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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 931975, 17 pages doi:10.1155/2012/931975 Research Article Exact Null Controllability for Fractional Nonlocal Integrodifferential Equations via Implicit Evolution System Amar Debbouche1 and Dumitru Baleanu2, 3 1 Department of Mathematics, Faculty of Science, Guelma University, Guelma, Algeria Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey 3 Institute of Space Sciences, P.O. Box, MG-23, 76900 Magurele-Bucharest, Romania 2 Correspondence should be addressed to Dumitru Baleanu, [email protected] Received 1 May 2012; Revised 15 July 2012; Accepted 16 July 2012 Academic Editor: F. Marcellán Copyright q 2012 A. Debbouche and D. Baleanu. 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. We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodiﬀerential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodiﬀerential control system in Banach space. As an application that illustrates the abstract results, two examples are provided. 1. Introduction In this paper, we study the fractional nonlocal integrodiﬀerential system of the form t dα ut At, B utut ft, B ut gt, s, B3 usds, 1 2 dtα 0 a B4 u0 − u0 hutdt, 1.1 1.2 0 where 0 < α ≤ 1, t ∈ 0, a. Let −A be the infinitesimal generator of a C0 -semigroup in a Banach space X, and {Bi t : i 1, 2, 3, 4} is a family of linear closed operators defined on dense sets Si ⊃ DA, i 1, 2, 3, 4, respectively, in X into X. It is assumed that u0 ∈ X, 2 Journal of Applied Mathematics f : I × X → X, g : Δ × X → X and h : CI : X → X are given abstract functions. Here, I 0, a and Δ {s, t : 0 ≤ s ≤ t ≤ a}. Basic researches in diﬀerential equations have showed that many phenomena in nature are modeled more accurately using fractional derivatives and integrals; for more detail, we can refer to 1–13 and the references therein. There are many applications where the fractional calculus can be used, for example, viscoelasticity, electrochemistry, diﬀusion processes, control theory, heat conduction, electricity, mechanics, chaos, and fractals 14. Controllability is a fundamental concept in mathematical control theory and plays an important role in both finite and infinite dimensional spaces, that is, systems represented by ordinary diﬀerential equations and partial diﬀerential equations, respectively. So it is natural to extend this concept to dynamical systems represented by fractional diﬀerential equations. Several fractional partial diﬀerential equations and integrodiﬀerential equations can be expressed abstractly in some Banach spaces, in many cases, the accurate analysis, design and assessment of systems subjected to realistic environments must take into account the potential of random loads and randomness in the system properties. Randomness is intrinsic to the mathematical formulation of many phenomena such as fluctuations in the stock market or noise in communication networks. Fu studied the controllability results of some kinds of neutral functional diﬀerential systems, see 15, 16. In our previous work 17, we established the controllability of fractional evolution nonlocal impulsive quasilinear delay integrodiﬀerential systems. The existence results to evolution equations with nonlocal conditions in Banach space were studied first by Byszewski 18, 19; subsequently, many authors have been studied the same question, see for instance 20–23. Deng 24 indicated that, using the nonlocal condition u0 hu u0 to describe for instance, the diﬀusion phenomenon of a small amount of gas in a transparent tube can give better result than using the usual local Cauchy problem u0 u0 . Let us observe also that, since Deng’s papers, the function h is considered 1.3 hu p ck utk , k1 where ck , k 1, 2, . . . , p are given constants and 0 ≤ t1 < · · · < tp ≤ a. In this paper, we introduce a new concept in the theory of Semigroup named “implicit evolution system” to show the reader “what is the main diﬀerence between the solutions of fractional 0 < α < 1 and classical first order homogeneous evolution