Complete Controllability of Fractional Neutral Differential Systems in Abstract Space

Abstract

By using fractional power of operators and Sadovskii fixed point theorem, we study the complete controllability of fractional neutral differential systems in abstract space without involving the compactness of characteristic solution operators introduced by us.

1. Introduction

Recently, fractional differential systems have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, and so forth (see [1–5]). There has been a great deal of interest in the solutions of fractional differential systems in analytic and numerical sense. One can see the monographs of Kilbas et al. [6], Miller and Ross [7], Podlubny [8], Lakshmikantham et al. [9], Tarasov [10], Wang et al. [11–13] and the survey of Agarwal et al. [14] and the reference therein. In order to study the fractional systems in the infinite dimensional space, the first important step is how to introduce a new concept of mild solutions. A pioneering work has been reported by EI-Borai [15] and Zhou and Jiao [16].

In recent years, controllability problems for various types of nonlinear fractional dynamical systems in infinite dimensional spaces have been considered in many publications. An extensive list of these publications focused on the complete and approximate controllability of the fractional dynamical systems can be found (see [17–34]). Although the controllability of fractional differential systems in abstract space has been discussed, Hernández et al. [35] point out that some papers on controllability of abstract control systems contain a similar technical error when the compactness of semigroup and other hypotheses is satisfied, more precisely, in this case the application of controllability results are restricted to the finite dimensional space. Ji et al. [32] find some conditions guaranteeing the controllability of impulsive differential system when the Banach space is nonseparable and evolution systems are not compact, by means of Möch fixed point theorem and the measure of noncompactness. Meanwhile, Wang et al. [19, 20] have researched the complete controllability of fractional evolution systems without involving the compactness of characteristic solution operators. Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention in the last decades. Sakthivel and Ren [29] have established a new set of sufficient conditions for the complete controllability for a class of fractional order neutral systems with bounded delay under the natural assumption that the associated linear control is completely controllable. To the author’s knowledge, there are few papers on the complete controllability of the abstract neutral fractional differential systems with unbounded delay.

In the present paper, we introduce a suitable concept of the mild solutions including characteristic solution operators 𝜑(⋅) and 𝑆(⋅) which are associated with operators semigroup {𝑇(𝑡);𝑡≥0} and some probability density functions 𝜉_𝑞. Then also without involving the compactness of characteristic solution operators, we obtain the controllability of the following abstract neutral fractional differential systems with unbounded delay: ^𝑐𝐷^𝑞_𝑡𝑥(𝑡)+𝐹𝑡,𝑥_𝑡+𝐴𝑥(𝑡)=𝐶𝑢(𝑡)+𝐺𝑡,𝑥_𝑡,(],𝑥𝑡∈0,𝑎₀(𝜗)=𝜙(𝜗)∈𝐵,𝜗∈(−∞,0],(1) where the state variable 𝑥(⋅) takes values in Banach space 𝑋,  𝑥_𝑡∶(−∞,0]→𝑋,𝑥_𝑡(𝜗)=𝑥(𝑡+𝜗) belongs to some abstract phase space 𝐵, and 𝐵 is the phase space to be specified later. The control function 𝑢(⋅) is given in 𝐿²([0,𝑎];𝑈), with 𝑈 as a Banach spaces. 𝐶 is a bounded linear operator from 𝑈 to 𝑋. The operator −𝐴 is a generator of a uniformly bounded analytic semigroup {𝑇(𝑡),𝑡≥0} in which 𝑋, 𝐹,𝐺∶[0,𝑎]×𝐵→𝑋 are appropriate functions.

2. Preliminaries

Throughout this paper 𝑋 will be a Banach space with norm ‖⋅‖ and 𝑌 is another Banach space, 𝐿_𝑏(𝑋,𝑌) denote the space of bounded linear operators from 𝑋 to 𝑌. We also use ‖𝑓‖_𝐿^𝑝_{([0,𝑎],𝑅}⁺₎ to denote the 𝐿^𝑝([0,𝑎],𝑅⁺) of norm of 𝑓 whenever 𝑓∈𝐿^𝑝([0,𝑎],𝑅⁺) for some 𝑝 with 1≤𝑝<∞. Let 𝐿^𝑝([0,𝑎],𝑅⁺) denote the Banach space of functions 𝑓:[0,𝑎]→𝑋 which are Bochner integrable normed by ‖𝑓‖_𝐿^𝑝_{([0,𝑎],𝑅}⁺₎. Let −𝐴:𝐷(𝐴)→𝑋 be the infinitesimal generator of a uniformly bounded analytic semigroup 𝑇(𝑡). Let 0∈𝜌(𝐴), then it is possible to define the fractional power 𝐴^𝛼, for 0<𝛼≤1, as a closed linear operator on its domain 𝐷(𝐴^𝛼). Furthermore, the subspace 𝐷(𝐴^𝛼) is dense in 𝑋 and the expression ‖𝑥‖_𝛼=𝐴^𝛼𝑥,𝑥∈𝐷𝐴^𝛼(2) defines a norm on 𝐷(𝐴^𝛼). Hereafter we denote by 𝑋_𝛼 the Banach space 𝐷(𝐴^𝛼) normed with ‖𝑥‖_𝛼. Then for each 0<𝛼≤1, 𝑋_𝛼 the Banach space, and ‖𝑥‖_𝛼↪‖𝑥‖_𝛽 for 0<𝛽<𝛼≤1 and the imbedding is compact whenever the resolvent operator of 𝐴 is compact. For a uniformly bounded analytic semigroup {𝑇(𝑡);𝑡≥0} the following properties will be used:(a)there is a 𝑀≥0 such that ‖𝑇(𝑡)‖≤𝑀 for all 𝑡≥0.(b)for any 𝛼≥0, there exists a positive constant 𝐶_𝛼 such that 𝐴^𝛼𝑇(𝑡≤𝐶)^𝛼𝑡^𝛼,0<𝑡≤𝑎.(3)For more details about the above preliminaries, we can refer to [16].

Although the semigroup {𝑇(𝑡);𝑡≥0} is only the uniformly bounded analytic semigroup but not compact, we can also give the definition of mild solution for our problem by using the similar method introduced in [36].

