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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 529025, 11 pages http://dx.doi.org/10.1155/2013/529025 Research Article Complete Controllability of Fractional Neutral Differential Systems in Abstract Space Fang Wang,1 Zhen-hai Liu,2 and Jing Li3 School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan Province 410076, China School of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, Guangxi Province 530006, China Changsha University of Science and Technology, Changsha, Hunan, China Correspondence should be addressed to Fang Wang; wangfang811209@tom.com Received 10 September 2012; Revised November 2012; Accepted 10 November 2012 Academic Editor: Yong Zhou Copyright © 2013 Fang Wang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited 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 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, Hern´andez 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 Măoch 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 𝜉𝑞 Then also without involving the compactness of characteristic solution operators, we obtain the controllability of the Abstract and Applied Analysis following abstract neutral fractional differential systems with unbounded delay: 𝑐 𝑥 (𝑡) = 𝜑 (𝑡) [𝜙 (0) + 𝐹 (0, 𝜙)] − 𝐹 (𝑡, 𝑥𝑡 ) 𝑞 𝐷𝑡 (𝑥 (𝑡) + 𝐹 (𝑡, 𝑥𝑡 )) + 𝐴𝑥 (𝑡) = 𝐶𝑢 (𝑡) + 𝐺 (𝑡, 𝑥𝑡 ) , 𝑡 ∈ (0, 𝑎] , 𝑥0 (𝜗) = 𝜙 (𝜗) ∈ 𝐵, function 𝐴𝑆(𝑡 − 𝑠)𝐹(𝑠, 𝑥𝑠 ), 𝑠 ∈ [0, 𝑡] is integrable and satisfies the following integral equation: (1) 𝑡 − ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑥𝑠 ) 𝑑𝑠 𝜗 ∈ (−∞, 0] , (4) 𝑡 where the state variable 𝑥(⋅) takes values in Banach space 𝑋, 𝑥𝑡 : (−∞, 0] → 𝑋, 𝑥𝑡 (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 + ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑥𝑠 )] 𝑑𝑠, where 𝜑(𝑡) and 𝑆(𝑡) are called characteristic solution operators and are given by ∞ 𝜑 (𝑡) = ∫ 𝜉𝑞 (𝜃) 𝑇 (𝑡𝑞 𝜃) 𝑑𝜃, ∞ (5) 𝑞 𝑆 (𝑡) = 𝑞 ∫ 𝜃𝜉𝑞 (𝜃) 𝑇 (𝑡 𝜃) 𝑑𝜃, and for 𝜃 ∈ (0, ∞), 𝜉𝑞 (𝜃) = (1/𝑞)𝜃−1−1/𝑞 𝑤𝑞 (𝜃−1/𝑞 ) ≥ 0, 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‖𝐿𝑝 ([0,𝑎],𝑅+ ) to denote the 𝐿𝑝 ([0, 𝑎], 𝑅+ ) of norm of 𝑓 whenever 𝑓 ∈ 𝐿𝑝 ([0, 𝑎], 𝑅+ ) for some 𝑝 with ≤ 𝑝 < ∞ Let 𝐿𝑝 ([0, 𝑎], 𝑅+ ) denote the Banach space of functions 𝑓: [0, 𝑎] → 𝑋 which are Bochner integrable normed by ‖𝑓‖𝐿𝑝 ([0,𝑎],𝑅+ ) Let −𝐴 : 𝐷(𝐴) → 𝑋 be the infinitesimal generator of a uniformly bounded analytic semigroup 𝑇(𝑡) Let ∈ 𝜌(𝐴), then it is possible to define the fractional power 𝐴𝛼 , for < 𝛼 ≤ 1, as a closed linear operator on its domain 𝐷(𝐴𝛼 ) Furthermore, the subspace 𝐷(𝐴𝛼 ) is dense in 𝑋 and the expression ‖𝑥‖𝛼 = 𝐴𝛼 𝑥 , 𝛼 𝑥 ∈ 𝐷 (𝐴 ) (2) defines a norm on 𝐷(𝐴𝛼 ) Hereafter we denote