑠₀(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)𝑑𝜏^𝑠=𝑡1_𝑠=0+1Γ(𝛼)Γ()×1−𝛼_𝑡𝑡1(𝑡−𝑠)^{𝛼−1𝑡1}₀(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)+1𝑑𝜏𝑑𝑠Γ()Γ()×)𝛼+11−𝛼(𝑡−𝑠^𝛼𝑡1₀(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)𝑑𝜏^𝑠=𝑡_𝑠=𝑡1+1Γ(𝛼)Γ()×1−𝛼_𝑡𝑡1(𝑡−𝑠)^𝛼−1𝑠_𝑡1(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)+1𝑑𝜏𝑑𝑠Γ()Γ()×)𝛼+11−𝛼(𝑡−𝑠^𝛼𝑠_𝑡1(𝑠−𝜏)^1−𝛼−1𝑥(𝜏)𝑑𝜏^𝑠=𝑡_𝑠=𝑡1=1Γ(𝛼)Γ()×1−𝛼_𝑡10𝑥(𝜏)𝑑𝜏^𝑡1_𝜏(𝑡−𝑠)^𝛼−1(𝑠−𝜏)^1−𝛼−1+1𝑑𝑠Γ(𝛼)Γ()×1−𝛼_𝑡10𝑥(𝜏)𝑑𝜏^𝑡_𝑡1(𝑡−𝑠)^𝛼−1(𝑠−𝜏)^1−𝛼−1+1𝑑𝑠Γ(𝛼)Γ()×1−𝛼_𝑡𝑡1𝑥(𝜏)𝑑𝜏^𝑡_𝜏(𝑡−𝑠)^𝛼−1(𝑠−𝜏)^1−𝛼−1−𝑥𝑑𝑠_1−𝛼(0)Γ()𝑡𝛼+1^𝛼−Δ𝑥_1−𝛼𝑡₁Γ(𝛼+1)𝑡−𝑡₁^𝛼=^𝑡₀𝑥(𝜏)𝑥𝑑𝜏−^1−𝛼(0)Γ()𝑡𝛼+1^𝛼−Δ𝑥_1−𝛼𝑡₁Γ(𝛼+1)𝑡−𝑡₁^𝛼. (14) Thus, by (13) and (14), we get 𝐼^𝛼_𝑡𝐷^𝛼_𝑡𝑥(𝑡)(𝑡)−𝑥=𝑥^1−𝛼(0)Γ(𝛼)𝑡^𝛼−1−Δ𝑥_1−𝛼𝑡₁Γ(𝛼)𝑡−𝑡₁^𝛼−1. (15) Similarly, if 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘=2,…,𝑚, we can get 𝐼^𝛼_𝑡𝐷^𝛼_𝑡𝑥(𝑡)=𝑥(𝑡)−^𝑚_𝑘=1Δ𝑥_1−𝛼𝑡_𝑘Γ(𝛼)𝑡−𝑡_𝑘^𝛼−1−𝑥_1−𝛼(𝑡)_𝑡=0𝑡^𝛼−1Γ(𝛼). (16) The proof is completed.

The Laplace transform formula for the Riemann-Liouville fractional integral is defined by 𝐿𝐼^𝛼_𝑡𝑥(𝑡)=1;𝜆𝜆^𝛼𝑥(𝜆), (17) where 𝑥(𝜆) is the Laplace of 𝑥 defined by 𝑥(𝜆)=^∞₀𝑒^−𝜆𝑡𝑥(𝑡)|𝑥(𝑡𝑑𝑡,𝑅𝑒𝜆>𝜔,)|≤𝑐𝑒^𝜔𝑡,𝑐isaconstant. (18)

Lemma 5. Let 𝛼∈(0,1] and ℎ∈𝐿^𝑝(𝐽,𝑋), 𝑝>1/𝛼; if 𝑥(𝑡)∈𝑃𝐶_1−𝛼(𝐽,𝑋), 𝑥_1−𝛼(𝑡)∈𝑃𝐶(𝐽,𝑋), and 𝑥 is a solution of the following problem: 𝐷^𝛼_𝑡𝑥(𝑡)(𝑡)(𝑡),(]=𝐴𝑥+ℎ𝑡∈0,𝑏,𝑡≠𝑡_𝑘,𝑘=1,2,…,𝑚,Δ𝐼_𝑡^1−𝛼𝑥(𝑡)_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡⁻_𝑘𝐼,𝑘=1,2,…,𝑚,_𝑡1−𝛼𝑥(𝑡)_𝑡=0=𝑥₀∈𝑋, (19) then, 𝑥 satisfies the following equation: 𝑥(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+∑^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1Δ𝐼₀^1−𝛼+𝑥𝑡_𝑘+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()ℎ(𝑠)𝑡𝑡−𝑠𝑑𝑠,𝑡∈_𝑘,𝑡_𝑘+1, (20) where 𝑘=1,2,…,𝑚, 𝑇_𝛼(𝑡)=𝛼^∞₀𝜃𝜉_𝛼(𝜃)𝑇𝑡^𝛼𝜃𝜉𝑑𝜃,_𝛼(𝜃)=1𝛼𝜃^{−1−(1/𝛼)}𝜛_𝛼𝜃^−1/𝛼,𝜛_𝛼(𝜃)=1𝜋^∞_𝑛=1(−1)^𝑛−1𝜃^{−𝑛𝛼−1}Γ(𝑛𝛼+1)(),(),𝑛!sin𝑛𝜋𝛼𝜃∈0,∞ (21) where 𝜉_𝛼 is a probability density function defined on (0,∞); that is, 𝜉_𝛼(𝜃)≥0,𝜃∈(0,∞),^∞₀𝜉_𝛼(𝜃)𝑑𝜃=1. (22)