Definition 1. We say that a function 𝑥(⋅):(−∞,𝑎]→𝑋 is a mild solution of the system (1) if 𝑥₀=𝜙, the restriction of 𝑥(⋅) to the interval [0,𝑎] is continuous and for each 0≤𝑡≤𝑎, the function 𝐴𝑆(𝑡−𝑠)𝐹(𝑠,𝑥_𝑠),𝑠∈[0,𝑡] is integrable and satisfies the following integral equation: 𝑥(𝑡)=𝜑(𝑡)𝜙(0)+𝐹0,𝜙−𝐹𝑡,𝑥_𝑡−^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑥_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐶𝑢(𝑠)+𝐺𝑠,𝑥_𝑠𝑑𝑠,(4) where 𝜑(𝑡) and 𝑆(𝑡) are called characteristic solution operators and are given by 𝜑(𝑡)=^∞₀𝜉_𝑞(𝜃)𝑇𝑡^𝑞𝜃𝑆(𝑡)𝑑𝜃,=𝑞^∞₀𝜃𝜉_𝑞(𝜃)𝑇𝑡^𝑞𝜃𝑑𝜃,(5) and for 𝜃∈(0,∞), 𝜉_𝑞(𝜃)=(1/𝑞)𝜃^{−1−1/𝑞}𝑤_𝑞(𝜃^−1/𝑞)≥0, 𝑤_𝑞(𝜃)=1𝜋^∞_𝑛=1(−1)^𝑛−1𝜗^{−𝑞𝑛−1}Γ𝑛𝑞+1.𝑛!sin𝑛𝜋𝑞(6) Here, 𝜉_𝑞 is a probability density function defined on (0,∞), that is, 𝜉_𝑞(𝜃)≥0, 𝜃∈(0,∞), and ∫^∞₀𝜉_𝑞(𝜃)𝑑𝜃=1.

Definition 2 (complete controllability). The fractional system (1) is said to be completely controllable on the interval [0,𝑎] if, for every initial function 𝜙∈𝐵 and 𝑥₁∈𝑋 there exists a control 𝑢∈𝐿²([0,𝑎],𝑈) such that the mild solution 𝑥(⋅) of (1) satisfies 𝑥(𝑎)=𝑥₁.

The following results of 𝜑(𝑡) and 𝑆(𝑡) will be used throughout this paper.

Lemma 3. The operators 𝜑(𝑡) and 𝑆(𝑡) have the following properties:(i) for any fixed 𝑡≥0, 𝜑(𝑡) and 𝑆(𝑡) are linear and bounded operators, that is, for any 𝑥∈𝑋, 𝜑(𝑡)𝑥≤𝑀₀‖𝑥‖,‖𝑆(𝑡)𝑥‖≤𝑞𝑀₀Γ‖𝑥‖;1+𝑞(7)(ii){𝜑(𝑡),𝑡≥0} and {𝑆(𝑡),𝑡≥0} are strongly continuous and there exists 𝑀₁,𝑀₂ such that ‖𝜑(𝑡)‖≤𝑀₁, ‖𝑆(𝑡)‖≤𝑀₂ for any 𝑡∈[0,𝑎];(iii) for 𝑡∈[0,𝑎] and any bounded subsets 𝐷⊂𝑋, 𝑡→{𝜑(𝑡)𝑥∶𝑥∈𝐷} and 𝑡→{𝑆(𝑡)𝑥∶𝑥∈𝐷} are equicontinuous if ‖𝑇(𝑡^𝑞₂𝜃)𝑥−𝑇(𝑡^𝑞₁𝜃)𝑥‖→0 with respect to 𝑥∈𝐷 as 𝑡₂→𝑡₁ for each fixed 𝜃∈[0,∞].

The proof of Lemma 3 we can see in [33].

To end this section, we recall Kuratowski’s measure of noncompactness, which will be used in the next section to study the complete controllability via the fixed points of condensing operator.

Definition 4. Let 𝑋 be a Banach space and Ω_𝑋 the bounded set of 𝑋. The Kuratowski’s measure of noncompactness is the map 𝛼∶Ω_𝑋→[0,∞) defined by 𝛼(𝐷)=inf𝑑>0∶𝐷⊆^𝑛_𝑖=1𝐷_𝑖,diam𝐷_𝑖≤𝑑,(8) here 𝐷∈Ω_𝑋.

One will use the following basic properties of the 𝛼 measure and Sadovskii's fixed point theorem here (see [37–39]).

Lemma 5. Let 𝐷₁ and 𝐷₂ be two bounded sets of a Banach space 𝑋. Then(i)𝛼(𝐷₁)=0 if and only if 𝐷₁ is relatively compact;(ii)𝛼(𝐷₁)≤𝛼(𝐷₂) if 𝐷₁⊆𝐷₂;(iii)𝛼(𝐷₁+𝐷₂)≤𝛼(𝐷₁)+𝛼(𝐷₂).

Lemma 6 (sadovskii’s fixed point theorem). Let 𝑁 be a condensing operator on a Banach space 𝑋, that is, 𝑁 is continuous and takes bounded sets into bounded sets, and 𝛼(𝑁(𝐷))<𝛼(𝐷) for every bounded set 𝐷 of 𝑋 with 𝛼(𝐷)>0. If 𝑁(𝑆)⊂𝑆 for a convex closed and bounded set 𝑆 of 𝑋, then 𝑁 has a fixed point in 𝑆.

3. Complete Controllability Result

To study the system (1), we assume the function 𝑥_𝑡 represents the history of the state from −∞ up to the present time 𝑡 and 𝑥_𝑡∶(−∞,0]→𝑋,𝑥_𝑡(𝜗)=𝑥(𝑡+𝜗) belongs to some abstract phase space 𝐵, which is defined axiomatically. In this article, we will employ an axiomatic definition of the phase space 𝐵 introduced by Hale and Kato [40] and follow the terminology used in [41]. Thus, 𝐵 will be a linear space of functions mapping (−∞,0] into 𝑋 endowed with a seminorm ‖⋅‖_𝐵. We will assume that 𝐵 satisfies the following axioms:(A) If 𝑥∈(−∞,𝜎+𝑎)→𝑋, 𝑎>0, is continuous on [𝜎,𝜎+𝑎] and 𝑥_𝜎∈𝐵, then for every 𝑡∈[𝜎,𝜎+𝑎] the following conditions hold:(i)𝑥_𝑡 is in 𝐵;(ii)‖𝑥(𝑡)‖≤𝐻‖𝑥_𝑡‖_𝐵; (iii)‖𝑥_𝑡‖_𝐵≤𝐾(𝑡−𝜎)sup{‖𝑥(𝑡)‖∶𝜎≤𝑠≤𝑡}+𝑀(𝑡−𝜎)‖𝑥_𝜎‖_𝐵.

Here 𝐻≥0 is a constant, 𝐾,𝑀∶[0,+∞)→[0,+∞), 𝐾 is continuous and 𝑀 is locally bounded, and 𝐻,𝐾,𝑀 are independent of 𝑥(𝑡).(B) For the function 𝑥(⋅) in (A), 𝑥_𝑡 is a 𝐵-valued continuous function on [𝜎,𝜎+𝑎].(C) The space 𝐵 is complete.

Now we give the basic assumptions on the system (1).