by 𝑋𝛼 the Banach space 𝐷(𝐴𝛼 ) normed with ‖𝑥‖𝛼 Then for each < 𝛼 ≤ 1, 𝑋𝛼 the Banach space, and ‖𝑥‖𝛼 → ‖𝑥‖𝛽 for < 𝛽 < 𝛼 ≤ 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 ≥ such that ‖𝑇(𝑡)‖ ≤ 𝑀 for all 𝑡 ≥ (b) for any 𝛼 ≥ 0, there exists a positive constant 𝐶𝛼 such that 𝐶 𝛼 𝐴 𝑇 (𝑡) ≤ 𝛼𝛼 , 𝑡 < 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 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 ≤ 𝑡 ≤ 𝑎, the 𝑤𝑞 (𝜃) = Γ (𝑛𝑞 + 1) 1∞ ∑ (−1)𝑛−1 𝜗−𝑞𝑛−1 sin (𝑛𝜋𝑞) 𝜋 𝑛=1 𝑛! (6) Here, 𝜉𝑞 is a probability density function defined on (0, ∞), ∞ that is, 𝜉𝑞 (𝜃) ≥ 0, 𝜃 ∈ (0, ∞), and ∫0 𝜉𝑞 (𝜃)𝑑𝜃 = Definition (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 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‖ ≤ 𝑞𝑀0 ‖𝑥‖ ; Γ (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 𝑞 𝑞 equicontinuous if ‖𝑇(𝑡2 𝜃)𝑥 − 𝑇(𝑡1 𝜃)𝑥‖ → with respect to 𝑥 ∈ 𝐷 as 𝑡2 → 𝑡1 for each fixed 𝜃 ∈ [0, ∞] The proof of Lemma 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 Abstract and Applied Analysis Definition Let 𝑋 be a Banach space and Ω𝑋 the bounded set of 𝑋 The Kuratowski’s measure of noncompactness is the map 𝛼 : Ω𝑋 → [0, ∞) defined by 𝑛 𝛼 (𝐷) = inf {𝑑 > : 𝐷 ⊆ ⋃ 𝐷𝑖 , diam (𝐷𝑖 ) ≤ 𝑑} , (8) 𝑖=1 𝑞 (𝐻1 ) 𝐹: [0, 𝑎] × 𝐵 → 𝑋 is continuous function, and there exists a constant 𝛽 ∈ (0, 1) and 𝐿, 𝐿 > such that the function 𝐹 is 𝑋𝛽 -valued and satisfies the Lipschitz condition: 𝛽 𝐴 𝐹 (𝑠1 , 𝜙1 ) − 𝐴𝛽 𝐹 (𝑠2 , 𝜙2 ) ≤ 𝐿 (𝑠1 − 𝑠2 + 𝜙1 − 𝜙2 𝐵 ) , here 𝐷 ∈ Ω𝑋 One will use the following basic properties of the 𝛼 measure and Sadovskii’s fixed point theorem here (see [37– 39]) Lemma Let 𝐷1 and 𝐷2 be two bounded sets of a Banach space 𝑋 Then (iii) 𝛼(𝐷1 + 𝐷2 ) ≤ 𝛼(𝐷1 ) + 𝛼(𝐷2 ) Lemma (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) > If 𝑁(𝑆) ⊂ 𝑆 for a convex closed and bounded set 𝑆 of 𝑋, then 𝑁 has a fixed point in 𝑆 𝛽 𝐴 𝐹 (𝑡, 𝜙) ≤ 𝐿 (𝜙𝐵 + 1) (A) If 𝑥 ∈ (−∞, 𝜎 + 𝑎) → 𝑋, 𝑎 > 0, is continuous on [𝜎, 𝜎 + 𝑎] and 𝑥𝜎 ∈ 𝐵, then for every 𝑡 ∈ [𝜎, 𝜎 + 𝑎] the following conditions hold: (i) 𝑥𝑡 is in 𝐵; (ii) ‖𝑥(𝑡)‖ ≤ 𝐻‖𝑥𝑡 ‖𝐵 ; (iii) ‖𝑥𝑡 ‖𝐵 ≤ 𝐾(𝑡 − 𝜎) sup{‖𝑥(𝑡)‖ : 𝜎 ≤ 𝑠 ≤ 𝑡} + 𝑀(𝑡 − 𝜎)‖𝑥𝜎 ‖𝐵 Here 𝐻 ≥ 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), 𝑥𝑡 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 (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 𝑔𝑘 ∈ 𝐿1/𝑞1 ([0, 𝑎]), < 𝑞1 < 𝑞 such that Complete Controllability Result To study the system (1), we assume the function 𝑥𝑡 represents the history of the state from −∞ up to the present time 𝑡 and 𝑥𝑡 : (−∞, 0] → 𝑋, 𝑥𝑡 (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: (9) for ≤ 𝑠1 , 𝑠2 ≤ 𝑎, 𝜙1 , 𝜙2 ∈ 𝐵, and the inequality (i) 𝛼(𝐷1 ) = if and only if 𝐷1 is relatively compact; (ii) 𝛼(𝐷1 ) ≤ 𝛼(𝐷2 ) if 𝐷1 ⊆ 𝐷2 ; 𝑞 𝐷 ⊂ 𝑋 