equation?” which is based on the work 17 and Pazy 25. A new form of nonlocal condition is also presented. Our paper is organized as follows. Section 2 is devoted to a review of some essential results which will be used in this work to obtain our main results. In Section 3, we use the theory of semigroups 25 in order to introduce our new concept that is called implicit evolution system. In Section 4, we establish the existence, uniqueness, and regularity of mild solutions of a class of fractional evolution nonlinear integrodiﬀerential systems with nonlocal conditions in Banach space. In Section 5, we prove the exact null controllability of a class of fractional evolution nonlocal integrodiﬀerential control systems; the last section deals to give examples that provide the abstract results. Journal of Applied Mathematics 3 2. Preliminary Results Definition 2.1. The fractional integral of order α with the lower limit zero for a function f ∈ C0, ∞ is defined as 1 I ft Γα α t 0 fs t − s1−α ds, t > 0, 0 < α < 1, 2.1 provided the right side is pointwise defined on 0, ∞, where Γ is the gamma function. Riemann-Liouville derivative of order α with the lower limit zero for a function f ∈ C0, ∞ can be written as L d 1 D ft Γ1 − α dt α t 0 fs ds, t − sα t > 0, 0 < α < 1. 2.2 The Caputo derivative of order α for a function f ∈ C0, ∞ can be written as C Dα ft L Dα ft − f0 , t > 0, 0 < α < 1. 2.3 Remark 2.2. 1 If f ∈ C1 0, ∞, then C Dα ft 1 Γ1 − α t 0 f s ds I 1−α f t, t − sα t > 0, 0 < α < 1. 2.4 2 The Caputo derivative of a constant is equal to zero. 3 If f is an abstract function with values in X, then integrals which appear in Definition 2.1 are taken in Bochner’s sense. Definition 2.3. By a strong solution of the nonlocal Cauchy problem 1.1, 1.2, we mean a function u with values in X such that i u is a continuous function in t ∈ I and ut ∈ DA, ii dα u/dtα exists and continuous on 0, a, 0 < α < 1, and u satisfies 1.1 on 0, a and 1.2. It is suitable to rewrite 1.1, 1.2 in the form t α−1 1 ut u0 t−η Γα 0 η × −A η, B1 u η u η f η, B2 u η g η, s, B3 us ds dη, 2.5 0 see also 26, 27. Let X and Y be two Banach spaces such that Y is densely and continuously embedded in X. We denote by Z every Banach space ZI, X endowed with the usual norm, which is given by uZ supt∈I ut, for u ∈ Z. The space of all bounded linear operators from X to Y is denoted by BX, Y . We recall some definitions and known facts from Pazy 25. 4 Journal of Applied Mathematics Definition 2.4. Let S be a linear operator in X, and let Y be a subspace of X. The operator S {x ∈ DS ∩ Y : Sx ∈ Y } and Sx Sx for x ∈ DS is called the part of S defined by DS in Y . Definition 2.5. Let Ω be a subset of X and for every t ∈ I and B1 u ∈ Ω, and let −At, B1 u be the infinitesimal generator of a C0 -semigroup St,B1 u s, s ≥ 0, on X. The family of operators {At, B1 u, t, B1 u ∈ I × Ω} is stable if there are constants M ≥ 1 and ω such that ρAt, B1 u ⊃ ω, ∞ k R λ : A tj , B1 u ≤ Mλ − ω−k j1 2.6 for λ > ω every finite sequences 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ a, 1 ≤ j ≤ k. The stability of {At, B1 u, t, B1 u ∈ I × Ω} implies that ⎛ ⎞ k k St ,B u sj ≤ M exp⎝ω sj ⎠, j 1 j1 j1 sj ≥ 0, 2.7 and any finite sequences 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ a, 1 ≤ j ≤ k, k 1, 2, . . .. Definition 2.6. Let St,B1 u s, s ≥ 0 be the C0 -semigroup generated by At, B1 u, t, B1 u ∈ I × Ω. A subspace Y of X is called At, B1 u-admissible if Y is invariant subspace of St,B1 u s, and the restriction of St,B1 u s to Y is a C0 -semigroup in Y . Let Ω ⊂ X be a subset of X such that for every t, B1 u ∈ I × Ω, At, B1 u is the infinitesimal generator of a C0 -semigroup St,B1 u s, s ≥ 0 on X. We make the following assumptions. H1 The family {At, B1 u, t, B1 u ∈ I × Ω} is stable. B1 u, t, B1 u ∈ H2 Y is At, B1 u-admissible for t, B1 u ∈ I × Ω, and the family {At, B1 u of At, B1 u in Y is stable in Y . I × Ω} of parts At, H3 For t, B1 u ∈ I ×Ω, DAt, B1 u ⊃ Y , At, B1 u is a bounded linear operator from Y to X and t → At, B1 u is continuous in the BY, X norm · . H4 There is a constant L > 0 such that At, B1 u − At, B1 vY → X ≤ Lu − vX 2.8 holds for every B1 u, B1 v ∈ Ω, and t ∈ I. In the next section, we will introduce a new concept in the theory of semigroups. 