(𝐻₀) (i) 𝐴 generates a uniformly bounded analytic semigroup {𝑇(𝑡),𝑡≥0} in 𝑋; (ii) for all bounded subsets 𝐷⊂𝑋 and 𝑥∈𝐷, ‖𝑇(𝑡^𝑞₂𝜃)𝑥−𝑇(𝑡^𝑞₁𝜃)𝑥‖→0 as 𝑡₂→𝑡₁ for each fixed 𝜃∈[0,∞].(𝐻₁)𝐹:[0,𝑎]×𝐵→𝑋 is continuous function, and there exists a constant 𝛽∈(0,1) and 𝐿,𝐿₁>0 such that the function 𝐹 is 𝑋_𝛽-valued and satisfies the Lipschitz condition: 𝐴^𝛽𝐹𝑠₁,𝜙₁−𝐴^𝛽𝐹𝑠₂,𝜙₂≤𝐿𝑠₁−𝑠₂+𝜙₁−𝜙_2𝐵,(9) for 0≤𝑠₁, 𝑠₂≤𝑎, 𝜙₁,𝜙₂∈𝐵, and the inequality 𝐴^𝛽𝐹𝑡,𝜙≤𝐿₁𝜙_𝐵+1(10) holds for 𝑡∈[0,𝑎],𝜙∈𝐵.(𝐻₂)The function 𝐺:[0,𝑎]×𝐵→𝑋 satisfies the following conditions:(i)for each 𝑡∈[0,𝑎], the function 𝐺(𝑡,⋅)∶𝐵→𝑋 is continuous and for each 𝜙∈𝐵 the function 𝐺(⋅,𝜙)∶[0,𝑎]→𝑋 is strongly measureable;(ii) for each positive number 𝑘, there is a positive function 𝑔_𝑘∈𝐿^1/𝑞1([0,𝑎]),0<𝑞₁<𝑞 such that sup_{𝜙𝐵≤𝑘}𝐺𝑡,𝜙≤𝑔_𝑘(𝑡),1liminf𝑘𝑔_{𝑘𝐿11/𝑞[0,𝑎]}=𝛾<∞.(11)

(𝐻₃)The linear operator 𝐶 is bounded, 𝑊 from 𝑈 into 𝑋 is defined by 𝑊𝑢=^𝑎₀(𝑎−𝑠)^𝑞−1𝑆(𝑎−𝑠)𝐶𝑢(𝑠)𝑑𝑠(12) and there exists a bounded invertible operator 𝑊⁻¹ defined on 𝐿²([0,𝑎];𝑈)/ker𝑊 and there exist two positive constants 𝑀₃,𝑀₄>0 such that ‖𝐵‖_{𝐿𝑏(𝑈,𝑋)}≤𝑀₃, ‖𝑊⁻¹‖_{𝐿𝑏(𝑋,𝐿}²_{([0,𝑎],𝑈)/ker𝑊)}≤𝑀₄.(𝐻₄) For all bounded subsets 𝐷⊆𝑋, the set Π_ℎ,𝛿(𝑡)=𝑄_2,ℎ,𝛿𝑧(𝑡)∣𝑧∈𝐷,(13) where 𝑄_2,ℎ,𝛿𝑧(𝑡)=₀^{𝑡−ℎ∞}_𝛿(𝑡−𝑠)^𝑞−1𝜃𝜉_𝑞(𝜃)𝑞𝑇(𝑡−𝑠)^𝑞𝜃×𝐶𝑢(𝑠)+𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝜃𝑑𝑠(14) is relatively compact in 𝑋 for arbitrary ℎ∈(0,𝑡) and 𝛿>0.

Theorem 7. Let 𝜙∈𝐵. If the assumptions (𝐻₀)–(𝐻₄) are satisfied, then the system (1) is controllable on interval [0,𝑎] provided that 𝑀₅𝐿𝐾_𝑎+𝐶_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝛽Γ1+𝑞𝛽𝐿𝐾_𝑎<1,(15)1+𝑎𝑀₂𝑀₃𝑀₄×𝐿₁𝑀₅𝐾_𝑎+𝑀₂1−𝑞₁𝑞−𝑞₁𝑎^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}𝐾_𝑎𝛾+𝐶_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝐾_𝑎<1,(16) where 𝑀₅=‖𝐴^−𝛽‖, 𝐾_𝑎=sup{𝐾(𝑡):0≤𝑡≤𝑎} and 𝐶_1−𝛽 is from (3).