and 𝑥 ∈ 𝐷, ‖𝑇(𝑡2 𝜃)𝑥 − 𝑇(𝑡1 𝜃)𝑥‖ → as 𝑡2 → 𝑡1 for each fixed 𝜃 ∈ [0, ∞] sup 𝐺 (𝑡, 𝜙) ≤ 𝑔𝑘 (𝑡) , ‖𝜙‖𝐵 ≤𝑘 1 lim inf 𝑔𝑘 𝐿1/𝑞1 [0,𝑎] = 𝛾 < ∞ 𝑘 (11) (𝐻3 ) The linear operator 𝐶 is bounded, 𝑊 from 𝑈 into 𝑋 is defined by 𝑎 𝑊𝑢 = ∫ (𝑎 − 𝑠)𝑞−1 𝑆 (𝑎 − 𝑠) 𝐶𝑢 (𝑠) 𝑑𝑠 (12) and there exists a bounded invertible operator 𝑊−1 defined on 𝐿2 ([0, 𝑎]; 𝑈)/ ker 𝑊 and there exist two positive constants 𝑀3 , 𝑀4 > such that ‖𝐵‖𝐿 𝑏 (𝑈,𝑋) ≤ 𝑀3 , ‖𝑊−1 ‖𝐿 𝑏 (𝑋,𝐿2 ([0,𝑎],𝑈)/ ker 𝑊) ≤ 𝑀4 (𝐻4 ) For all bounded subsets 𝐷 ⊆ 𝑋, the set Πℎ,𝛿 (𝑡) = {𝑄2,ℎ,𝛿 𝑧 (𝑡) | 𝑧 ∈ 𝐷} , (13) where 𝑄2,ℎ,𝛿 𝑧 (𝑡) = ∫ 𝑡−ℎ ∞ ∫ (𝑡 − 𝑠)𝑞−1 𝜃𝜉𝑞 (𝜃) 𝑞𝑇 ((𝑡 − 𝑠)𝑞 𝜃) 𝛿 (14) × [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝜃 𝑑𝑠 is relatively compact in 𝑋 for arbitrary ℎ ∈ (0, 𝑡) and 𝛿 > 4 Abstract and Applied Analysis Theorem Let 𝜙 ∈ 𝐵 If the assumptions (𝐻0 )–(𝐻4 ) are satisfied, then the system (1) is controllable on interval [0, 𝑎] provided that 𝑀5 𝐿𝐾𝑎 + 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) 𝐿𝐾𝑎 < 1, (15) 𝑧 (𝑡) = 𝜑 (𝑡) 𝐹 (0, 𝜙) − 𝐹 (𝑡, 𝑧𝑡 + 𝑦𝑡 ) 𝑡 − ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝑠 (1 + 𝑎𝑀2 𝑀3 𝑀4 ) − 𝑞1 (𝑞−𝑞1 )/(1−𝑞1 ) 1−𝑞1 )𝑎 ) 𝐾𝑎 𝛾 × (𝐿 𝑀5 𝐾𝑎 + 𝑀2 (( 𝑞 − 𝑞1 + If 𝑥(⋅) satisfies (18), we can decompose it as 𝑥(𝑡) = 𝑧(𝑡) + 𝑦(𝑡), ≤ 𝑡 ≤ 𝑎, which implies 𝑥𝑡 = 𝑧𝑡 + 𝑦𝑡 for every ≤ 𝑡 ≤ 𝑎 and the function 𝑧(⋅) satisfies 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) 𝑡 + ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )] 𝑑𝑠 (21) Moreover 𝑧0 = Let 𝑄 be the operator on 𝐶([0, 𝑎], 𝑋) defined by 𝐿 𝐾𝑎 ) < 1, (16) −𝛽 𝑄𝑧 (𝑡) = 𝜑 (𝑡) 𝐹 (0, 𝜙) − 𝐹 (𝑡, 𝑧𝑡 + 𝑦𝑡 ) where 𝑀5 = ‖𝐴 ‖, 𝐾𝑎 = sup{𝐾(𝑡) : ≤ 𝑡 ≤ 𝑎} and 𝐶1−𝛽 is from (3) − ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝑠 Proof Using the assumption (𝐻3 ), for arbitrary function 𝑥(⋅) define the control + ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )] 𝑑𝑠 𝑢 (𝑡) = 𝑊−1 [𝑥1 − 𝜑 (𝑎) (𝜙 (0) + 𝐹 (0, 𝜙)) + 𝐹 (𝑎, 𝑥𝑎 ) + ∫ (𝑎 − 𝑠)𝑞−1 𝐴𝑆 (𝑎 − 𝑠) 𝐹 (𝑠, 𝑥𝑠 ) 𝑑𝑠 𝑎 − ∫ (𝑎 − 𝑠)𝑞−1 𝑆 (𝑎 − 𝑠) 𝐺 (𝑠, 𝑥𝑠 ) 𝑑𝑠] (𝑡) (17) It will be shown that when using this control the operator 𝑃 defined by 𝑃𝑥 (𝑡) = 𝜑 (𝑡) [𝜙 (0) + 𝐹 (0, 𝜙)] − 𝐹 (𝑡, 𝑥𝑡 ) 𝑡 − ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑥𝑠 ) 𝑑𝑠 0 𝑡 (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 𝑎 𝑡 (18) 𝐵𝑘 = {𝑧 ∈ 𝐶 ([0, 𝑎] : 𝑋) : 𝑧 (0) = 0, ‖𝑧 (𝑡)‖ ≤ 𝑘, ≤ 𝑡 ≤ 𝑎} , (23) then for each 𝑘, 𝐵𝑘 is clearly a bounded closed convex set in 𝐶([0, 𝑎] : 𝑋) Since by (3) and (10) the following relation holds: 1−𝛽 𝛽 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) ≤ 𝐴 𝑆 (𝑡 − 𝑠) 𝐴 𝐹 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) ≤ 𝑡 + ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑥𝑠 )] 𝑑𝑠 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 𝜑 (𝑡) 