Proof. We observe that 𝑥(⋅) can be decomposed to 𝑝(⋅)+𝑞(⋅), where 𝑝 is the continuous mild solution for 𝐷^𝛼_𝑡𝑝(𝑡)(𝑡)(𝑡)(],𝐼=𝐴𝑝+ℎ,𝑡∈0,𝑏_𝑡1−𝛼𝑝(𝑡)_𝑡=0=𝑥₀∈𝑋, (23) and 𝑞 is the 𝑃𝐶_1−𝛼-mild solution for 𝐷^𝛼_𝑡𝑞(𝑡)(𝑡),(]=𝐴𝑞𝑡∈0,𝑏,𝑡≠𝑡_𝑘,𝑘=1,2,…,𝑚,Δ𝐼_𝑡^1−𝛼𝑞(𝑡)_{𝑡=𝑡𝑘}=𝑦_𝑘𝐼,𝑘=1,2,…,𝑚,_𝑡1−𝛼𝑞(𝑡)_𝑡=0=0∈𝑋. (24) Indeed, by adding together (23) and (24), it follows by (19). Since 𝑝 is continuous, then 𝑝(𝑡⁺_𝑘)=𝑝(𝑡⁻_𝑘), 𝑘=1,2,…,𝑚. On the other hand, any solution of (19) can be decomposed to (23) and (24). So we show the results by the following.
At first, we calculate the mild solution of (23).
Apply Riemann-Liouville fractional integral operator on both sides of (23); then, by Lemma 3, we get 𝑝(𝑡)=𝐼_𝑡^1−𝛼𝑝(𝑡)_𝑡=0Γ(𝛼)𝑡^𝛼−1+𝐼^𝛼_𝑡𝐴𝑝(𝑡)+𝐼^𝛼_𝑡ℎ(𝑡)=𝑡^𝛼−1Γ(𝛼)𝑥₀+𝐼^𝛼_𝑡𝐴𝑝(𝑡)+𝐼^𝛼_𝑡ℎ(𝑡). (25) That is, 𝑝(𝑡)=𝑡^𝛼−1Γ(𝛼)𝑥₀+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑝(𝑠)+ℎ(𝑠)𝑑𝑠. (26) Let 𝜆>0; taking the Laplace transformations 𝑝(𝜆)=^∞₀𝑒^−𝜆𝑡𝑝(𝑡)𝑑𝑡,ℎ(𝜆)=^∞₀𝑒^−𝜆𝑡ℎ(𝑡)𝑑𝑡 (27) to (26), we obtain 𝑝(𝜆)=1𝜆^𝛼𝑥₀+1𝜆^𝛼𝐴𝑝(𝜆)+1𝜆^𝛼ℎ(𝜆)=𝜆^𝛼𝐼−𝐴⁻¹𝑥₀+𝜆^𝛼𝐼−𝐴⁻¹ℎ(𝜆)=^∞₀𝑒^{−𝜆𝛼𝑡}𝑇(𝑡)𝑥₀𝑑𝑡+^∞₀𝑒^{−𝜆𝛼𝑡}𝑇(𝑡)ℎ(𝜆)𝑑𝑡. (28) Consider the one-sided stable probability density 𝜛_𝛼(𝜃)=1𝜋^∞_𝑛=1(−1)^𝑛−1𝜃^{−𝑛𝛼−1}Γ(𝑛𝛼+1)(),(),𝑛!sin𝑛𝜋𝛼𝜃∈0,∞ (29) whose Laplace transformation is given by ^∞₀𝑒^−𝜆𝜃𝜛_𝛼(𝜃)𝑑𝜃=𝑒^−𝜆𝛼,𝛼∈(0,1). (30) Hence, it follows from (28) and (30) that ^∞₀𝑒^{−𝜆𝛼𝑡}𝑇(𝑡)𝑥₀=𝑑𝑡_∞0𝑒^{−(𝜆𝑠)𝛼}𝑇𝑠^𝛼𝑥₀𝑑𝑠^𝛼𝑡=𝑠^𝛼=𝛼^∞₀𝑒^{−(𝜆𝑠)𝛼}𝑇𝑠^𝛼𝑠^𝛼−1𝑥₀𝑑𝑠=𝛼^∞₀𝜛_𝛼(𝜃)𝑒^{−𝜆𝑠𝜃}𝑇𝑠^𝛼𝑠^𝛼−1𝑥₀𝑑𝜃𝑑𝑠=𝛼^∞₀𝜛_𝛼(𝜃)𝑒^−𝜆𝑢𝑇𝑢^𝛼𝜃^𝛼𝑠^𝛼−1𝜃^𝛼𝑥₀()=𝑑𝑢𝑑𝜃𝑢=𝜃𝑠_∞0𝑒^−𝜆𝑢𝛼^∞₀𝜛_𝛼(𝜃)𝑇𝑢^𝛼𝜃^𝛼𝑠^𝛼−1𝜃^𝛼𝑥₀𝑑𝜃𝑑𝑢,_∞0𝑒^{−𝜆𝛼𝑡}𝑇(𝑡)ℎ(𝜆)=𝑑𝑡_∞0𝑒^{−𝜆𝛼𝑡}𝑇(𝑡)^∞₀𝑒^−𝜆𝑠ℎ(𝑠)=𝑑𝑠𝑑𝑡_∞0𝑒^{−𝜆𝛼𝑡}𝑇(𝑡)𝑒^−𝜆𝑠ℎ(𝑠)𝑑𝑠𝑑𝑡=^∞₀𝑒^{−(𝜆𝜇)𝛼}𝑇𝜇^𝛼𝑞𝜇^𝛼−1𝑒^−𝜆𝑠ℎ(𝑠)𝑑𝑠𝑑𝜇𝑡=𝜇^𝛼=^∞₀𝛼𝜛_𝛼(𝜃)𝑒^{−𝜆𝜇𝜃}𝑇𝜇^𝛼𝜇^𝛼−1𝑒^−𝜆𝑠ℎ(𝑠)=𝑑𝜃𝑑𝑠𝑑𝜇_∞0𝛼𝜛_𝛼(𝜃)𝑒^−𝜆𝜈𝑇𝜈^𝛼𝜃^𝛼×𝜈^𝛼−1𝜃^𝛼𝑒^−𝜆𝑠ℎ(𝑠)=𝑑𝜈𝑑𝜃𝑑𝑠𝜈=𝜇𝜃_∞0𝛼𝜛_𝛼(𝜃)𝑒^{−𝜆(𝜈+𝑠)}𝑇𝜈^𝛼𝜃^𝛼𝜈^𝛼−1𝜃^𝛼ℎ(𝑠)=𝑑𝜈𝑑𝜃𝑑𝑠_∞0∞𝑠𝛼𝜛_𝛼(𝜃)𝑒^−𝜆𝜏𝑇(𝜏−𝑠)^𝛼𝜃^𝛼×(𝜏−𝑠)^𝛼−1𝜃^𝛼ℎ(𝑠)()=𝑑𝜏𝑑𝜃𝑑𝑠𝜏=𝜈+𝑠_∞0𝑒^−𝜆𝜏𝛼^𝜏₀^∞₀𝜛_𝛼(𝜃)𝑇(𝜏−𝑠)^𝛼𝜃^𝛼×(𝜏−𝑠)^𝛼−1𝜃^𝛼ℎ(𝑠)𝑑𝜃𝑑𝑠𝑑𝜏. (31)
According to the above work, we get 𝑝(𝜆)=^∞₀𝑒^−𝜆𝑡𝛼^∞₀𝜛_𝛼(𝜃)𝑇𝑡^𝛼𝜃^𝛼𝑡^𝛼−1𝜃^𝛼𝑥₀𝑑𝜃+𝛼^𝑡₀^∞₀𝜛_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃^𝛼×(𝑡−𝑠)^𝛼−1𝜃^𝛼ℎ(𝑠)𝑑𝜃𝑑𝑠𝑑𝑡. (32)
Now, we can invert the Laplace transform to (20) and obtain 𝑝(𝑡)=𝛼^∞₀𝜛_𝛼(𝜃)𝑇𝑡^𝛼𝜃^𝛼𝑡^𝛼−1𝜃^𝛼𝑥₀𝑑𝜃+𝛼^𝑡₀^∞₀𝜛_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃^𝛼(𝑡−𝑠)^𝛼−1𝜃^𝛼ℎ(𝑠)𝑑𝜃𝑑𝑡=𝛼^∞₀1𝛼𝜃^{−1−(1/𝛼)}𝜛_𝛼𝜃^−1/𝛼𝜃𝑇𝑡^𝛼𝜃𝑡^𝛼−1𝑥₀𝑑𝜃+𝛼^𝑡₀^∞₀(𝑡−𝑠)^𝛼−11𝛼𝜃^{−1−(1/𝛼)}𝜛_𝛼𝜃^−1/𝛼×()𝜃𝑇𝑡−𝑠^𝛼𝜃ℎ(𝑠)𝑑𝜃𝑑𝑡. (33)
Let 𝜉_𝛼(𝜃)=1𝛼𝜃^{−1−(1/𝛼)}𝜛_𝛼𝜃^−1/𝛼,𝑇_𝛼(𝑡)=𝛼^∞₀𝜃𝜉_𝛼(𝜃)𝑇𝑡^𝛼𝜃𝑑𝜃. (34)
Then, we get 𝑝(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)ℎ(𝑠)𝑑𝑠. (35)
Now we calculate the 𝑃𝐶_1−𝛼-mild solution of (24).
Applying Riemann-Liouville fractional integral operator on both sides of (24), then by Lemma 4, we get 𝑞(𝑡)=1Γ(𝛼)∫^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑞(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,_𝑚𝑘∑₌₁𝐼_𝑘𝑥𝑡⁻_𝑘Γ(𝛼)𝑡−𝑡_𝑖^𝛼−1+1Γ(𝛼)∫^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑞(𝑠)𝑑𝑠,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (36) The above equation (36) can be rewritten as 𝑞(𝑡)=^𝑚_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘Γ(𝛼)𝑡−𝑡_𝑘^𝛼−1𝜒_𝑘(𝑡)+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑞(𝑠)𝑑𝑠, (37) where 𝜒_𝑘(𝑡)=0,𝑡≤𝑡_𝑘,1,𝑡>𝑡_𝑘. (38)
Let 𝜆>0; taking the Laplace transformation to (37), we obtain 𝑞(𝜆)=^𝑚_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘𝑒^{−𝜆𝑡𝑘}𝜆^𝛼+1𝜆^𝛼𝐴𝑞(𝜆). (39) That is, 𝑞(𝜆)=^𝑚_𝑘=1𝜆^𝛼𝐼−𝐴⁻¹𝐼_𝑘𝑥𝑡⁻_𝑘𝑒^{−𝜆𝑡𝑘}. (40) Notice that the Laplace transform of 𝑡^𝛼−1𝑇_𝛼(𝑡)𝑦_𝑘 is (𝜆^𝛼𝐼−𝐴)⁻¹𝑦_𝑘. Thus one can calculate the mild solution of (24) as 𝑞(𝑡)=^𝑚_𝑘=1𝜒_𝑘(𝑡)𝑡−𝑡_𝑖^𝛼−1𝑇_𝛼𝑡−𝑡_𝑖𝐼_𝑘𝑥𝑡⁻_𝑘. (41)
By the above work, the 𝑃𝐶_1−𝛼-mild solution of (19) is given by 𝑥(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑚_𝑘=1𝜒_𝑘(𝑡)𝑡−𝑡_𝑘^𝛼−1𝑇_𝛼𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡⁻_𝑘+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)ℎ(𝑠)𝑑𝑠. (42) That is, 𝑥(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+∑^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()ℎ(𝑠)𝑡𝑡−𝑠𝑑𝑠,𝑡∈_𝑘,𝑡_𝑘+1, (43) where 𝑘=1,2,…,𝑚, 𝑇_𝛼(𝑡)=𝛼^∞₀𝜃𝜉_𝛼(𝜃)𝑇𝑡^𝛼𝜃𝜉𝑑𝜃,_𝛼(𝜃)=1𝛼𝜃^{−1−(1/𝛼)}𝜛_𝛼𝜃^−1/𝛼,𝜛_𝛼(𝜃)=1𝜋^∞_𝑛=1(−1)^𝑛−1𝜃^{−𝑛𝛼−1}Γ(𝑛𝛼+1)(),(),𝑛!sin𝑛𝜋𝛼𝜃∈0,∞ (44) where 𝜉_𝛼 is a probability density function defined on (0,∞); that is, 𝜉_𝛼(𝜃)≥0,𝜃∈(0,∞),^∞₀𝜉_𝛼(𝜃)𝑑𝜃=1. (45) This completes the proof of the lemma.