Proof. Using the assumption (𝐻₃), for arbitrary function 𝑥(⋅) define the control 𝑢(𝑡)=𝑊⁻¹𝑥₁−𝜑(𝑎)𝜙(0)+𝐹0,𝜙+𝐹𝑎,𝑥_𝑎+^𝑎₀(𝑎−𝑠)^𝑞−1𝐴𝑆(𝑎−𝑠)𝐹𝑠,𝑥_𝑠−𝑑𝑠_𝑎0(𝑎−𝑠)^𝑞−1𝑆(𝑎−𝑠)𝐺𝑠,𝑥_𝑠𝑑𝑠(𝑡).(17) It will be shown that when using this control the operator 𝑃 defined by 𝑃𝑥(𝑡)=𝜑(𝑡)𝜙(0)+𝐹0,𝜙−𝐹𝑡,𝑥_𝑡−^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑥_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐶𝑢(𝑠)+𝐺𝑠,𝑥_𝑠𝑑𝑠(18) has a fixed point 𝑥(⋅). Then 𝑥(⋅) is a mild solution of system (1), and it is easy to verify that 𝑥(𝑎)=𝑃𝑥(𝑎)=𝑥₁, which implies that the system is controllable.
Next we will prove that 𝑃 has a fixed point using the fixed point theorem of Sadovskii [38].
Let 𝑦(⋅):(−∞,𝑎]→𝑋 be the function defined by 𝑦(𝑡)=𝜑(𝑡)𝜙(0)[],𝜙(𝑡),𝑡∈0,𝑎,−∞<𝑡<0,(19) then 𝑦₀=𝜙 and the map 𝑡→𝑦_𝑡 is continuous. We can assume 𝑁=sup{‖𝑦_𝑡‖:0≤𝑡≤𝑎}. For each 𝑧∈𝐶([0,𝑎]∶𝑋),𝑧(0)=0. We can denote by 𝑧 the function defined by 𝑧(𝑡)=𝑧(𝑡),0≤𝑡≤𝑎,0,−∞<𝑡<0.(20) If 𝑥(⋅) satisfies (18), we can decompose it as 𝑥(𝑡)=𝑧(𝑡)+𝑦(𝑡), 0≤𝑡≤𝑎, which implies 𝑥_𝑡=𝑧_𝑡+𝑦_𝑡 for every 0≤𝑡≤𝑎 and the function 𝑧(⋅) satisfies 𝑧(𝑡)=𝜑(𝑡)𝐹0,𝜙−𝐹𝑡,𝑧_𝑡+𝑦_𝑡−^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑧_𝑠+𝑦_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐶𝑢(𝑠)+𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝑠.(21) Moreover 𝑧₀=0. Let 𝑄 be the operator on 𝐶([0,𝑎],𝑋) defined by 𝑄𝑧(𝑡)=𝜑(𝑡)𝐹0,𝜙−𝐹𝑡,𝑧_𝑡+𝑦_𝑡−^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑧_𝑠+𝑦_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐶𝑢(𝑠)+𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝑠.(22) Obviously the operator 𝑃 has a fixed point is equivalent to 𝑄 has a fixed point, so it turns out to prove that 𝑄 has a fixed point. For each positive number 𝑘, let 𝐵_𝑘={𝑧∈𝐶([0,𝑎]∶𝑋)∶𝑧(0)=0,‖𝑧(𝑡)‖≤𝑘,0≤𝑡≤𝑎},(23) then for each 𝑘, 𝐵_𝑘 is clearly a bounded closed convex set in 𝐶([0,𝑎]:𝑋). Since by (3) and (10) the following relation holds: 𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑧_𝑠+𝑦_𝑠≤𝐴^1−𝛽𝑆(𝑡−𝑠)𝐴^𝛽𝐹𝑠,𝑧_𝑠+𝑦_𝑠≤𝐶_1−𝛽𝑞Γ1+𝛽Γ()1+𝑞𝛽𝑡−𝑠^{−(1−𝛽)𝑞}×𝐿₁𝑧_𝑠+𝑦_𝑠𝐵≤𝐶+1_1−𝛽𝑞Γ1+𝛽Γ()1+𝑞𝛽𝑡−𝑠^{−(1−𝛽)𝑞}×𝐿₁𝑧_𝑠𝐵+𝑦_𝑠𝐵≤𝐶+1_1−𝛽𝑞Γ1+𝛽Γ()1+𝑞𝛽𝑡−𝑠^{−(1−𝛽)𝑞}×𝐿₁𝑘𝐾_𝑎+𝑁+1,(24) then from Bocher’s theorem [42] it follows that 𝐴𝑆(𝑡−𝑠)𝐹(𝑠,𝑧_𝑠+𝑦_𝑠) is integrable on [0,𝑎], so 𝑄 is well defined on 𝐵_𝑘.
In order to make the following process clear we divide it into several steps.
Step  1. We claim that there exists a positive number 𝑘 such that 𝑄(𝐵_𝑘)⊆𝐵_𝑘.
If it is not true, then for each positive number 𝑘, there is a function 𝑧_𝑘(⋅)∈𝐵_𝑘, but 𝑄𝑧_𝑘∉𝐵_𝑘, that is, ‖𝑄𝑧_𝑘(𝑡)‖>𝑘 for some 𝑡∈[0,𝑎]. However, on the other hand, we have 𝑘<𝑄𝑧_𝑘(𝑡=𝜑(𝑡)𝐹)0,𝜙−𝐹𝑡,𝑧_𝑘,𝑡+𝑦_𝑡−^𝑡₀(𝑡−𝑠)^𝑞−1×𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑧_𝑘,𝑠+𝑦_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆()×𝑡−𝑠𝐶𝑢_𝑘(𝑠)+𝐺𝑠,𝑧_𝑘,𝑠+𝑦_𝑠=𝜑(𝑡)𝐹𝑑𝑠0,𝜙−𝐹𝑡,𝑧_𝑘,𝑡+𝑦_𝑡−^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)×𝐹𝑠,𝑧_𝑘,𝑠+𝑦_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆()×𝑡−𝑠𝐶𝑊⁻¹𝑥₁−𝜑(𝑎)𝜙(0)+𝐹0,𝜙+𝐹𝑎,𝑧_𝑘,𝑎+𝑦_𝑎+^𝑎₀(𝑎−𝜏)^𝑞−1𝐴𝑆(𝑎−𝜏)×𝐹𝜏,𝑧_𝑘,𝜏+𝑦_𝜏−𝑑𝜏_𝑎0(𝑎−𝜏)^𝑞−1𝑆(𝑎−𝜏)×𝐺𝜏,𝑧_𝑘,𝜏+𝑦_𝜏𝑠)+𝑑𝜏(𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)×𝐺𝑠,𝑧_𝑘,𝑠+𝑦_𝑠𝑑𝑠≤𝑀₁𝐹0,𝜙+𝐹𝑡,𝑧_𝑘,𝑡+𝑦_𝑡+^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑧_𝑘,𝑠+𝑦_𝑠+𝑑𝑠_𝑡0𝑀₂𝑀₃𝑀₄×𝑥₁+𝑀₁𝜙(0)+𝐹is+𝐹0,𝜙𝑎,𝑧_𝑘,𝑎+𝑦_𝑎+^𝑎₀(𝑎−𝜏)^𝑞−1×𝐴𝑆(𝑎−𝜏)𝐹𝜏,𝑧_𝑘,𝜏+𝑦_𝜏+𝑑𝜏_𝑎0𝑀₂(𝑎−𝜏)^𝑞−1×𝐺𝜏,𝑧_𝑘,𝜏+𝑦_𝜏𝑑𝜏(𝑠)𝑑𝑠+^𝑡₀𝑀₂(𝑡−𝑠)^𝑞−1𝐺𝑠,𝑧_𝑘,𝑠+𝑦_𝑠𝑑𝑠,(25) where 𝑢_𝑘 is the corresponding control of 𝑥_𝑘, 𝑥_𝑘=𝑧_𝑘+𝑦. Since ^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑧_𝑘,𝑠+𝑦_𝑠≤𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝐴^1−𝛽𝑆(𝑡−𝑠)𝐴^𝛽𝐹𝑠,𝑧_𝑘,𝑠+𝑦_𝑠≤𝐶𝑑𝑠_1−𝛽𝑞Γ1+𝛽Γ×1+𝑞𝛽_𝑡0(𝑡−𝑠)^𝑞−1(𝑡−𝑠)^{−(1−𝛽)𝑞}𝐿₁𝑘𝐾_𝑎≤𝐶+𝑁+1_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝑘𝐾_𝑎,𝐹+𝑁+1𝑡,𝑧_𝑘,𝑡+𝑦_𝑡=𝐴^−𝛽𝐴^𝛽𝐹𝑡,𝑧_𝑘,𝑡+𝑦_𝑡≤𝑀₅𝐿₁𝑘𝐾_𝑎+𝑁+1_𝑡0,(𝑡−𝑠)^𝑞−1𝐺𝑠,𝑧_𝑘,𝑠+𝑦_𝑠≤𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑔_{𝑘𝐾𝑎+𝑁}(𝑠)𝑑𝑠,(26) there