𝜙 (0) , 𝑡 ∈ [0, 𝑎] , 𝑦 (𝑡) = { 𝜙 (𝑡) , −∞ < 𝑡 < 0, (19) then 𝑦0 = 𝜙 and the map 𝑡 → 𝑦𝑡 is continuous We can assume 𝑁 = sup{‖𝑦𝑡 ‖ : ≤ 𝑡 ≤ 𝑎} For each 𝑧 ∈ 𝐶([0, 𝑎] : 𝑋), 𝑧(0) = We can denote by 𝑧 the function defined by 𝑧 (𝑡) , 𝑧 (𝑡) = { 0, ≤ 𝑡 ≤ 𝑎, −∞ < 𝑡 < (20) ≤ ≤ 𝐶1−𝛽 𝑞Γ (1 + 𝛽) (𝑡 − 𝑠)−(1−𝛽)𝑞 Γ (1 + 𝑞𝛽) × 𝐿 (𝑧𝑠 + 𝑦𝑠 𝐵 + 1) 𝐶1−𝛽 𝑞Γ (1 + 𝛽) (𝑡 − 𝑠)−(1−𝛽)𝑞 Γ (1 + 𝑞𝛽) × 𝐿 (𝑧𝑠 𝐵 + 𝑦𝑠 𝐵 + 1) 𝐶1−𝛽 𝑞Γ (1 + 𝛽) Γ (1 + 𝑞𝛽) (𝑡 − 𝑠)−(1−𝛽)𝑞 × 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) , (24) then from Bocher’s theorem [42] it follows that 𝐴𝑆(𝑡 − 𝑠)𝐹(𝑠, 𝑧𝑠 + 𝑦𝑠 ) is integrable on [0, 𝑎], so 𝑄 is well defined on 𝐵𝑘 In order to make the following process clear we divide it into several steps Step We claim that there exists a positive number 𝑘 such that 𝑄(𝐵𝑘 ) ⊆ 𝐵𝑘 Abstract and Applied Analysis + 𝐹 (𝑎, 𝑧𝑘,𝑎 + 𝑦𝑎 ) If it is not true, then for each positive number 𝑘, there is a function 𝑧𝑘 (⋅) ∈ 𝐵𝑘 , but 𝑄𝑧𝑘 ∉ 𝐵𝑘 , that is, ‖𝑄𝑧𝑘 (𝑡)‖ > 𝑘 for some 𝑡 ∈ [0, 𝑎] However, on the other hand, we have 𝑎 + ∫ (𝑎 − 𝜏)𝑞−1 𝑘 < 𝑄𝑧𝑘 (𝑡) 𝑎 + ∫ 𝑀2 (𝑎 − 𝜏)𝑞−1 = 𝜑 (𝑡) 𝐹 (0, 𝜙) − 𝐹 (𝑡, 𝑧𝑘,𝑡 + 𝑦𝑡 ) × 𝐺 (𝜏, 𝑧𝑘,𝜏 + 𝑦𝜏 ) 𝑑𝜏} (𝑠) 𝑑𝑠 𝑡 − ∫ (𝑡 − 𝑠)𝑞−1 𝑡 + ∫ 𝑀2 (𝑡 − 𝑠)𝑞−1 𝐺 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 ) 𝑑𝑠, 0 × 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 ) 𝑑𝑠 𝑡 + ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) × [𝐶𝑢𝑘 (𝑠) + 𝐺 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 )] 𝑑𝑠 (25) where 𝑢𝑘 is the corresponding control of 𝑥𝑘 , 𝑥𝑘 = 𝑧𝑘 +𝑦 Since 𝑡 𝑞−1 ∫ (𝑡 − 𝑠) 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 ) 𝑑𝑠 𝑡 ≤ ∫ (𝑡 − 𝑠)𝑞−1 𝐴1−𝛽 𝑆 (𝑡 − 𝑠) 𝐴𝛽 𝐹 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 ) 𝑑𝑠 = 𝜑 (𝑡) 𝐹 (0, 𝜙) − 𝐹 (𝑡, 𝑧𝑘,𝑡 + 𝑦𝑡 ) 𝑡 ≤ − ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) 𝐶1−𝛽 𝑞Γ (1 + 𝛽) Γ (1 + 𝑞𝛽) 𝑡 × 𝐹 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 ) 𝑑𝑠 𝑡 + ∫ (𝑡 − 𝑠) 𝑞−1 × ∫ (𝑡 − 𝑠)𝑞−1 (𝑡 − 𝑠)−(1−𝛽)𝑞 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) 𝑆 (𝑡 − 𝑠) ≤ −1 × [𝐶𝑊 × 𝐴𝑆 (𝑎 − 𝜏) 𝐹 (𝜏, 𝑧𝑘,𝜏 + 𝑦𝜏 ) 𝑑𝜏 {𝑥1 − 𝜑 (𝑎) [𝜙 (0) + 𝐹 (0, 𝜙)] + 𝐹 (𝑎, 𝑧𝑘,𝑎 + 𝑦𝑎 ) 𝑎 + ∫ (𝑎 − 𝜏)𝑞−1 𝐴𝑆 (𝑎 − 𝜏) × 𝐹 (𝜏, 𝑧𝑘,𝜏 + 𝑦𝜏 ) 𝑑𝜏 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) , 𝛽Γ (1 + 𝑞𝛽) 𝐹 (𝑡, 𝑧𝑘,𝑡 + 𝑦𝑡 ) = 𝐴−𝛽 𝐴𝛽 𝐹 (𝑡, 𝑧𝑘,𝑡 + 𝑦𝑡 ) ≤ 𝑀5 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) , 𝑡 ∫ (𝑡 − 𝑠)𝑞−1 𝐺 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 ) 𝑑𝑠 𝑡 𝑎 ≤ ∫ (𝑡 − 𝑠)𝑞−1 𝑔𝑘𝐾𝑎 +𝑁 (𝑠) 𝑑𝑠, − ∫ (𝑎 − 𝜏)𝑞−1 𝑆 (𝑎 − 𝜏) 0 ×𝐺 (𝜏, 𝑧𝑘,𝜏 + 𝑦𝜏 ) 𝑑𝜏} (𝑠) 𝑑𝑠 𝑡 + ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) there holds 𝑘 < 𝑀1 𝐹 (0, 𝜙) + 𝑀5 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) × 𝐺 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 ) ] 𝑑𝑠 ≤ 𝑀1 𝐹 (0, 𝜙) + 𝐹 (𝑡, 𝑧𝑘,𝑡 + 