According to Lemma 5, we give the following definition.

Definition 6. A function 𝑥∈𝑃𝐶_1−𝛼(𝐽,𝑋) is called a mild solution of (4) if 𝐼_𝑡^1−𝛼𝑥(𝑡)|_𝑡=0=𝑥₀ and there exits 𝑓∈𝐿¹(𝐽<𝑋) such that 𝑓(𝑡)∈𝐹(𝑡,𝑥(𝑡)) a.e. on 𝑡∈𝐽 and 𝑥(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢(𝑠)+𝑓(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈0,𝑡₁,𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+∑^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼((𝑠)((𝑠𝑡𝑡−𝑠)𝐵𝑢+𝑓𝑠,𝑥))𝑑𝑠,𝑡∈_𝑘,𝑡_𝑘+1, (46) where 𝑘=1,2,…,𝑚, 𝑇_𝛼(𝑡)=𝛼^∞₀𝜃𝜉_𝛼(𝜃)𝑇𝑡^𝛼𝜃𝜉𝑑𝜃,_𝛼(𝜃)=1𝛼𝜃^{−1−(1/𝛼)}𝜛_𝛼𝜃^−1/𝛼,𝜛_𝛼(𝜃)=1𝜋^∞_𝑛=1(−1)^𝑛−1𝜃^{−𝑛𝛼−1}Γ(𝑛𝛼+1)(),(),𝑛!sin𝑛𝜋𝛼𝜃∈0,∞ (47) where 𝜉_𝛼 is a probability density function defined on (0,∞); that is, 𝜉_𝛼(𝜃)≥0,𝜃∈(0,∞),^∞₀𝜉_𝛼(𝜃)𝑑𝜃=1. (48)
Due to the work of the paper [31], we have the following result.

Lemma 7. The operator 𝑇_𝛼(𝑡) has the following properties.(i)For any fixed 𝑡≥0, 𝑇_𝛼(𝑡) is linear and bounded operator; that is, for any 𝑥∈𝑋, 𝑇_𝛼(𝑡)𝑥≤𝑀Γ(𝛼)‖𝑥‖. (49) (ii)𝑇_𝛼(𝑡) (𝑡≥0) is strongly continuous.

Now, we also introduce some basic definitions on multivalued maps. For more details, see [36–38].

A multivalued map 𝐺∶𝑋→𝒫(𝑋) is convex (closed) valued if 𝐺(𝑥) is convex (closed) for all 𝑥∈𝑋. 𝐺 is bounded on bounded sets if 𝐺(𝐵)=⋃_𝑥∈𝐵. 𝐺(𝑥) is bounded on 𝑋 for any bounded set 𝐵 of 𝑋; that is, sup_𝑥∈𝐵{sup{‖𝑦‖∶𝑦∈𝐺(𝑥)}}<∞.

𝐺 is called upper semicontinuous (u.s.c.) on 𝑋 if, for each 𝑥₀∈𝑋, the set 𝐺(𝑥₀) is a nonempty closed subset of 𝑋, and if, for each open set 𝑈 of 𝑋 containing 𝐺(𝑥₀), there exists an open neighborhood 𝑉 of 𝑥₀ such that 𝐺(𝑉)⊆𝑈.

𝐺 is said to be completely continuous if 𝐺(𝐵) is relatively compact for every 𝐵∈𝑃_𝑏(𝑋).

If the multivalued map 𝐺 is completely continuous with nonempty compact values, then 𝐺 is u.s.c. if and only if 𝐺 has a closed graph (i.e., 𝑥_𝑛→𝑥_∗, 𝑦_𝑛→𝑦_∗, 𝑦_𝑛∈𝐺(𝑥_𝑛) imply 𝑦_∗∈𝐺(𝑥_∗)).

We say that 𝐺 has a fixed point if there is a 𝑥∈𝑋 such that 𝑥∈𝐺(𝑥).

A multivalued map 𝐺∶𝐽→𝑃_𝑐𝑙(𝑋) is said to be measurable if for each 𝑥∈𝑋 the function 𝑌∶𝐽→𝑅⁺ defined by 𝑌(𝑡)=𝑑(𝑥,𝐺(𝑡))=inf{‖𝑥−𝑧‖∶𝑧∈𝐺(𝑡)} is measurable.

Definition 8. The system (4) is said to be exactly controllable on 𝐽, if, for all 𝑥₀,𝑥₁∈𝑋, there exists a control 𝑢∈𝐿^𝑝(𝐽,𝑈) (𝑝>1/𝛼) such that the mild solution of (4) satisfies 𝑥(0;𝑢)=𝑥₀ and 𝑥(𝑏;𝑢)=𝑥₁.