holds 𝑘<𝑀₁𝐹0,𝜙+𝑀₅𝐿₁𝑘𝐾_𝑎+𝐶+𝑁+1_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝑘𝐾_𝑎+𝑁+1+𝑎𝑀₂𝑀₃𝑀₄𝑥₁+𝑀₁𝜙(0)+𝐹0,𝜙+𝑀₅𝐿₁𝑘𝐾_𝑎+𝐶+𝑁+1_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝑘𝐾_𝑎+𝑁+1+𝑀₂^𝑎₀(𝑎−𝜏)^𝑞−1𝑔_{𝑘𝐾𝑎+𝑁}(𝜏)𝑑𝜏+𝑀₂^𝑎₀(𝑎−𝑠)^𝑞−1𝑔_{𝑘𝐾𝑎+𝑁}(𝑠)𝑑𝑠=𝑀₅𝐿₁𝑘𝐾_𝑎1+𝑎𝑀₂𝑀₃𝑀₄+𝑀₂1+𝑎𝑀₂𝑀₃𝑀₄^𝑎₀(𝑎−𝑠)^𝑞−1𝑔_{𝑘𝐾𝑎+𝑁}(𝑠)+𝐶𝑑𝑠_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝑘𝐾_𝑎+𝑁+11+𝑎𝑀₂𝑀₃𝑀₄+𝑀₁𝐹0,𝜙+𝑀₅𝐿₁𝑁+𝑀₅𝐿₁+𝑎𝑀₂𝑀₃𝑀₄𝑥₁+𝑎𝑀₂𝑀₃𝑀₄𝜙(0)+𝐹0,𝜙+𝑀₅𝐿₁𝑁𝑎𝑀₂𝑀₃𝑀₄+𝑀₅𝐿₁𝑎𝑀₂𝑀₃𝑀₄=𝑀^∗+1+𝑎𝑀₂𝑀₃𝑀₄×𝑀₅𝐿₁𝐾_𝑎𝑘+𝑀₂^𝑎₀(𝑎−𝑠)^{(𝑞−1)/(𝑞−𝑞1)}𝑑𝑠^1−𝑞1×^𝑎₀𝑔_{𝑘𝐾𝑎+𝑁}(𝑠)^1/𝑞1𝑑𝑠^𝑞1+𝐶_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝑘𝐾_𝑎+𝑁+1=𝑀^∗+1+𝑎𝑀₂𝑀₃𝑀₄×𝑀₅𝐿₁𝐾_𝑎𝑘+𝑀₂1−𝑞₁𝑞−𝑞₁𝑎^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}×𝑔_{𝑘𝐾𝑎+𝑁𝐿11/𝑞[0,𝑎]}+𝐶_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝑘𝐾_𝑎+𝑁+1,(27) where 𝑀^∗=𝑀₁𝐹0,𝜙+𝑀₅𝐿₁𝑁+𝑀₅𝐿₁+𝑎𝑀₂𝑀₃𝑀₄𝑥₁+𝑎𝑀₂𝑀₃𝑀₄𝜙(0)+𝐹0,𝜙+𝑀₅𝐿₁𝑁𝑎𝑀₂𝑀₃𝑀₄+𝑀₅𝐿₁𝑎𝑀₂𝑀₃𝑀₄.(28) Dividing on both sides by 𝑘 and taking the low limit, we get 1+𝑎𝑀₂𝑀₃𝑀₄×𝐿₁𝑀₀𝐾_𝑎+𝑀₂1−𝑞₁𝑞−𝑞₁𝑎^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}𝐾_𝑎𝛾+𝐶_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝐾_𝑎≥1.(29) This contradicts (16). Hence for some positive number 𝑘, 𝑄𝐵_𝑘⊆𝐵_𝑘.
Now, we define operator 𝑄₁ and 𝑄₂ on 𝐵_𝑘 as 𝑄₁𝑧(𝑡)=𝜑(𝑡)𝐹0,𝜙−𝐹𝑡,𝑧_𝑡+𝑦_𝑡−^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑧_𝑠+𝑦_𝑠𝑄𝑑𝑠,₂𝑧(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1𝑆()×(𝑠)𝑡−𝑠𝐶𝑢+𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝑠,(30) for all 𝑡∈[0,𝑎], respectively.
We prove that 𝑄₁ is contraction, while 𝑄₂ is completely continuous.
Step  2. 𝑄₁ is contraction.
Let 𝑧₁,𝑧₂∈𝐵_𝑘. Then, for each 𝑡∈[0,𝑎], and by axiom (A)-(iii) and (15), we have 𝑄₁𝑧₁(𝑡)−𝑄₁𝑧₂(𝑡=𝜑(𝑡)𝐹)0,𝜙−𝐹𝑡,𝑧_1,𝑡+𝑦_𝑡−^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑧_1,𝑠+𝑦_𝑠(𝑡)𝐹𝑑𝑠−𝜑0,𝜙+𝐹𝑡,𝑧_2,𝑡+𝑦_𝑡+^𝑡₀(𝑡−𝑠)^𝑞−1𝐴𝑆(𝑡−𝑠)𝐹𝑠,𝑧_2,𝑠+𝑦_𝑠≤𝐹𝑑𝑠𝑡,𝑧_1,𝑡+𝑦_𝑡−𝐹𝑡,𝑧_2,𝑡+𝑦_𝑡+^𝑡₀(𝑡−𝑠)^𝑞−1()×𝐹𝐴𝑆𝑡−𝑠𝑠,𝑧_1,𝑠+𝑦_𝑠−𝐹𝑠,𝑧_2,𝑠+𝑦_𝑠≤𝐴𝑑𝑠_−𝛽𝐴^𝛽𝐹𝑡,𝑧_1,𝑡+𝑦_𝑡−𝐴^−𝛽𝐴^𝛽𝐹𝑡,𝑧_2,𝑡+𝑦_𝑡+^𝑡₀(𝑡−𝑠)^𝑞−1×𝐴^1−𝛽𝑆(𝑡−𝑠)𝐴^𝛽𝐹𝑠,𝑧_1,𝑠+𝑦_𝑠−𝐹𝑠,𝑧_2,𝑠+𝑦_𝑠𝑑𝑠≤𝑀₅𝐿𝐾_𝑎𝑧_1,𝑡−𝑧_2,𝑡𝐵+^𝑎₀(𝑡−𝑠)^𝑞−1𝐶_1−𝛽𝑞Γ1+𝛽Γ()1+𝑞𝛽𝑡−𝑠^{−(1−𝛽)𝑞}×𝐿𝑧_1,𝑠−𝑧_2,𝑠𝐵𝑑𝑠≤𝑀₅𝐿𝐾_𝑎sup_{0≤𝑠≤𝑎}𝑧₁(𝑠)−𝑧₂(𝑠+𝐶_1−𝛽)Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝐾_𝑎sup_{0≤𝑠≤𝑎}𝑧₁(𝑠)−𝑧₂(𝑠≤𝑀₅)𝐿𝐾_𝑎+𝐶_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝐾_𝑎×sup_{0≤𝑠≤𝑎}𝑧₁(𝑠)−𝑧₂(𝑠).(31) Thus 𝑄₁𝑧₁(𝑡)−𝑄₁𝑧₂(𝑡)<𝑧₁−𝑧₂,(32) and 𝑄₁ is contraction.
Step  3. 𝑄₂ is completely continuous.
Let {𝑧_𝑛}⊆𝐵_𝑘 with 𝑧_𝑛→𝑧 in 𝐵_𝑘, then for each 𝑠∈[0,𝑎],𝑧_𝑛,𝑠→𝑧_𝑠, and by (𝐻₁) and (𝐻₂)-(i), we have 𝐺𝑠,𝑧_𝑛,𝑠+𝑦_𝑠−𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝐹→0,𝑠,𝑧_𝑛,𝑠+𝑦_𝑠−𝐹𝑠,𝑧_𝑠+𝑦_𝑠𝑢→0,_𝑛(𝑠)−𝑢(𝑠)→0,(33) as 𝑛→∞.
Since ‖𝐺(𝑠,𝑧_𝑛,𝑠+𝑦_𝑠)−𝐺(𝑠,𝑧_𝑠+𝑦_𝑠)‖≤2𝑔_{𝑘𝐾𝑎+𝑁(𝑠)}, then by the dominated convergence theorem we have 𝑄₂𝑧_𝑛(𝑡)−𝑄₂𝑧(𝑡)=sup_{0≤𝑡≤𝑎}^𝑡₀(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐶𝑢_𝑛(𝑠)+𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐺𝑠,𝑧_𝑛,𝑠+𝑦_𝑠−𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐶𝑢(𝑠)𝑑𝑠−^𝑡₀(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐺𝑠,𝑧_𝑠+𝑦_𝑠≤𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐶𝑢_𝑛(𝑠)(𝑠)+−𝐶𝑢𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑆()×𝐺𝑡−𝑠𝑠,𝑧_𝑛,𝑠+𝑦_𝑠−𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝑠→0,(34) as 𝑛→∞, that is, 𝑄₂ is continuous.