𝑦𝑡 ) 𝑡 + ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 ) 𝑑𝑠 𝑡 + ∫ 𝑀2 𝑀3 𝑀4 × { 𝑥1 + 𝑀1 𝜙 (0) is + 𝐹 (0, 𝜙) (26) + 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) + 𝑎𝑀2 𝑀3 𝑀4 { 𝑥1 + 𝑀1 𝜙 (0) + 𝐹 (0, 𝜙) + 𝑀5 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) + 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) 𝑎 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) +𝑀2 ∫ (𝑎 − 𝜏)𝑞−1 𝑔𝑘𝐾𝑎 +𝑁 (𝜏) 𝑑𝜏} Abstract and Applied Analysis 𝑎 Now, we define operator 𝑄1 and 𝑄2 on 𝐵𝑘 as + 𝑀2 ∫ (𝑎 − 𝑠)𝑞−1 𝑔𝑘𝐾𝑎 +𝑁 (𝑠) 𝑑𝑠 = 𝑀5 𝐿 𝑘𝐾𝑎 (1 + 𝑎𝑀2 𝑀3 𝑀4 ) (𝑄1 𝑧) (𝑡) = 𝜑 (𝑡) 𝐹 (0, 𝜙) − 𝐹 (𝑡, 𝑧𝑡 + 𝑦𝑡 ) 𝑎 + 𝑀2 (1 + 𝑎𝑀2 𝑀3 𝑀4 ) ∫ (𝑎 − 𝑠)𝑞−1 𝑔𝑘𝐾𝑎 +𝑁 (𝑠) 𝑑𝑠 𝑡 − ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝑠, + 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) (1 + 𝑎𝑀2 𝑀3 𝑀4 ) 𝛽Γ (1 + 𝑞𝛽) + 𝑀1 𝐹 (0, 𝜙) + 𝑀5 𝐿 𝑁 + 𝑀5 𝐿 + 𝑎𝑀2 𝑀3 𝑀4 𝑥1 + 𝑎𝑀2 𝑀3 𝑀4 𝜙 (0) + 𝐹 (0, 𝜙) + 𝑀5 𝐿 𝑁𝑎𝑀2 𝑀3 𝑀4 + 𝑀5 𝐿 𝑎𝑀2 𝑀3 𝑀4 = 𝑀∗ + (1 + 𝑎𝑀2 𝑀3 𝑀4 ) 𝑎 × [𝑀5 𝐿 𝐾𝑎 𝑘 + 𝑀2 (∫ (𝑎 − 𝑠)(𝑞−1)/(𝑞−𝑞1 ) 𝑑𝑠) 1−𝑞1 𝑎 × (∫ (𝑔𝑘𝐾𝑎 +𝑁 (𝑠)) 1/𝑞1 + 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) 𝑑𝑠) 𝑞1 𝑡 (𝑄2 𝑧) (𝑡) = ∫ (𝑡 − 𝑠) for all 𝑡 ∈ [0, 𝑎], respectively We prove that 𝑄1 is contraction, while 𝑄2 is completely continuous Step 𝑄1 is contraction Let 𝑧1 , 𝑧2 ∈ 𝐵𝑘 Then, for each 𝑡 ∈ [0, 𝑎], and by axiom (A)-(iii) and (15), we have 𝑄1 𝑧1 (𝑡) − 𝑄1 𝑧2 (𝑡) = 𝜑 (𝑡) 𝐹 (0, 𝜙) − 𝐹 (𝑡, 𝑧1,𝑡 + 𝑦𝑡 ) 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1)] 𝑡 − ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑧1,𝑠 + 𝑦𝑠 ) 𝑑𝑠 = 𝑀 + (1 + 𝑎𝑀2 𝑀3 𝑀4 ) 1−𝑞1 − 𝑞1 (𝑞−𝑞1 )/(1−𝑞1 ) 𝑎 ) 𝑞 − 𝑞1 − 𝜑 (𝑡) 𝐹 (0, 𝜙) + 𝐹 (𝑡, 𝑧2,𝑡 + 𝑦𝑡 ) × 𝑔𝑘𝐾𝑎 +𝑁𝐿1/𝑞1 [0,𝑎] + 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1)] , (27) where 𝑀∗ = 𝑀1 𝐹 (0, 𝜙) + 𝑀5 𝐿 𝑁 + 𝑀5 𝐿 + 𝑎𝑀2 𝑀3 𝑀4 𝑥1 + 𝑎𝑀2 𝑀3 𝑀4 𝜙 (0) + 𝐹 (0, 𝜙) 𝑡 + ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) 𝐹 (𝑠, 𝑧2,𝑠 + 𝑦𝑠 ) 𝑑𝑠 ≤ 𝐹 (𝑡, 𝑧1,𝑡 + 𝑦𝑡 ) − 𝐹 (𝑡, 𝑧2,𝑡 + 𝑦𝑡 ) 𝑡 + ∫ (𝑡 − 𝑠)𝑞−1 𝐴𝑆 (𝑡 − 𝑠) × (𝐹 (𝑠, 𝑧1,𝑠 + 𝑦𝑠 ) − 𝐹 (𝑠, 𝑧2,𝑠 + 𝑦𝑠 )) 𝑑𝑠 ≤ 𝐴−𝛽 𝐴𝛽 𝐹 (𝑡, 𝑧1,𝑡 + 𝑦𝑡 ) − 𝐴−𝛽 𝐴𝛽 𝐹 (𝑡, 𝑧2,𝑡 + 𝑦𝑡 ) 𝑡 + ∫ (𝑡 − 𝑠)𝑞−1 + 𝑀5 𝐿 𝑁𝑎𝑀2 𝑀3 𝑀4 + 𝑀5 𝐿 𝑎𝑀2 𝑀3 𝑀4 (28) Dividing on both sides by 𝑘 and taking the low limit, we get (1 + 𝑎𝑀2 𝑀3 𝑀4 ) − 𝑞1 (𝑞−𝑞1 )/(1−𝑞1 ) 1−𝑞1 )𝑎 ) 𝐾𝑎 𝛾 × (𝐿 𝑀0 𝐾𝑎 + 𝑀2 (( 𝑞 − 𝑞1 + 𝑆 (𝑡 − 𝑠) × [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )] 𝑑𝑠, ∗ × [𝑀5 𝐿 𝐾𝑎 𝑘 + 𝑀2 ( (30) 𝑞−1 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) 𝐿 𝐾𝑎 ) ≥ (29) This contradicts (16) Hence for some positive number 𝑘, 𝑄𝐵𝑘 ⊆ 𝐵𝑘 × 𝐴1−𝛽 𝑆 (𝑡−𝑠) 𝐴𝛽 (𝐹 (𝑠, 𝑧1,𝑠 +𝑦𝑠 )−𝐹 (𝑠, 𝑧2,𝑠 +𝑦𝑠 )) 𝑑𝑠 ≤ 𝑀5 𝐿𝐾𝑎 𝑧1,𝑡 − 𝑧2,𝑡 𝐵 𝑎 + ∫ (𝑡 − 𝑠)𝑞−1 𝐶1−𝛽 𝑞Γ (1 + 𝛽) Γ (1 + 𝑞𝛽) × 𝐿𝑧1,𝑠 − 𝑧2,𝑠 𝐵 𝑑𝑠 (𝑡 − 𝑠)−(1−𝛽)𝑞 ≤ 𝑀5 𝐿𝐾𝑎 sup 𝑧1 (𝑠) − 𝑧2 (𝑠) 0≤𝑠≤𝑎 + 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) 