Definition 9. The system (4) is said to be approximately controllable on the interval 𝐽, if, for all 𝑥₀∈𝑋, one has ℛ(𝑏,𝑥₀)=𝑋, where ℛ(𝑏,𝑥₀)={𝑥(𝑏;𝑢)∶𝑢∈𝐿^𝑝(𝐽,𝑈)(𝑝>1/𝛼),𝑥(0;𝑢)=𝑥₀} is the reachable set of system (4) with the initial values 𝑥₀,𝑥₁ at the terminal time 𝑏.
It is convenient at this point to introduce two relevant operators: Γ^𝑏₀=^𝑏₀(𝑏−𝑠)^2𝛼−2𝑇_𝛼(𝑏−𝑠)𝐵𝐵^∗𝑇^∗_𝛼()1𝑏−𝑠𝑑𝑠,2𝑅<𝛼≤1,𝑎,Γ^𝑏₀=𝑎𝐼+Γ^𝑏₀⁻¹,𝑎>0, (50) where 𝐵^∗ denotes the adjoint of 𝐵 and 𝑇^∗_𝛼(𝑡) is the adjoint of 𝑇_𝛼(𝑡). It is straightforward that the operator Γ^𝑏₀ is a linear bounded operator.
We consider the following linear fractional differential system: ^𝑐𝐷^𝛼𝑥(𝑡)(𝑡)(𝑡),1=𝐴𝑥+𝐵𝑢2(],𝐼<𝛼≤1,𝑡∈𝐽=0,𝑏_𝑡1−𝛼𝑥(𝑡)_𝑡=0=𝑥₀∈𝑋. (51)

Lemma 10. The linear fractional differential system (51) is approximately controllable on 𝐽 if and only if 𝑎𝑅(𝑎,Γ^𝑏₀)→0 as 𝑎→0⁺ in the strong operator topology.

The proof of this lemma is a straightforward adaptation of the proof of [3].

Lemma 11. Let 𝐸 be a Banach space and let 𝒲⊂𝐿¹(𝐽,𝐸) be integrably bounded. If, for all 𝑡∈𝐽, there is a relatively weakly compact set 𝐶(𝑡)⊂𝐸 such that 𝑤(𝑡)∈𝐶(𝑡) for every 𝑤∈𝒲, then 𝒲 is relatively weakly compact in 𝐿¹(𝐽,𝐸).

Lemma 12 (Lasota and Opial [39]). Let 𝐽 be a compact real interval and let 𝑋 be a Banach space. The multivalued map 𝐹∶𝐽×𝑋→𝒫_{𝑏,𝑐𝑙,𝑐𝑣}(𝑋) is measurable to 𝑡 for each fixed 𝑥∈𝑋, u.s.c. to 𝑥 for each 𝑡∈𝐽, and for each 𝑥∈𝐶(𝐽,𝑋) the set 𝑆_𝐹,𝑥={𝑓∈𝐿¹(𝐽,𝑋)∶𝑓(𝑡)∈𝐹(𝑡,𝑥(𝑡)),𝑡∈𝐽} is nonempty. Let Γ be a linear continuous mapping from 𝐿¹(𝐽,𝑋) to 𝐶(𝐽,𝑋); then, the operator Γ∘𝑆_𝐹∶𝐶(𝐽,𝑋)⟶𝒫_{𝑏,𝑐𝑙,𝑐𝑣}(𝐶(𝐽,𝑋)),𝑥⟼Γ∘𝑆_𝐹(𝑥)=Γ𝑆_𝐹,𝑥 (52) is a closed graph operator in 𝐶(𝐽,𝑋)×𝐶(𝐽,𝑋).

Lemma 13 (see [37]). Let 𝐷 be a bounded, convex, and closed subset in the Banach space 𝐸 and let 𝐺∶𝐷→2^𝑋⧵{∅} be a u.s.c. condensing multivalued map. If, for every 𝑥∈𝐷, 𝐺(𝑥) is a closed and convex set in 𝐷, then 𝐺 has a fixed point.

3. Main Results

In this section, we present our main result on approximate controllability of system (4). To do this, we first prove the existence of solutions for fractional control system. Secondly, we show that, under certain assumptions, the approximate controllability of (4) is implied by the approximate controllability of the corresponding linear system.

For convenience, let us introduce some notations: 𝑀_𝑏=‖𝐵‖,𝛽=1−𝛾𝑏𝛼−𝛾^{(𝛼−𝛾)/(1−𝛾)1−𝛾},Λ=𝑀𝛽Γ(𝛼)+𝑀³𝑀²_𝐵𝑏^2𝛼−1𝛽𝑎(2𝛼−1)(Γ(𝛼))³. (53)

Before stating and proving our main results, we introduce the following assumptions. 𝐻(1): 𝑇_𝛼(𝑡) is compact, ‖𝑎𝑅(𝑎,Γ^𝑏₀)‖≤1. 𝐻(2): 𝐹 is a multivalued map satisfying 𝐹∶𝐽×𝑋→𝒫_{𝑐𝑝,𝑐𝑣}(𝑋) which is measurable to 𝑡 for each fixed 𝑥∈𝐷, u.s.c. to 𝑥 for each 𝑡∈𝐽, and for each 𝑥∈𝑃𝐶_1−𝛼(𝐽,𝑋) the set 𝑆_𝐹,𝑥=𝑓∈𝐿¹(𝐽,𝑋)∶𝑓(𝑡)∈𝐹(𝑡,𝑥(𝑡)) (54) is nonempty. 𝐻(3): There exist a function 𝑃(𝑡)∈𝐿^1/𝛾(𝐽,𝑅⁺), 𝛾∈(0,𝛼), and a nondecreasing continuous function 𝜓∶→𝑅⁺, such that ‖𝐹(𝑡,𝑥(𝑡))‖_𝑋=sup𝑓(𝑡)_𝑋(𝑡)((𝑡(𝑡)𝜓‖𝑥‖∶𝑓∈𝐹𝑡,𝑥))≤𝑃_𝐷, (55) for a.e. 𝑡∈𝐽, for all 𝑥∈𝐷, and for each 𝑟>0, there exists 0<𝜌<1, such that lim_𝑟→∞𝜓(𝑟)inf𝑟‖𝑃‖_𝐿²=𝜌<1. (56)

Theorem 14. If the conditions 𝐻(1)--𝐻(4) are held, then the system (4) has a mild solution.