Next we prove that the family {𝑄₂𝑧∶𝑧∈𝐵_𝑘} is an equicontinuous family of functions. To do this, let 0≤𝑡₁<𝑡₂≤𝑎, then 𝑄₂𝑧𝑡₂−𝑄₂𝑧𝑡₁=^𝑡2₀𝑡₂−𝑠^𝑞−1𝑆𝑡₂(𝑠)+−𝑠𝐶𝑢𝑑𝑠_𝑡20𝑡₂−𝑠^𝑞−1𝑆𝑡₂−𝑠𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝑠−^𝑡1₀𝑡₁−𝑠^𝑞−1𝑆𝑡₁(𝑠)−−𝑠𝐶𝑢𝑑𝑠_𝑡10𝑡₁−𝑠^𝑞−1𝑆𝑡₁−𝑠𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝑠≤𝑀₃^𝑡1₀𝑡₂−𝑠^𝑞−1𝑆𝑡₂−𝑠−𝑆𝑡₁−𝑠𝑢(𝑠)𝑑𝑠+𝑀₃^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1×𝑆𝑡₁𝑢(𝑠+−𝑠‖)‖𝑑𝑠_𝑡10𝑡₂−𝑠^𝑞−1×𝑆𝑡₂−𝑠−𝑆𝑡₁−𝑠𝐺𝑠,𝑧_𝑠+𝑦_𝑠+𝑑𝑠_𝑡10𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1×𝑆𝑡₁−𝑠𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝑠+𝑀₃^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝑆𝑡₂𝑢(𝑠+−𝑠‖)‖𝑑𝑠_𝑡2𝑡1𝑡₂−𝑠^𝑞−1𝑆𝑡₂−𝑠𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝑠.(35) Noting that ‖𝑢(𝑠)‖≤𝑀₄𝑥₁+𝑀₁𝜙(0)+𝐹0,𝜙+𝐹𝑎,𝑥_𝑎+^𝑎₀(𝑎−𝑠)^𝑞−1𝑆(𝑎−𝑠)𝐹𝑠,𝑥_𝑠+𝑑𝑠_𝑎0(𝑎−𝑠)^𝑞−1𝑆(𝑎−𝑠)𝐺𝑠,𝑥_𝑠𝑑𝑠≤𝑀₄𝑥₁+𝑀₁𝜙(0)+𝐹+𝐹0,𝜙𝑎,𝑥_𝑎+𝑀₅𝐿₁𝑘𝐾_𝑎+𝐶+𝑁+1_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝑘𝐾_𝑎+𝑁+1+𝑀₂1−𝑞₁𝑞−𝑞₁𝑎^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}×𝑔_{𝑘𝐾𝑎+𝑁𝐿11/𝑞[0,𝑎]},_𝑡0(𝑡−𝑠)^𝑞−1𝐺𝑠,𝑧_𝑘,𝑠+𝑦_𝑠≤𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑔_{𝑘𝐾𝑎+𝑁}(𝑠)≤𝑑𝑠_𝑎0(𝑎−𝑠)^𝑞−1𝑔_{𝑘𝐾𝑎+𝑁}(𝑠)≤𝑑𝑠1−𝑞₁𝑞−𝑞₁𝑎^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}𝑔_{𝑘𝐾𝑎+𝑁𝐿11/𝑞[0,𝑎]}.(36) We see that ‖𝑄₂𝑧(𝑡₂)−𝑄₂𝑧(𝑡₁)‖ tends to zero independently of 𝑧∈𝐵_𝑘 as 𝑡₂→𝑡₁ since for 𝑡∈[0,𝑎] and any bounded subsets 𝐷⊂𝑋, 𝑡→{𝑆(𝑡)𝑥:𝑥∈𝐷} is equicontinuous.
Hence, 𝑄₂ maps 𝐵_𝑘 into an equicontinuous family functions.
It remains to prove that 𝑉(𝑡)={(𝑄₂𝑧)(𝑡):𝑧∈𝐵_𝑘} is relatively compact in 𝑋. let 0≤𝑡≤𝑎 be fixed, 0<𝜖<𝑡, for 𝑧∈𝐵_𝑘, we define Π=𝑄₂𝐵_𝑘 and Π(𝑡)={𝑄₂𝑧(𝑡)∣𝑧∈𝐵_𝑘}, for 𝑡∈[0,𝑎].
Clearly, Π(0)={𝑄₂𝑧(0)∣𝑧∈𝐵_𝑘}={0} is compact, and hence, it is only to consider 0<𝑡≤𝑎. For each ℎ∈(0,𝑡),𝑡∈(0,𝑎], arbitrary 𝛿>0, define Π_ℎ,𝛿(𝑡)=𝑄_2,ℎ,𝛿𝑧(𝑡)∣𝑧∈𝐵_𝑘,(37) where 𝑄_2,ℎ,𝛿𝑧(𝑡)=₀^{𝑡−ℎ∞}_𝛿(𝑡−𝑠)^𝑞−1𝜃𝜉_𝑞(𝜃)𝑞𝑇(𝑡−𝑠)^𝑞𝜃×𝐶𝑢(𝑠)+𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝜃𝑑𝑠.(38) Then the sets {𝑄_2,ℎ,𝛿𝑧(𝑡)∣𝑧∈𝐵_𝑘} are relatively compact in 𝑋 since the condition (𝐻₄). It comes from the following inequalities: 𝑄₂𝑧(𝑡)−𝑄_2,ℎ,𝛿𝑧(𝑡=_𝑡0∞0)(𝑡−𝑠)^𝑞−1𝜃𝜉_𝑞(𝜃)𝑞𝑇(𝑡−𝑠)^𝑞𝜃×𝐶𝑢(𝑠)+𝐺𝑠,𝑧_𝑠+𝑦_𝑠−𝑑𝜃𝑑𝑠_{0𝑡−ℎ∞𝛿}(𝑡−𝑠)^𝑞−1𝜃𝜉_𝑞(𝜃)𝑞𝑇(𝑡−𝑠)^𝑞𝜃×𝐶𝑢(𝑠)+𝐺𝑠,𝑧_𝑠+𝑦_𝑠=𝑑𝜃𝑑𝑠_𝑡0𝛿0(𝑡−𝑠)^𝑞−1𝜃𝜉_𝑞(𝜃)𝑞𝑇(𝑡−𝑠)^𝑞𝜃×𝐶𝑢(𝑠)+𝐺𝑠,𝑧_𝑠+𝑦_𝑠+𝑑𝜃𝑑𝑠_𝑡0∞𝛿(𝑡−𝑠)^𝑞−1𝜃𝜉_𝑞(𝜃)𝑞𝑇(𝑡−𝑠)^𝑞𝜃×𝐶𝑢(𝑠)+𝐺𝑠,𝑧_𝑠+𝑦_𝑠−𝑑𝜃𝑑𝑠_{0𝑡−ℎ∞𝛿}(𝑡−𝑠)^𝑞−1𝜃𝜉_𝑞(𝜃)𝑞𝑇(𝑡−𝑠)^𝑞𝜃−₀^{𝑡−ℎ∞}_𝛿(𝑡−𝑠)^𝑞−1𝜃𝜉_𝑞(𝜃)𝑞𝑇(𝑡−𝑠)^𝑞𝜃×𝐶𝑢(𝑠)+𝐺𝑠,𝑧_𝑠+𝑦_𝑠𝑑𝜃𝑑𝑠≤𝑀𝑀₃‖𝑢(𝑠)‖𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑑𝑠^𝛿₀𝜃𝜉_𝑞(𝜃)𝑑𝜃+𝑀𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑔_{𝑘𝐾𝑎+𝑁}(𝑠)𝑑𝑠^𝛿₀𝜃𝜉_𝑞(𝜃)𝑑𝜃+𝑀𝑀₃‖𝑢(𝑠)‖^𝑡_𝑡−ℎ(𝑡−𝑠)^𝑞−1𝑑𝑠⋅𝑞^∞_𝛿𝜃𝜉_𝑞(𝜃)𝑑𝜃+𝑀^𝑡_𝑡−ℎ(𝑡−𝑠)^𝑞−1𝑔_{𝑘𝐾𝑎+𝑁}(𝑠)𝑑𝑠⋅𝑞^∞_𝛿𝜃𝜉_𝑞(𝜃)≤𝑑𝜃𝑀𝑀₃𝑀₄𝑥₁+𝑀₁𝜙(0)+𝐹+𝐹0,𝜙𝑎,𝑥_𝑎+𝑀₅𝐿₁𝑘𝐾_𝑎+𝐶+𝑁+1_1−𝛽Γ1+𝛽𝑎^𝑞𝛽𝐿𝛽Γ1+𝑞𝛽¹𝑘𝐾_𝑎+𝑁+1+𝑀1−𝑞₁𝑞−𝑞₁𝑎^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}×𝑔_{𝑘𝐾𝑎+𝑁𝐿11/𝑞[0,𝑎]}𝑎^𝑞+𝑀1−𝑞₁𝑞−𝑞₁𝑎^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}𝑔_{𝑘𝐾𝑎+𝑁𝐿11/𝑞[0,𝑎]}𝑞×^𝛿₀𝜃𝜉_𝑞(𝜃)𝑑𝜃+𝑀1−𝑞₁𝑞−𝑞₁ℎ^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}×𝑔_{𝑘𝐾𝑎+𝑁𝐿11/𝑞[0,𝑎]}𝑞^∞₀𝜃𝜉_𝑞(𝜃)𝑑𝜃.(39)
Therefore, Π(𝑡)={𝑄₂𝑧(𝑡)∣𝑧∈𝐵_𝑘} is relatively compact in 𝑋 for all 𝑡∈[0,𝑎].
Thus, the continuity of 𝑄₂ and relatively compact of {𝑄₂𝑧(𝑡)∣𝑧∈𝐵_𝑘} imply that 𝑄₂ is a completely continuous operator.
These arguments enable us to conclude that 𝑄=𝑄₁+𝑄₂ is a condense mapping on 𝐵_𝑘, and by the fixed point theorem of Sadovskii there exists a fixed point 𝑧(⋅) for 𝑄 on 𝐵_𝑘. In fact, by Step  1–Step  3 and Lemma 3, we can conclude that 𝑄=𝑄₁+𝑄₂ is continuous and takes bounded sets into bounded sets. Meanwhile, it is easy to see 𝛼(𝑄₂(𝐵_𝑘))=0 since 𝑄₂(𝐵_𝑘) is relatively compact. Since 𝑄₁(𝐵_𝑘))⊆𝐵_𝑘 and 𝛼(𝑄₂(𝐵_𝑘))=0, we can obtain 𝛼(𝑄(𝐵_𝑘))≤𝛼(𝑄₁(𝐵_𝑘))+𝛼(𝑄₂(𝐵_𝑘))≤𝛼(𝐵_𝑘) for every bounded set 𝐵_𝑘 of 𝑋 with 𝛼(𝐵_𝑘)>0, that is, 𝑄=𝑄₁+𝑄₂ is a condense mapping on 𝐵_𝑘. If we define 𝑥(𝑡)=𝑧(𝑡)+𝑦(𝑡),−∞<𝑡≤𝑎, it is easy to see that 𝑥(⋅) is a mild solution of (1) satisfying 𝑥₀=𝜙,𝑥(𝑎)=𝑥₁. Then the proof is completed.