𝐿 𝐾𝑎 sup 𝑧1 (𝑠) − 𝑧2 (𝑠) 0≤𝑠≤𝑎 Abstract and Applied Analysis ≤ (𝑀5 𝐿𝐾𝑎 + 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) Next we prove that the family {𝑄2 𝑧 : 𝑧 ∈ 𝐵𝑘 } is an equicontinuous family of functions To this, let ≤ 𝑡1 < 𝑡2 ≤ 𝑎, then 𝐿 𝐾𝑎 ) × sup 𝑧1 (𝑠) − 𝑧2 (𝑠) 0≤𝑠≤𝑎 (31) 𝑄2 𝑧 (𝑡2 ) − 𝑄2 𝑧 (𝑡1 ) 𝑡2 𝑞−1 = ∫ (𝑡2 − 𝑠) 𝑆 (𝑡2 − 𝑠) 𝐶𝑢 (𝑠) 𝑑𝑠 𝑡2 Thus 𝑞−1 𝑆 (𝑡2 − 𝑠) 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝑠 𝑞−1 𝑆 (𝑡1 − 𝑠) 𝐶𝑢 (𝑠) 𝑑𝑠 + ∫ (𝑡2 − 𝑠) 𝑄1 𝑧1 (𝑡) − 𝑄1 𝑧2 (𝑡) < 𝑧1 − 𝑧2 , 𝑡1 (32) − ∫ (𝑡1 − 𝑠) 𝑡1 and 𝑄1 is contraction Step 𝑄2 is completely continuous Let {𝑧𝑛 } ⊆ 𝐵𝑘 with 𝑧𝑛 → 𝑧 in 𝐵𝑘 , then for each 𝑠 ∈ [0, 𝑎], 𝑧𝑛,𝑠 → 𝑧𝑠 , and by (𝐻1 ) and (𝐻2 )-(i), we have 𝑞−1 − ∫ (𝑡1 − 𝑠) 𝑡1 ≤ 𝑀3 ∫ (𝑡2 − 𝑠) 𝑆 (𝑡1 − 𝑠) 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝑠 (𝑆 (𝑡2 − 𝑠) − 𝑆 (𝑡1 − 𝑠)) 𝑢 (𝑠) 𝑑𝑠 𝑞−1 𝑡1 + 𝑀3 ∫ ((𝑡2 − 𝑠) 𝑞−1 (33) 𝑢𝑛 (𝑠) − 𝑢 (𝑠) → 0, as 𝑛 → ∞ Since ‖𝐺(𝑠, 𝑧𝑛,𝑠 + 𝑦𝑠 ) − 𝐺(𝑠, 𝑧𝑠 + 𝑦𝑠 )‖ ≤ 2𝑔𝑘𝐾𝑎 +𝑁(𝑠) , then by the dominated convergence theorem we have 𝑄2 𝑧𝑛 (𝑡) − 𝑄2 𝑧 (𝑡) 𝑡 = sup ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) 𝐶𝑢𝑛 (𝑠) 𝑑𝑠 0≤𝑡≤𝑎 𝑡 + ∫ (𝑡 − 𝑠) 𝑞−1 ) × 𝑆 (𝑡1 − 𝑠) ‖𝑢 (𝑠)‖ 𝑑𝑠 𝐺 (𝑠, 𝑧𝑛,𝑠 + 𝑦𝑠 ) − 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) → 0, 𝐹 (𝑠, 𝑧𝑛,𝑠 + 𝑦𝑠 ) − 𝐹 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) → 0, 𝑞−1 − (𝑡1 − 𝑠) 𝑡1 𝑞−1 + ∫ (𝑡2 − 𝑠) × (𝑆 (𝑡2 − 𝑠) − 𝑆 (𝑡1 − 𝑠)) 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝑠 𝑡1 + ∫ ((𝑡2 − 𝑠) 𝑞−1 𝑞−1 − (𝑡1 − 𝑠) ) × 𝑆 (𝑡1 − 𝑠) 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝑠 𝑡2 𝑞−1 + 𝑀3 ∫ (𝑡2 − 𝑠) 𝑡1 𝑡2 𝑞−1 + ∫ (𝑡2 − 𝑠) 𝑡1 𝑆 (𝑡2 − 𝑠) ‖𝑢 (𝑠)‖ 𝑑𝑠 𝑆 (𝑡2 − 𝑠) 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝑠 (35) 𝑆 (𝑡 − 𝑠) 𝐺 (𝑠, 𝑧𝑛,𝑠 + 𝑦𝑠 ) 𝑑𝑠 Noting that 𝑡 − ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) 𝐶𝑢 (𝑠) 𝑑𝑠 (34) 𝑡 − ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 ) 𝑑𝑠 𝑡 ≤ ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) (𝐶𝑢𝑛 (𝑠) − 𝐶𝑢 (𝑠)) 𝑑𝑠 𝑡 + ∫ (𝑡 − 𝑠)𝑞−1 𝑆 (𝑡 − 𝑠) × (𝐺 (𝑠, 𝑧𝑛,𝑠 + 𝑦𝑠 ) − 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )) 𝑑𝑠 → 0, as 𝑛 → ∞, that is, 𝑄2 is continuous ‖𝑢 (𝑠)‖ ≤ 𝑀4 [ 𝑥1 + 𝑀1 𝜙 (0) + 𝐹 (0, 𝜙) + 𝐹 (𝑎, 𝑥𝑎 ) 𝑎 + ∫ (𝑎 − 𝑠)𝑞−1 𝑆 (𝑎 − 𝑠) 𝐹 (𝑠, 𝑥𝑠 ) 𝑑𝑠 𝑎 + ∫ (𝑎 − 𝑠)𝑞−1 𝑆 (𝑎 − 𝑠) 𝐺 (𝑠, 𝑥𝑠 ) 𝑑𝑠] ≤ 𝑀4 [ 𝑥1 + 𝑀1 𝜙 (0) + 𝐹 (0, 𝜙) + 𝐹 (𝑎, 𝑥𝑎 ) + 𝑀5 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) + 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) Abstract and Applied Analysis + 𝑀2 ( 𝑡 𝛿 = ∫ ∫ (𝑡 − 𝑠)𝑞−1 𝜃𝜉𝑞 (𝜃) 𝑞𝑇 ((𝑡 − 𝑠)𝑞 𝜃) 0 − 𝑞1 (𝑞−𝑞1 )/(1−𝑞1 ) 1−𝑞1 𝑎 ) 𝑞 − 𝑞1 × [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )] 𝑑𝜃 𝑑𝑠 × 𝑔𝑘𝐾𝑎 +𝑁𝐿1/𝑞1 [0,𝑎] ] , 𝑡 ∞ 𝛿 + ∫ ∫ (𝑡 − 𝑠)𝑞−1 𝜃𝜉𝑞 (𝜃) 𝑞𝑇 ((𝑡 − 𝑠)𝑞 𝜃) 𝑡 ∫ (𝑡 − 𝑠)𝑞−1 𝐺 (𝑠, 𝑧𝑘,𝑠 + 