Proof. We consider a set 𝐵_𝑟=𝑥∈𝑃𝐶_1−𝛼(𝐽,𝑋)∶‖𝑥‖≤𝑟,𝑟>0 (57) on the space 𝑃𝐶_1−𝛼(𝐽,𝑋). We easily know that 𝐵_𝑟 is a bounded, closed, and convex set in 𝑃𝐶_1−𝛼(𝐽,𝑋). For 𝑎>0, for all 𝑥(⋅)∈𝑃𝐶_1−𝛼(𝐽,𝑋), 𝑥₁∈𝑋, we take the control function as 𝑢(𝑡)=(𝑏−𝑡)^𝛼−1𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀𝑃(𝑥(⋅)), (58) where 𝑃(𝑥(⋅))=𝑥₁−𝑡^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀(𝑏−𝑠)^𝛼−1𝑇_𝛼(𝑏−𝑠)𝑓(𝑠)𝑑𝑠,𝑓∈𝑆_𝐹,𝑥. (59) By this control, we define the operator Φ_𝑎∶𝑃𝐶_1−𝛼(𝐽,𝑋)→𝒫(𝑃𝐶_1−𝛼(𝐽,𝑋)) as follows: Φ_𝑎(𝑥)=𝜏∈𝑃𝐶_1−𝛼()(𝑡)𝐽,𝑋∶𝜏=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()𝑓(𝑠)+𝑡−𝑠𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠,𝑓∈𝑆_𝐹,𝑥,𝑡∈0,𝑡₁,𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()𝑓(𝑠)+𝑡−𝑠𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1×𝑇_𝛼(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠,𝑓∈𝑆_𝐹,𝑥,𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=1,2,…,𝑚. (60) We will show that, for all 𝑎>0, the operator Φ_𝑎∶𝑃𝐶_1−𝛼(𝐽,𝑋)→𝒫(𝑃𝐶_1−𝛼(𝐽,𝑋)) has a fixed point. For the sake of convenience, we subdivide the proof into several steps.
Step  1. For each 𝑎>0, the operator Φ_𝑎(𝑥) is convex for each 𝑥∈𝑃𝐶_1−𝛼(𝐽,𝑋).
In fact, if 𝜏₁,𝜏₂∈Φ_𝑎(𝑥), then, for each 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘=1,2,…,𝑚, there exist 𝑓₁,𝑓₂∈𝑆_𝐹,𝑥 such that 𝜏_𝑖(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓_𝑖(𝑠)+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓_𝑖𝜇𝑑𝜇𝑑𝑠,𝑖=1,2. (61) Let 0≤𝜆≤1; then, for each 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘=1,2,…,𝑚, we have 𝜆𝜏₁(𝑡)+(1−𝜆)𝜏₂(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝜆𝑓₁(𝑠)+(1−𝜆)𝑓₂(𝑠+)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼×𝑏−𝜇𝜆𝑓₁𝜇+(1−𝜆)𝑓₂𝜇𝑑𝜇𝑑𝑠. (62) Since 𝑆_𝐹,𝑥 is convex (because 𝐹 has convex values), 𝜆𝑓₁+(1−𝜆)𝑓₂∈𝑆_𝐹,𝑥; thus, 𝜆𝜏₁(𝑡)+(1−𝜆)𝜏₂∈Φ_𝑎(𝑥).
Step  2. For each 𝑎>0, there is a positive constant 𝑟₀=𝑟(𝑎), such that Φ_𝑎(𝐵_𝑟0)⊂𝐵_𝑟0.
If this is not true, then there exists 𝑎>0 such that, for every 𝑟>0, there exists a 𝑥∈𝐵_𝑟 such that Φ_𝑎(𝑥)⊆𝐵̸_𝑟; that is, Φ_𝑎(𝑥)≡sup‖𝜏‖_{𝑃𝐶1−𝛼(𝐽,𝑋)}∶𝜏∈Φ_𝑎(𝑥𝜏(𝑡))>𝑟,=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()𝑓(𝑠)+𝑡−𝑠𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓𝜇𝑑𝜇𝑑𝑠, (63) for some 𝑆_𝐹,𝑥.
By using Holder’s inequality and 𝐻(3), we have ^𝑡₀(𝑡−𝑠)^𝛼−1‖𝑢(𝑠≤)‖𝑑𝑠𝑀𝑀_𝐵𝑏^2𝛼−1(𝛼)×𝑥𝑎Γ)(2𝛼−1₁+𝑀𝑥₀Γ(𝛼)+^𝑚_𝑘=1𝑀Γ(𝛼)𝑑_𝑘‖𝑥‖+𝑚𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘(0+(𝑟)𝛽)𝑀𝜓Γ(𝛼)‖𝑃‖_𝐿^1/𝛾. (64) Then, we obtain 𝑡−𝑡_𝑘^1−𝛼‖𝜏(𝑡≤)‖𝑡−𝑡_𝑘^1−𝛼𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+𝑡−𝑡_𝑘^1−𝛼×^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘+𝑡−𝑡_𝑘^1−𝛼×^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑓(𝑠+𝑡−𝑠))𝑑𝑠𝑡−𝑡_𝑘^1−𝛼×^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝐵𝑢(𝑠≤𝑡−𝑠)‖‖‖)‖𝑑𝑠𝑀𝑏^𝛼−1𝑏−𝑡_𝑘^1−𝛼Γ(𝛼)𝑥₀+𝑏−𝑡_𝑘^1−𝛼×^𝑚_𝑘=1𝑀Γ(𝛼)𝑑_𝑘‖𝑥‖+𝑚𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘(0+𝑀)𝑏−𝑡_𝑘^1−𝛼Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑃(𝑠)𝜓(𝑟)+𝑑𝑠𝑀𝑀_𝐵𝑏−𝑡_𝑘^1−𝛼Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑢(𝑠)≤𝑑𝑠𝑀𝑏^𝛼−1𝑏−𝑡_𝑘^1−𝛼Γ(𝛼)𝑥₀+𝑏−𝑡_𝑘^1−𝛼×^𝑚_𝑘=1𝑀Γ(𝛼)𝑑_𝑘‖𝑥‖+𝑚𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘(0+𝑏_2𝛼−1)𝑀²𝑀²_𝐵𝑏−𝑡_𝑘^1−𝛼𝑎(2𝛼−1)(Γ(𝛼))²×𝑥₁+𝑀𝑥₀Γ(𝛼)+^𝑚_𝑘=1𝑀Γ(𝛼)𝑑_𝑘‖𝑥‖+𝑚𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘(0(𝑟)Λ‖𝑃‖)+𝜓_𝐿^1/𝛾. (65) Thus, 𝑟≤𝑀𝑏^𝛼−1𝑏−𝑡_𝑘^1−𝛼Γ(𝛼)𝑥₀+𝑏−𝑡_𝑘^1−𝛼×^𝑚_𝑘=1𝑀Γ(𝛼)𝑑_𝑘‖𝑥‖+𝑚𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘(0+𝑏_2𝛼−1)𝑀²𝑀²_𝐵𝑏−𝑡_𝑘^1−𝛼𝑎(2𝛼−1)(Γ(𝛼))²×𝑥₁+𝑀𝑥₀Γ(𝛼)+^𝑚_𝑘=1𝑀Γ(𝛼)𝑑_𝑘‖𝑥‖+𝑚𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘(0(𝑟)Λ‖𝑃‖)+𝜓_𝐿^1/𝛾. (66) Dividing both sides by 𝑟 and taking the low limit as 𝑟→∞, we get 1≤lim_𝑟→∞𝜓(𝑟)inf𝑟‖𝑃‖_𝐿^1/𝛾, (67) which is a contradiction to 𝐻(3). Thus, for each 𝑎>0, there exists 𝑟₀ such that Φ_𝑎(𝑥) maps 𝐵_𝑟0 into itself.
Step  3. Φ_𝛼(𝑥) is closed for each 𝑥∈𝑃𝐶_1−𝛼(𝐽,𝑋).
Indeed, for each given 𝑥∈𝑃𝐶_1−𝛼(𝐽,𝑋), let {𝜏_𝑛}_𝑛≥0⊂Φ_𝑎(𝑥) such that 𝜏_𝑛→𝜏 in 𝑃𝐶_1−𝛼(𝐽,𝑋). Then, there exists 𝑓_𝑛∈𝑆_𝐹,𝑥 such that, for each 𝑡∈𝐽, 𝜏_𝑛(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡_𝑘+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓_𝑛(𝑠)+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝑛(𝑠)𝑑𝑠, (68) where 𝑢_𝑛(𝑡)=𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀(𝑏−𝑠)^𝛼−1𝑇_𝛼(𝑏−𝑠)𝑓_𝑛(𝑠)𝑑𝑠. (69) Because of [40, Proposition 3.1], 𝑆_𝐹,𝑥 is weakly compact in 𝐿¹(𝐽,𝑋) which implies that 𝑓_𝑛 converges weakly to some 𝑓∈𝑆_𝐹,𝑥 in 𝐿¹(𝐽,𝑋). Thus, 𝑢_𝑛⇀𝑢, and 𝑢(𝑡)=𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀(𝑏−𝑠)^𝛼−1𝑇_𝛼(𝑏−𝑠)𝑓(𝑠)𝑑𝑠. (70) Then, for each 𝑡∈𝐽, 𝜏_𝑛(𝑡)⟶𝜏(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓(𝑠)𝑑𝑠+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓𝜇𝑑𝜇𝑑𝑠. (71) Thus, we show that 𝜏∈Φ_𝑎(𝑥).
Step  4. Φ_𝑎 is u.s.c and condensing.
We decompose Φ_𝑎 as Φ_𝑎=Φ¹_𝑎+Φ²_𝑎, where the operators Φ¹_𝑎 and Φ²_𝑎 are defined by Φ¹_𝑎𝑥(𝑡)=^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘,Φ𝑡∈𝐽,𝑘=1,2,…,𝑚,_2𝑎(𝑥)=𝜏∈𝑃𝐶_1−𝛼(𝐽,𝑋)∶𝜏(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()𝑓(𝑠)+𝑡−𝑠𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)×𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓𝜇𝑑𝜇𝑑𝑠,𝑓∈𝑆_𝐹,𝑥,𝑡∈𝐽. (72)
According to [41, Corollary 2.2.1], we will prove that Φ¹_𝑎 is a contraction operator, while Φ²_𝑎 is a completely continuous operator.
Let us begin proving that Φ¹_𝑎 is a contraction operator. For any 𝑥,𝑦∈𝑋, we obtain Φ¹_𝑎𝑥(𝑡)−Φ¹_𝑎𝑦(𝑡≤_𝑚𝑘=1)𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑦𝑡⁻_𝑘≤𝑀Γ(𝛼)^𝑚_𝑘=1𝑑_𝑘𝑥−𝑦. (73) Then, Φ¹_𝑎 is a contraction operator, since (𝑀/Γ(𝛼))∑^𝑚_𝑘=1𝑑_𝑘<1.
Next, we prove that Φ²_𝑎 is u.s.c and completely continuous. We subdivide the proof into several claims.