Remark 8. 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 fractional differential equations always have been used in the system model. So we can also consider the complete controllability for (1) with impulses.

Remark 9. Since the complete controllability steers the systems to arbitrary final state while approximate controllability steers the system to arbitrary small neighborhood of final state. In view of the definition of approximate controllability in [28], we can deduce that the considered systems (1) is also approximate controllable on the interval [0,𝑎].

4. An Example

As an application of Theorem 7, we consider the following system: 𝜕^2/3𝜕𝑡^2/3𝑧(𝑡,𝑥)+^𝑡_−∞^𝜋₀𝑏𝑧−𝜕𝑠−𝑡,𝑦,𝑥𝑠,𝑦𝑑𝑦𝑑𝑠₂𝜕𝑥²𝑧()(𝑡)𝑡,𝑥=𝐶𝑢+𝑎₀(𝑥)𝑧()+𝑡,𝑥_𝑡−∞𝑎₁(𝑠,𝑡)𝑧(𝑠,𝑥)𝑑𝑠+𝑎₂(),𝑧()()()()𝑡,𝑥0≤𝑡≤𝑎,0≤𝑥≤𝜋,𝑡,0=𝑧𝑡,𝜋=0,𝑧𝜗,𝑥=𝜙𝜗,𝑥,𝜗≤0.(40)