𝑦𝑠 ) 𝑑𝑠 × [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )] 𝑑𝜃 𝑑𝑠 𝑡 ≤ ∫ (𝑡 − 𝑠)𝑞−1 𝑔𝑘𝐾𝑎 +𝑁 (𝑠) 𝑑𝑠 −∫ 𝑎 ≤ ∫ (𝑎 − 𝑠) 𝑞−1 𝑡−ℎ 𝑔𝑘𝐾𝑎 +𝑁 (𝑠) 𝑑𝑠 −∫ (36) We see that ‖𝑄2 𝑧(𝑡2 ) − 𝑄2 𝑧(𝑡1 )‖ tends to zero independently of 𝑧 ∈ 𝐵𝑘 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 𝐵𝑘 into an equicontinuous family functions It remains to prove that 𝑉(𝑡) = {(𝑄2 𝑧)(𝑡) : 𝑧 ∈ 𝐵𝑘 } is relatively compact in 𝑋 let ≤ 𝑡 ≤ 𝑎 be fixed, < 𝜖 < 𝑡, for 𝑧 ∈ 𝐵𝑘 , we define Π = 𝑄2 𝐵𝑘 and Π(𝑡) = {𝑄2 𝑧(𝑡) | 𝑧 ∈ 𝐵𝑘 }, for 𝑡 ∈ [0, 𝑎] Clearly, Π(0) = {𝑄2 𝑧(0) | 𝑧 ∈ 𝐵𝑘 } = {0} is compact, and hence, it is only to consider < 𝑡 ≤ 𝑎 For each ℎ ∈ (0, 𝑡), 𝑡 ∈ (0, 𝑎], arbitrary 𝛿 > 0, define Πℎ,𝛿 (𝑡) = {𝑄2,ℎ,𝛿 𝑧 (𝑡) | 𝑧 ∈ 𝐵𝑘 } , 𝛿 𝑡−ℎ − 𝑞1 (𝑞−𝑞1 )/(1−𝑞1 ) 1−𝑞1 ≤ (( )𝑎 ) 𝑔𝑘𝐾𝑎 +𝑁𝐿1/𝑞1 [0,𝑎] 𝑞 − 𝑞1 ∞ ∫ (𝑡 − 𝑠)𝑞−1 𝜃𝜉𝑞 (𝜃) 𝑞𝑇 ((𝑡 − 𝑠)𝑞 𝜃) ∞ ∫ (𝑡 − 𝑠)𝑞−1 𝜃𝜉𝑞 (𝜃) 𝑞𝑇 ((𝑡 − 𝑠)𝑞 𝜃) 𝛿 × [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )] 𝑑𝜃 𝑑𝑠 𝑄2,ℎ,𝛿 𝑧 (𝑡) = ∫ ∫ (𝑡 − 𝑠) 𝛿 𝑞−1 𝛿 𝑡 + 𝑀𝑀3 ‖𝑢 (𝑠)‖ ∫ + 𝑀∫ 𝑡−ℎ −∫ 𝑡−ℎ ∞ ∫ (𝑡 − 𝑠)𝑞−1 𝜃𝜉𝑞 (𝜃) 𝑞𝑇 ((𝑡 − 𝑠)𝑞 𝜃) 𝛿 × [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )] 𝑑𝜃 𝑑𝑠 (𝑡 − 𝑠)𝑞−1 𝑑𝑠 ⋅ 𝑞 ∫ 𝜃𝜉𝑞 (𝜃) 𝑑𝜃 𝛿 ∞ (𝑡 − 𝑠)𝑞−1 𝑔𝑘𝐾𝑎 +𝑁 (𝑠) 𝑑𝑠 ⋅ 𝑞 ∫ 𝜃𝜉𝑞 (𝜃) 𝑑𝜃 𝛿 ≤ {𝑀𝑀3 𝑀4 [ 𝑥1 + 𝑀1 𝜙 (0) + 𝐹 (0, 𝜙) + 𝐹 (𝑎, 𝑥𝑎 ) + 𝑀5 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) 𝐶1−𝛽 Γ (1 + 𝛽) 𝑎𝑞𝛽 𝛽Γ (1 + 𝑞𝛽) + 𝑀(( 𝑞 (38) Then the sets {𝑄2,ℎ,𝛿 𝑧(𝑡) | 𝑧 ∈ 𝐵𝑘 } are relatively compact in 𝑋 since the condition (𝐻4 ) It comes from the following inequalities: × [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )] 𝑑𝜃 𝑑𝑠 ∞ 𝑡−ℎ 𝑡 𝐿 (𝑘𝐾𝑎 + 𝑁 + 1) − 𝑞1 (𝑞−𝑞1 )/(1−𝑞1 ) 1−𝑞1 )𝑎 ) 𝑞 − 𝑞1 ×𝑔𝑘𝐾𝑎 +𝑁𝐿1/𝑞1 [0,𝑎] ] 𝑎𝑞 × [𝐶𝑢 (𝑠) + 𝐺 (𝑠, 𝑧𝑠 + 𝑦𝑠 )] 𝑑𝜃 𝑑𝑠 𝑄2 𝑧 (𝑡) − 𝑄2,ℎ,𝛿 𝑧 (𝑡) 𝑡 ∞ = ∫ ∫ (𝑡 − 𝑠)𝑞−1 𝜃𝜉𝑞 (𝜃) 𝑞𝑇 ((𝑡 − 𝑠)𝑞 𝜃) 0 0 (37) 𝜃𝜉𝑞 (𝜃) 𝑞𝑇 ((𝑡 − 𝑠) 𝜃) + 𝑀𝑞 ∫ (𝑡 − 𝑠)𝑞−1 𝑔𝑘𝐾𝑎 +𝑁 (𝑠) 𝑑𝑠 ∫ 𝜃𝜉𝑞 (𝜃) 𝑑𝜃 where ∞ 𝛿 𝑡 + 𝑡−ℎ 𝑡 ≤ 𝑀𝑀3 ‖𝑢 (𝑠)‖ 𝑞 ∫ (𝑡 − 𝑠)𝑞−1 𝑑𝑠 ∫ 𝜃𝜉𝑞 (𝜃) 𝑑𝜃 +𝑀(( − 𝑞1 (𝑞−𝑞1 )/(1−𝑞1 ) 1−𝑞1 )𝑎 ) 𝑔𝑘𝐾𝑎 +𝑁𝐿1/𝑞1 [0,𝑎] } 𝑞 𝑞 − 𝑞1 𝛿 × ∫ 𝜃𝜉𝑞 (𝜃) 𝑑𝜃 + 𝑀(( − 𝑞1 (𝑞−𝑞1 )/(1−𝑞1 ) 1−𝑞1 )ℎ ) 𝑞 − 𝑞1 ∞ × 𝑔𝑘𝐾𝑎 +𝑁𝐿1/𝑞1 [0,𝑎] 𝑞 ∫ 𝜃𝜉𝑞 (𝜃) 𝑑𝜃 (39) Therefore, Π(𝑡) = {𝑄2 𝑧(𝑡) | 𝑧 ∈ 𝐵𝑘 } is relatively compact in 𝑋 for all 𝑡 ∈ [0, 𝑎] Thus, the continuity of 𝑄2 and relatively compact of {𝑄2 𝑧(𝑡) | 𝑧 ∈ 𝐵𝑘 } imply that 𝑄2 is a completely continuous operator Abstract and Applied Analysis These arguments enable us to conclude that 𝑄 = 𝑄1 +𝑄2 is a condense mapping on 𝐵𝑘 , and by the fixed point theorem of Sadovskii there exists a fixed point 𝑧(⋅) for 𝑄 on 𝐵𝑘 In fact, by Step 1–Step 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 (𝐵𝑘 )) = since 𝑄2 (𝐵𝑘 ) is relatively compact Since 𝑄1 (𝐵𝑘 )) ⊆ 𝐵𝑘 and 𝛼(𝑄2 (𝐵𝑘 )) = 0, we can obtain 𝛼(𝑄(𝐵𝑘 )) ≤ 𝛼(𝑄1 (𝐵𝑘 )) + 𝛼(𝑄2 (𝐵𝑘 )) ≤ 𝛼(𝐵𝑘 ) for every bounded set 