Claim  1. There exists a positive constant 𝑟 such that Φ²_𝑎(𝐵_𝑟)⊆𝐵_𝑟.
By employing the technique used in Step 2, one can easily show that there exists 𝑟>0 such that Φ²_𝑎(𝐵_𝑟)⊆𝐵_𝑟.
Claim  2. Φ²_𝑎(𝐵_𝑟) is a family of equicontinuous functions.
Let 0≤𝑠≤𝑡₁≤𝑡₂≤𝑏. For each 𝑥∈𝐵_𝑟, 𝜑∈Φ²_𝑎(𝑥), there exists 𝑓∈𝑆_𝐹,𝑥 such that 𝜏(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()𝑓(𝑠)+𝑡−𝑠𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠,𝑡∈𝐽. (74) Then, we have 𝜏𝑡₂−𝜏𝑡₁≤^𝑡1₀𝑡₂−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝑇_𝛼𝑡₂𝑓(𝑠)×−𝑠𝑑𝑠_𝑡10𝑡₁−𝑠^𝛼−1𝑇_𝛼𝑡₂−𝑠−𝑇_𝛼𝑡₁𝑓(𝑠)+−𝑠𝑑𝑠_𝑡2𝑡1𝑡₂−𝑠^𝛼−1𝑇_𝛼𝑡₂𝑓(𝑠)×−𝑠𝑑𝑠_𝑡10𝑡₂−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝑇_𝛼𝑡₂𝑢(𝑠)×−𝑠𝑑𝑠_𝑡10𝑡₁−𝑠^𝛼−1𝑇_𝛼𝑡₂−𝑠−𝑇_𝛼𝑡₁𝑢(𝑠)+−𝑠𝑑𝑠_𝑡2𝑡1𝑡₂−𝑠^𝛼−1𝑇_𝛼𝑡₂−𝑠𝑢(𝑠)𝑑𝑠≤𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅+𝐼₆. (75) By using Hölder’s inequality and assumption 𝐻(3), we get 𝐼₁=𝑀𝜓(𝑟)‖𝑃‖_𝐿^1/𝛾Γ(𝛼)×1−𝛾𝑡𝛼−𝛾^{1(𝛼−𝛾)/(1−𝛾)1−𝛾}+1−𝛾𝑡𝛼−𝛾²−𝑡₁^{(𝛼−𝛾)/(1−𝛾)1−𝛾}−1−𝛾𝑡𝛼−𝛾^{2(𝛼−𝛾)/(1−𝛾)1−𝛾},𝐼₂=𝜓(𝑟)‖𝑃‖_𝐿^1/𝛾1−𝛾𝑡𝛼−𝛾^{1(𝛼−𝛾)/(1−𝛾)1−𝛾}×sup_{𝑠∈0,𝑡1}𝑇_𝛼𝑡₂−𝑠−𝑇_𝛼𝑡₁,𝐼−𝑠₃=𝑀𝜓(𝑟)‖𝑃‖_𝐿^1/𝛾Γ(𝛼)1−𝛾𝑡𝛼−𝛾²−𝑡₁^{(𝛼−𝛾)/(1−𝛾)1−𝛾},𝐼₄=𝑀𝑀_𝐵Γ(𝛼)^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝑢(𝑠𝐼‖)‖𝑑𝑠,₅=𝑀_𝐵sup_{𝑠∈0,𝑡1}𝑇_𝛼𝑡₂−𝑠−𝑇_𝛼𝑡₁−𝑠×^𝑡1₀𝑡₁−𝑠^𝛼−1‖𝑢(𝑠𝐼)‖𝑑𝑠,₆=𝑀𝑀_𝐵Γ(𝛼)^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1‖𝑢(𝑠)‖𝑑𝑠. (76) It is easy to see that 𝐼₂ tends to zero independently of 𝑥∈𝐵_𝑟 as 𝑡₂→𝑡₁. Note that, from Lemma 10, 𝑇_𝛼(𝑡) is continuous in the uniform operator topology for 𝑡>0; we can directly obtain 𝐼₁ and 𝐼₃ tending to zero independently of 𝑥∈𝐵_𝑟 as 𝑡₂→𝑡₁. Applying the absolute continuity of the Lebesgue integral, we have 𝐼₄, 𝐼₅, and 𝐼₆ tending to zero independently of 𝑥∈𝐵_𝑟 as 𝑡₂→𝑡₁.
Therefore, Φ²_𝑎(𝐵_𝑟)⊂𝑃𝐶_1−𝛼(𝐽,𝑋) is equicontinuous.
Claim  3. The set Π(𝑡)={𝜏(𝑡)∶𝜏∈Φ²_𝑎(𝐵_𝑟)}⊂𝑋 is relatively compact for each 𝑡∈𝐽.
Let 0<𝑡≤𝑏 be fixed. For 𝑥∈𝐵_𝑟 and 𝜏∈Φ²_𝑎(𝑥), there exists 𝑓∈𝑆_𝐹,𝑥 such that, for each 𝑡∈𝐽, 𝜏(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()𝑓(𝑠)+𝑡−𝑠𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓𝜇𝑑𝜇𝑑𝑠,𝑡∈𝐽. (77)
For all 𝜀∈(0,𝑡) and for all 𝛿>0, define 𝜏^𝜀,𝛿(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+𝛼₀^{𝑡−𝜀∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃𝑓(𝑠)𝑑𝜃𝑑𝑠+𝛼₀^{𝑡−𝜀∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃×𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑠)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓𝜇𝑑𝜇𝑑𝜃𝑑𝑠=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+𝑇𝜀^𝛼𝛿×𝛼₀^{𝑡−𝜀∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇×(𝑡−𝑠)^𝛼𝜃−𝜀^𝛼𝛿𝑓(𝑠)𝑑𝜃𝑑𝑠+𝛼₀^{𝑡−𝜀∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃−𝜀^𝛼𝛿×𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑠)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓𝜇𝑑𝜇𝑑𝜃𝑑𝑠. (78)
By the compactness of 𝑇(𝜀^𝛼𝛿) (𝜀^𝛼𝛿>0), we obtain the set Π^𝜀,𝛿(𝑡)=𝜑^𝜀,𝛿(𝑡)∶𝜏∈Φ²_𝑎𝐵_𝑟 (79) which is relatively compact in 𝑋 for all 𝜀∈(0,𝑡) and 𝛿>0. Moreover, we have 𝜏(𝑡)−𝜏^𝜀,𝛿(𝑡≤(𝑟)𝛽)𝛼𝑀𝜓Γ(𝛼)‖𝑃‖_𝐿^1/𝛾+𝛼𝑀²𝑀²_𝐵𝑏^2𝛼−1𝑎()Γ(𝛼)×𝑥2𝛼−1₁+𝑀𝑥₀Γ(𝛼)+^𝑚_𝑘=1𝑀Γ(𝛼)𝑑_𝑘‖𝑥‖+𝑚𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘(0+(𝑟)𝛽)𝑀𝜓Γ(𝛼)‖𝑃‖_𝐿^1/𝛾𝛿₀𝜃𝜉_𝛼(𝜃)+(𝑟)𝑑𝜃𝑀𝜓Γ(𝛼)‖𝑃‖_𝐿^1/𝛾1−𝛾𝜀𝛼−𝛾^{(𝛼−𝛾)/(1−𝛾)1−𝛾}+𝑀²𝑀²_𝐵𝜀^2𝛼−1𝑎(2𝛼−1)(Γ(𝛼))²×𝑥₁+𝑀𝑥₀Γ(𝛼)+^𝑚_𝑘=1𝑀Γ(𝛼)𝑑_𝑘‖𝑥‖+𝑚𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘(0+(𝑟)𝛽)𝑀𝜓Γ(𝛼)‖𝑃‖_𝐿^1/𝛾. (80) The right-hand side of the above inequality tends to zero as 𝜀→0. Therefore, there are relatively compact sets arbitrarily close to the set Π(𝑡), 𝑡>0. Hence the set Π(𝑡), 𝑡>0 is also relatively compact in 𝑋. As a consequence of Claims 1–3 together with the Arzola-Ascoli theorem, we can conclude that Φ²_𝑎 is completely continuous.
Claim  4. Φ²_𝛼 has a closed graph.
Let 𝑥_𝑛→𝑥^∗ (𝑛→∞), 𝜏_𝑛∈Φ²_𝑎(𝑥_𝑛), 𝜏_𝑛→𝜏^∗ (𝑛→∞). We will prove that 𝜏^∗∈Φ²_𝑎(𝑥^∗). Since 𝜏_𝑛∈Φ²_𝑎(𝑥_𝑛), there exists 𝑓_𝑛∈𝑆_{𝐹,𝑥𝑛}, such that, for each 𝑡∈𝐽, 𝜏_𝑛(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓_𝑛(𝑠)+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓_𝑛𝜇𝑑𝜇𝑑𝑠. (81) We must prove that there exists 𝑓^∗∈𝑆_𝐹,𝑥∗, such that, for each 𝑡∈𝐽, 𝜏^∗(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓^∗(𝑠)+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓^∗𝜇𝑑𝜇𝑑𝑠. (82) Since 𝜏_𝑛→𝜏^∗ (𝑛→∞), we can obtain 𝜏_𝑛(𝑡)−𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀−^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−𝜏𝑑𝑠_∗(𝑡)−𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀−^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑡)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1×𝐼_𝑘𝑥𝑡⁻_𝑘𝑑𝑠⟶0,as𝑛⟶∞. (83) Consider the linear continuous operator Γ∶𝐿^1/𝛾(𝐽,𝑋)⟶𝐶_1−𝛼(),(𝑡)=𝐽,𝑋Γ𝑓^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()×𝑓(𝑠)()𝑡−𝑠−𝐵𝑏−𝑠^𝛼−1𝐵^∗𝑇^∗_𝛼(𝑏−𝑠)𝑅𝑎,Γ^𝑏₀×^𝑏₀𝑇_𝛼𝑏−𝜂𝑓𝜂𝑑𝜂𝑑𝑠. (84) Clearly it follows from Lemma 13 that Γ∘𝑆_𝐹 is a closed graph operator. Moreover, we have 𝜏_𝑛(𝑡)−𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀−^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑠)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘𝑑𝑠∈Γ𝑆_{𝐹,𝑥𝑛}. (85) Since 𝑥_𝑛→𝑥_∗, it follows from Lemma 13 that 𝜏_∗(𝑡)−𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀−^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑠)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘𝑑𝑠∈Γ𝑆_𝐹,𝑥∗. (86) Therefore, Φ²_𝑎 has a closed graph. Since Φ²_𝑎 is a completely continuous multivalued map with compact value, we have that Φ²_𝑎 is u.s.c.
Thus Φ_𝑎=Φ¹_𝑎+Φ²_𝑎 is u.s.c and condensing. Therefore, applying Lemma 13, we conclude that Φ_𝑎 has a fixed point 𝑥(⋅) on 𝐵_𝑟0. Thus, the fractional control system (4) has a mild solution on 𝐽.
The proof is complete.