To write system (40) to the form of (1), let 𝑋=𝐿²([0,𝜋]) and 𝐴 defined by 𝐴𝑓=−𝑓^′′ with domain 𝐷(𝐴)={𝑓(⋅)∈𝑋:𝑓,𝑓^′absolutelycontinuous,𝑓^′′∈𝑋,𝑓(0)=𝑓(𝜋)=0}, the −𝐴 generates a uniformly bounded analytic semigroup which satisfies the condition (𝐻₀). Furthermore, 𝐴 has a discrete spectrum, the eigenvalues are −𝑛², 𝑛∈𝑁, with the corresponding normalized eigenvectors 𝑧_𝑛(𝑥)=(2/𝜋)^1/2sin(𝑛𝑥). Then the following properties hold.

(i) If 𝐴∈𝐷(𝐴), then 𝐴𝑓=^∞_𝑛=1𝑛²𝑓,𝑧_𝑛𝑧_𝑛.(41)

(ii) For each 𝑓∈𝑋, 𝐴^−1/2𝑓=^∞_𝑛=11𝑛𝑓,𝑧_𝑛𝑧_𝑛.(42) In particular, ‖𝐴^−1/2‖=1.

(iii) The operator 𝐴^1/2 is given by 𝐴^1/2𝑓=^∞_𝑛=1𝑛𝑓,𝑧_𝑛𝑧_𝑛(43) on the space 𝐷(𝐴^1/2)={𝑓(⋅)∈𝑋,𝐴^1/2𝑓∈𝑋}.

Here we take the phase space 𝐵=𝐶₀×𝐿²(𝑔,𝑋), which contains all classes of functions 𝜙:(−∞,0]→𝑋 such that 𝜙 is Lebesgue measurable and 𝑔(⋅)‖𝜙(⋅)‖² is Lebesgue integrable on (−∞,0) where 𝑔:(−∞,0)→𝑅 is a positive integrable function. The seminorm in 𝐵 is defined by 𝜙_𝐵=𝜙(0)+⁰_−∞𝑔(𝜗)𝜙(𝜗)²𝑑𝜗^1/2.(44) From [41], under some conditions 𝐵 is a phase space verifying (A), (B), (C), and in this case 𝐾(𝑡)=1+(∫⁰_−𝑡𝑔(𝜗)𝑑𝜗)^1/2 (see [41] for the details).

We assume the following conditions hold.(a) The function 𝑏 is measurable and ∫^𝜋₀∫⁰_−∞∫^𝜋₀(𝑏²(𝜗,𝑦,𝑥)/𝑔(𝜗))𝑑𝑦𝑑𝜗𝑑𝑥<∞.(b) The function (𝜕/𝜕𝑥)𝑏(𝜗,𝑦,𝑥) is measurable, 𝑏(𝜗,𝑦,0)=𝑏(𝜗,𝑦,𝜋)=0 and let 𝑁₁=∫^𝜋₀∫⁰_−∞∫^𝜋₀(1/𝑔(𝜗))((𝜕/𝜕𝑥)𝑏(𝜗,𝑦,𝑥))²𝑑𝑦𝑑𝜗𝑑𝑥<∞.(c) The function 𝑎₀(⋅)∈𝐿^∞([0,𝜋]), 𝑎(⋅) is measurable, with ∫⁰_−∞(𝑎²₁(𝜗))/𝑔(𝜗)𝑑𝜗<∞, the function 𝑎₂(𝑡,⋅)∈𝐿²([0,𝜋]) for each 𝑡≥0 is measurable in 𝑡.(d)The function 𝜙 defined by 𝜙(𝜗)(𝑥)=𝜙(𝜗,𝑥) belongs to 𝐵.(e)The linear operator 𝑊:𝑈→𝑋 is defined by 𝑊𝑢=^𝑎₀(𝑎−𝑠)^−1/3𝑆(𝑎−𝑠)𝐶𝑢(𝑠)𝑑𝑠(45)

and has a bounded invertible operator 𝑊⁻¹ defined 𝐿²([0,𝑎]);𝑈)/ker𝑊.

We define 𝐹,𝐺:[0,𝑎]×𝐵→𝑋 by 𝐹(𝑡,𝜙)=𝑍₁(𝜙) and 𝐺(𝑡,𝜙)=𝑍₂(𝜙)+ℎ(𝑡), where 𝑍₁𝜙=⁰_−∞^𝜋₀𝑏𝜙()𝑍𝜗,𝑦,𝑥𝜗,𝑥𝑑𝑦𝑑𝜗,₂𝜙=𝑎₀(𝑥)𝜙(0,𝑥)+⁰_−∞𝑎₁(𝜗)𝜙()ℎ(𝑡)𝜗,𝑥𝑑𝜗,=𝑎₂(𝑡,⋅).(46) From (a) and (c) it is clear that 𝑍₁ and 𝑍₂ are bounded linear operators on 𝐵. Furthermore, 𝑍₁(𝜙)∈𝐷(𝐴^1/2), and ‖𝐴^1/2𝑍₁‖≤𝑁₁. In fact, from the definition of 𝑍₁ and (b) it follows that ⟨𝑍₁(𝜙),𝑧_𝑛⟩=(1/𝑛)(2/𝜋)^1/2⟨𝑍(𝜙),con(𝑛𝑥)⟩, where 𝑍(𝜙)=∫⁰_−∞∫^𝜋₀(𝜕/𝜕𝑥)𝑏(𝜗,𝑦,𝑥)𝜙(𝜗,𝑥)𝑑𝜗. From (b) we know that 𝑍:𝐵→𝑋 is a bounded linear operator with ‖𝑍‖≤𝑁₁. Hence ‖𝐴^1/2𝑍₁(𝜙)‖=‖𝑍(𝜙)‖, which implies the assertion. Therefore, from Theorem 7, the system (40) is completely controllable on [0,𝑎] under the above assumptions.

5. Conclusion

In this paper, by using the uniformly boundedness, analyticity, and equicontinuity of characteristic solution operators and the Sadovskii fixed point theorem, we obtained the complete controllability of the abstract neutral fractional differential systems with unbounded delay in a Banach space. It shows that the compactness of the characteristic solution operators can be weakened to equicontinuity. Our theorem guarantees the effectiveness of complete controllability results under some weakly compactness conditions.

Acknowledgments

The authors are highly grateful for the referee’s careful reading and comments on this paper. The present Project is supported by NNSF of China Grant no. 11271087, no. 61263006, and Guangxi Scientific Experimental (China-ASEAN Research) Centre no. 20120116.