𝐵𝑘 of 𝑋 with 𝛼(𝐵𝑘 ) > 0, that is, 𝑄 = 𝑄1 + 𝑄2 is a condense mapping on 𝐵𝑘 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 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 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] An Example (ii) For each 𝑓 ∈ 𝑋, ∞ 𝐴−1/2 𝑓 = ∑ ⟨𝑓, 𝑧𝑛 ⟩ 𝑧𝑛 𝑛=1 𝑛 (42) In particular, ‖𝐴−1/2 ‖ = (iii) The operator 𝐴1/2 is given by ∞ 𝐴1/2 𝑓 = ∑ 𝑛 ⟨𝑓, 𝑧𝑛 ⟩ 𝑧𝑛 (43) 𝑛=1 on the space 𝐷(𝐴1/2 ) = {𝑓(⋅) ∈ 𝑋, 𝐴1/2 𝑓 ∈ 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 2 𝜙𝐵 = 𝜙 (0) + (∫ 𝑔 (𝜗) 𝜙 (𝜗) 𝑑𝜗) −∞ 𝜕2 𝑧 (𝑡, 𝑥) 𝜕𝑥2 = 𝐶𝑢 (𝑡) + 𝑎0 (𝑥) 𝑧 (𝑡, 𝑥) +∫ 𝑡 −∞ (40) 𝑧 (𝑡, 0) = 𝑧 (𝑡, 𝜋) = 0, 𝑎 𝜗 ≤ 0 ∞ 𝑛=1 (c) The function 𝑎0 (⋅) ∈ 𝐿∞ ([0, 𝜋]), 𝑎(⋅) is measurable, with ∫−∞ (𝑎12 (𝜗))/𝑔(𝜗)𝑑𝜗 < ∞, the function 𝑎2 (𝑡, ⋅) ∈ 𝐿2 ([0, 𝜋]) for each 𝑡 ≥ is measurable in 𝑡 𝑊𝑢 = ∫ (𝑎 − 𝑠)−1/3 𝑆 (𝑎 − 𝑠) 𝐶𝑢 (𝑠) 𝑑𝑠 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 absolutely continuous, 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 𝑧𝑛 (𝑥) = (2/𝜋)1/2 sin(𝑛𝑥) Then the following properties hold (i) If 𝐴 ∈ 𝐷(𝐴), then 𝐴𝑓 = ∑ 𝑛2 ⟨𝑓, 𝑧𝑛 ⟩ 𝑧𝑛 𝜋 (e) The linear operator 𝑊: 𝑈 → 𝑋 is defined by ≤ 𝑥 ≤ 𝜋, 𝑧 (𝜗, 𝑥) = 𝜙 (𝜗, 𝑥) , (a) The function 𝑏 is measurable and ∫0 ∫−∞ ∫0 (𝑏2 (𝜗, 𝑦, 𝑥)/𝑔(𝜗))𝑑𝑦 𝑑𝜗 𝑑𝑥 < ∞ (d) The function 𝜙 defined by 𝜙(𝜗)(𝑥) = 𝜙(𝜗, 𝑥) belongs to 𝐵 𝑎1 (𝑠, 𝑡) 𝑧 (𝑠, 𝑥) 𝑑𝑠 + 𝑎2 (𝑡, 𝑥) , ≤ 𝑡 ≤ 𝑎, (44) (b) The function (𝜕/𝜕𝑥)𝑏(𝜗, 𝑦, 𝑥) is measurable, 𝑏(𝜗, 𝑦, 𝜋 𝜋 0) = 𝑏(𝜗, 𝑦, 𝜋) = and let 𝑁1 = ∫0 ∫−∞ ∫0 (1/ 𝑔(𝜗))((𝜕/𝜕𝑥)𝑏(𝜗, 𝑦, 𝑥))2 𝑑𝑦 𝑑𝜗 𝑑𝑥 < ∞ 𝑡 𝜋 𝜕2/3 [𝑧 𝑥) + 𝑏 (𝑠 − 𝑡, 𝑦, 𝑥) 𝑧 (𝑠, 𝑦) 𝑑𝑦 𝑑𝑠] ∫ ∫ (𝑡, 𝜕𝑡2/3 −∞ − From [41], under some conditions 𝐵 is a phase space verifying (A), (B), (C), and in this case 𝐾(𝑡) = + (∫−𝑡 𝑔(𝜗)𝑑𝜗)1/2 (see [41] for the details) We assume the following conditions hold 𝜋 As an application of Theorem 7, we consider the following system: 1/2 (41) (45) and has a bounded invertible operator 𝑊−1 defined 𝐿2 ([0, 𝑎]); 𝑈)/ ker 𝑊 We define 𝐹, 𝐺: [0, 𝑎] × 𝐵 → 𝑋 by 𝐹(𝑡, 𝜙) = 𝑍1 (𝜙) and 𝐺(𝑡, 𝜙) = 𝑍2 (𝜙) + ℎ(𝑡), where 𝑍1 (𝜙) = ∫ 𝜋 ∫ 𝑏 (𝜗, 𝑦, 𝑥) 𝜙 (𝜗, 𝑥) 𝑑𝑦 𝑑𝜗, −∞ 𝑍2 (𝜙) = 𝑎0 (𝑥) 𝜙 (0, 𝑥) + ∫ −∞ 𝑎1 (𝜗) 𝜙 (𝜗, 𝑥) 𝑑𝜗, ℎ (𝑡) = 𝑎2 (𝑡, ⋅) (46) 10 From (a) and (c) it is clear that 𝑍1 and 𝑍2 are bounded linear operators on 𝐵 Furthermore, 𝑍1 (𝜙) ∈ 𝐷(𝐴1/2 ), and ‖𝐴1/2 𝑍1 ‖ ≤ 𝑁1 In fact, from the definition of 𝑍1 and (b) it follows that ⟨𝑍1 (𝜙), 𝑧𝑛 ⟩ = (1/𝑛)(2/𝜋)1/2 ⟨𝑍(𝜙), con(𝑛𝑥)⟩, 𝜋 where 𝑍(𝜙) = ∫−∞ ∫0 (𝜕/𝜕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 ‖𝐴1/2 𝑍1 (𝜙)‖ = ‖𝑍(𝜙)‖, which implies the assertion Therefore, from Theorem 7, the system (40) is completely controllable on [0, 𝑎] under the above assumptions Conclusion In this paper, by using the uniformly boundedness, 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