The following result concerns the approximate controllability of that problem (4). We assume that the following assumption be held. 𝐻^′(3): There exists a positive constant 𝐿 such that ‖𝐹(𝑡,𝑥(𝑡))‖≤𝐿 for all (𝑡,𝑥)∈𝐽×𝑋.

Theorem 15. Assume that assumptions 𝐻(1), 𝐻(2), 𝐻^′(3), and 𝐻(4) are satisfied and the linear system (24) is approximately controllable on 𝐽. Then system (4) is approximately controllable on 𝐽.

Proof. By employing the technique used in Theorem 14, we can easily show that, for all 0<𝑎<1, the operator Φ_𝑎 has a fixed point in 𝐵_𝑟0, where 𝑟₀=𝑟(𝑎). Let 𝑥^𝑎(⋅) be a fixed point of Φ_𝑎 in 𝐵_𝑟0. Any fixed point of Φ_𝑎 is a mild solution of (4); this means that there exists 𝑓^𝑎∈𝑆_{𝐹,𝑥𝑎} such that, for each 𝑡∈𝐽^′, 𝑥^𝑎(𝑡)=𝑡^𝛼−1𝑇_𝛼(𝑡)𝑥₀+^𝑚_𝑘=1𝑇_𝛼𝑡−𝑡_𝑘𝑡−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓^𝑎(𝑠)+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝐵𝐵^∗𝑇^∗_𝛼(𝑏−𝑠)𝑅𝑎,Γ^𝑏₀×𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀𝑏−𝜇^𝛼−1𝑇_𝛼𝑏−𝜇𝑓^𝑎𝜇𝑑𝜇𝑑𝑠. (87) Define 𝑃𝑓^𝑎=𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀(𝑏−𝑠)^𝛼−1𝑇_𝛼(𝑏−𝑠)𝑓^𝑎(𝑠)𝑑𝑠,forsome𝑓^𝑎∈𝑆_{𝐹,𝑥𝑎}. (88) Noting that 𝐼−Γ^𝑏₀𝑅(𝑎,Γ^𝑏₀)=𝑎𝑅(𝑎,Γ^𝑏₀), we get 𝑥^𝑎(𝑏)=𝑥₁−𝑎𝑅𝑎,Γ^𝑏₀𝑃𝑓^𝑎. (89) By assumption 𝐻^′(3), ^𝑏₀𝑓^𝑎(𝑠)²𝑑𝑠≤𝐿²𝑏. (90) Consequently the sequence {𝑓^𝑎} is uniformly bounded in 𝐿^1/𝛾(𝐽,𝑋). Thus, there is a subsequence, still denoted by {𝑓^𝑎}, that converges weakly to, say, 𝑓 in 𝐿^1/𝛾(𝐽,𝑋). Denoting ℎ=𝑥₁−𝑏^𝛼−1𝑇_𝛼(𝑏)𝑥₀−^𝑚_𝑘=1𝑇_𝛼𝑏−𝑡_𝑘𝑏−𝑡_𝑘^𝛼−1𝐼_𝑘𝑥𝑡⁻_𝑘−^𝑏₀(𝑏−𝑠)^𝛼−1𝑇_𝛼(𝑏−𝑠)𝑓(𝑠)𝑑𝑠, (91) we see that 𝑃𝑓^𝛼=−ℎ_𝑏0(𝑏−𝑠)^𝛼−1𝑇_𝛼(𝑏−𝑠)𝑓^𝑎(𝑠)−𝑓(𝑠)𝑑𝑠≤sup_{0≤𝑡≤𝑏}^𝑏₀(𝑏−𝑠)^𝛼−1𝑇_𝛼(𝑏−𝑠)𝑓^𝛼(𝑠)−𝑓(𝑠)𝑑𝑠. (92) By the Ascoli-Arzela theorem, we can show that the linear operator 𝑔→∫^⋅₀(⋅−𝑠)^𝛼−1𝑇_𝛼(⋅−𝑠)𝑔(𝑠)𝑑𝑠∶𝐿^1/𝛾(𝐽,𝑋)→𝑃𝐶_1−𝛼(𝐽,𝑋) is compact; consequently the right-hand side of (92) tends to zero as 𝑎→0⁺.
This implies that 𝑥^𝑎(𝑏)−𝑥₁=𝑎𝑅𝑎,Γ^𝑏₀𝑃𝑓^𝑎≤𝑎𝑅𝑎,Γ^𝑏₀(ℎ)+𝑎𝑅𝛼,Γ^𝑏₀𝑃𝑓^𝑎≤−ℎ𝑎𝑅𝑎,Γ^𝑏₀(ℎ)+𝑃𝑓^𝑎−ℎ⟶0,as𝑎⟶0⁺. (93) This proves the approximate controllability of system (4).

Remark 16. In [9], Ganesh et al. have studied the approximate controllability of fractional integrodifferential evolution equations. If our problem (4) can be changed as 𝐷^𝛼_𝑡𝑥(𝑡)(𝑡)(𝑡)((𝑡),(𝑡,(],1∈𝐴𝑥+𝐵𝑢+𝐹𝑡,𝑥𝐻𝑥)())𝑡∈0,𝑏2𝐼<𝛼≤1,_01−𝛼+𝑥(𝑡)_𝑡=0=𝑥₀+𝑔(𝑥)∈𝑋, (94) where 𝐷^𝛼_𝑡 denotes the Riemann-Liouville fractional derivative, the operator 𝐻 is defined by (𝐻𝑥)(𝑡)=∫^𝑏₀ℎ(𝑡,𝑠,𝑥(𝑠))𝑑𝑠. By a similar way, we can get the approximate controllability of the system (94).

Remark 17. In [15], Rathinasamy and Yong have researched the approximate controllability of fractional differential equations with state-dependent delay. Our problem (4) can be adopted to the impulsive fractional differential inclusion system with state-dependent delay: 𝐷^𝛼_𝑡𝑥(𝑡)∈𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐹𝑡,𝑥_{𝜌(𝑡,𝑥𝑡)},(],1𝑡∈0,𝑏2<𝛼≤1,Δ𝐼₀^1−𝛼+𝑥(𝑡)_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡⁻_𝑘𝐼,𝑘=1,2,…,𝑚,_01−𝛼+𝑥(𝑡)_𝑡=0=𝜙∈𝑋, (95) where 𝐷^𝛼_𝑡 is the Riemann-Liouville fractional derivative, 𝐴 is the infinitesimal generator of a 𝐶₀-semigroup {𝑇(𝑡),𝑡≥0} on a Banach space 𝑋, the state 𝑥(⋅) takes values in 𝑋, the control function 𝑢(⋅) is given in 𝐿^𝑝(𝐽,𝑈) (𝑝>1/𝛼), 𝑈 is a Banach space of admissible control, and 𝐵∶𝑈→𝑋 is a bounded linear operator. 𝐹∶𝐽×𝐷→𝒫(𝑋)∶=2^𝑋⧵{∅} with 𝐷∶=𝐶((−∞,0],𝑋) is a multivalued map. For a continuous function 𝑥∶𝐽^∗∶=(−∞,𝑏]→𝑋, 𝑥_𝑡 is the element of 𝐷 defined by 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠), −∞≤𝑠≤0, and 𝜙∈𝐷. Further, 𝜌∶[0,𝑏]×𝒫(𝑋)→(−∞,𝑏] are appropriate nonlinear functions.

4. An Example

In this final section, we give an example to illustrate our abstract results.

Example 1. Consider the following fractional partial differential inclusion with control 𝐷^𝛼_𝑡𝑥∈𝜕𝑡,𝑦²𝜕𝑦²𝑥𝑡,𝑦+𝐹𝑡,𝑥𝑡,𝑦+𝐵𝑢𝑡,𝑦,𝑡∈𝐽^′=[]⧵10,12,𝑦∈[0,𝜋],Δ𝐼₀^1−𝛼+𝑥12=𝑥𝑦,𝑦𝑥𝑦[],𝑥()()[],𝐼2𝜋+,𝑦∈0,𝜋𝑡,0=𝑥𝑡,𝜋=0,𝑡∈𝐽=0,1_01−𝛼+𝑥𝑡,𝑦_𝑡=0=𝑥₀𝑦,𝑡∈[0,1],𝑦∈[0,𝜋], (96) where 𝐷^𝛼 is the Riemann-Liouville fractional partial derivative of order 3/5, 𝐽=[0,1].
Let 𝑋=𝑈=𝐿²[0,𝜋], define; the operator 𝐴 by 𝐴𝑥=𝑥_𝑦𝑦 with the domain 𝐷(𝐴)=𝑥∈𝑋∶𝑥,𝑥^′𝑥areabsolutelycontinuous,_𝑦𝑦∈𝑋,𝑥(𝑡,0)=𝑥(𝑡,𝜋)=0. (97) Then, 𝐴𝑥=−^∞_𝑛=1𝑛²𝑥,𝑒_𝑛𝑒_𝑛,𝑥∈𝐷(𝐴), (98) where 𝑒_𝑛(𝑦)=2/𝜋sin𝑛𝑦, 0≤𝑦≤𝜋, 𝑛=1,2,…, is the orthogonal set of eigenvectors of 𝐴. It is easily shown that operator 𝐴 generates a strongly continuous semigroup {𝑇(𝑡)∶𝑡≥0} on 𝑋, which are compact, analytic, and self-adjoint in 𝑋.
It can be easily seen that the linear system corresponding to (96) is approximately controllable on [0,1] [42].
Define 𝑥(𝑡)(𝑦)=𝑥(𝑡,𝑦); let 𝐵∶𝑈→𝑋 be defined by 𝐵𝑢(𝑡)(𝑦)=𝐵𝑢(𝑡,𝑦). Then system (96) can be written in the abstract form given by (4). We assume that 𝐹∶𝐽×𝐷→𝒫(𝑋) satisfy the following conditions. (𝐹₁): 𝐹∶𝐽×𝐷→𝒫_{𝑐𝑝,𝑐𝑣}(𝑋) is measurable to 𝑡 for each fixed 𝑥∈𝐷, u.s.c. to 𝑥 for a.e. 𝑡∈𝐽, and for each 𝑥∈𝑃𝐶_1−𝛼(𝐽,𝑋) the set 𝑆_𝐹,𝑥=𝑓∈𝐿¹(𝐽,𝑋)∶𝑓(𝑡)∈𝐹(𝑡,𝑥) (99) is nonempty. (𝐹₂): There exists a positive constant 𝐿 such that ‖𝐹(𝑡,𝑥)‖≤𝐿 for all (𝑡,𝑥)∈𝐽×𝐷, since, for any 𝑥,𝑦∈𝐴𝐶(𝐽,𝑋), we have Δ𝐼_𝑡^1−𝛼𝑥𝑡_𝑘,𝑧−Δ𝐼_𝑡^1−𝛼𝑦𝑡_𝑘,𝑧≤𝑥−𝑦2. (100)
Obviously, the hypotheses of Theorem 15 are fulfilled. Thus, the system (96) is approximately controllable on [0,1].

5. Conclusions

In this paper, we study the approximate controllability for impulsive fractional differential inclusions with Riemann-Liouville fractional derivatives in Banach spaces. We mainly use fixed point theorem and semigroup theory; we obtain some new mild solutions for this kind of problems (4). An illustrative example is also discussed to show the effectiveness of the results in this paper. In the near future, we will consider Riemann-Liouville fractional partial differential problems, which will be more complicated.

Acknowledgments

The project was supported by NNSF of China Grant nos. 11271087 and 61263006, the Scientific Research Foundation of Guangxi University for Nationalities no. 2012QD014, and the Innovation Project of Guangxi University for Nationalities no. gxun-chx2013083.