3. Implicit Evolution System Let Ω be a subset of X and {At, B1 u, t, B1 u ∈ I × Ω} a family of operators satisfying the conditions H1 –H4 . If u ∈ CI : X has values in Ω, then there is a unique evolution system Uα t, s; B1 u, 0 < α ≤ 1, 0 ≤ s ≤ t ≤ a, in X satisfying Journal of Applied Mathematics 5 i Uα t, s; B1 u ≤ Meωt−s for 0 ≤ s ≤ t ≤ a, where M and ω are stability constants, ii ∂α /∂tα Uα t, s; B1 uy As, B1 usUα t, s; B1 uy for y ∈ Y and 0 ≤ s ≤ t ≤ a, iii ∂α /∂sα Uα t, s; B1 uy −Uα t, s; B1 uAs, B1 usy for y ∈ Y and 0 ≤ s ≤ t ≤ a. Remark 3.1. 1 If B1 is the identity and α 1, then Ut, s; u is the explicit evolution system given in Pazy 25 and in Zaidman 28. 2 Since, in our case, Ut, s; u is dependent of α and B1 , so we call it an implicit evolution system generated by −At, B1 u. 3 For nonautonomous diﬀerential equations in a Banach space, the implicit evolution system is similar to our concept α, u-resolvent family. 4 We can deduce that 1.1-1.2 is well posed if and only if −At, B1 u is the generator of the implicit evolution system Ut, s; u. Further, we assume the following. H5 For every u ∈ CI : X satisfying ut ∈ Ω for 0 ≤ t ≤ a, we have Ut, s; uY ⊂ Y, 0≤s≤t≤a 3.1 and Ut, s; u is strongly continuous in Y for 0 ≤ s ≤ t ≤ a. H6 Y is reflexive. H7 For every t, B2 u ∈ I × Ω, ft, B2 u ∈ Y . H8 The operator B4 t λα I−1 exists in BX for any λ with Re λ ≤ 0 and −1 B4 t λα I ≤ Cα , |λ| 1 t ∈ I, 3.2 where Cα is a positive constant independent of both t and λ. H9 h : CI : Ω → Y is Lipschitz continuous in X and bounded in Y , that is, there exist constants k1 > 0 and k2 > 0 such that huY ≤ k1 , 3.3 hu − hvY ≤ k2 max u − vX . t∈I For the conditions H9 and H10 , let Z be taken as both X and Y . H10 g : Δ × Z → Z is continuous, and there exist constants k3 > 0 and k4 > 0 such that t 0 gt, s, B3 u − gt, s, B3 v ds ≤ k3 u − v , Z Z k4 max t 0 u, v ∈ X, gt, s, 0 ds : t, s ∈ Δ . Z 3.4 6 Journal of Applied Mathematics H11 f : I × Z → Z is continuous, and there exist constants k5 > 0 and k6 > 0 such that ft, B2 u − ft, B2 v ≤ k5 u − v , Z Z k6 maxft, 0Z . u, v ∈ X, 3.5 t∈I Let us take M0 max Ut, s; uΩZ , 0 ≤ s ≤ t ≤ a, u ∈ Ω. H12 There exist positive constants r > 0 and 0 < λ < 1 such that M0 {u0 aCα k1 ark3 k5 k4 k6 } ≤ r, λ Kau0 Y a2 Cα k1 K aM0 Cα k2 a{Kk3 k5 r k4 k6 a M0 k3 k5 }. 3.6 By a mild solution of 1.1, 1.2, we mean a function u ∈ CI : X with values in Ω and u0 ∈ X satisfying the integral equation ut Ut, 0; uu0 Ut, 0; uB4−1 t 0 a hutdt 0 s Ut, s; u fs, B2 us g s, η, B3 u η dη ds. 3.7 0 H13 Further, there exists a constant K > 0 such that for every u, v ∈ CI : X with values in Ω and every ω ∈ Y we have Ut, s; uω − Ut, s; vω ≤ KωY t uτ − vτdτ. 3.8 s 4. Existence Results Theorem 4.1. Let u0 ∈ Y and Ω {u ∈ X : uY ≤ r}, r > 0. If −At, B1 u is the generator of an implicit evolution system Ut, s; u and the assumptions (H5 )∼(H13 ) are satisfied, then 1.1, 1.2 has a unique mild solution on I. Proof. Let S be a nonempty closed subset of CI : X defined by S {u : u ∈ CI : X, uY ≤ r}, t ∈ I. 4.1 Journal of Applied Mathematics 7 Consider a mapping P on S defined by P ut Ut, 0; uu0 Ut, 0; uB4−1 t a hutdt 0 4.2 s Ut, s; u fs, B2 us g s, η, B3 u η dη ds. 0 0 For u ∈ S, we have a −1 Ut, 0; uB hutdt P utY ≤ Ut, 0; uu0 4 t 0 Ut, s; u fs, B2 us − fs, 0 fs, 0 0 s s g s, η, B3 u η −g s, η, 0 dη g s, η, 0 dη ds. 0 0 ≤ M0 u0 aM0 Cα k1 t M0 {k5 us k6 k3 us k4 }ds 0 ≤ M0 u0 aM0 Cα k1 aM0 {k5 us k6 k3 us k4 } ≤ M0 {u0 aCα k1 ark3 k5 k4 k6 } ≤ r. 4.3 Thus, P maps S into itself. Now, we will show that P is a strict contraction on S which will ensure the existence of a unique continuous function satisfying 3.7 on I. If u, v ∈ S, then P ut − P vt a a −1 −1 ≤ Ut, 0; uu0 − Ut, 0; vu0 Ut, 0; uB hutdt − Ut, 0; vB hvtdt 4 4 0 t s Ut, s; u fs, B us g s, η, B u η dη 2 3 0 0 0 s −Ut, s; v fs, B2 vs g s, η, B3 v η dη ds 0 a a −1 −1 ≤ Ut, 0; uu0 − Ut, 0; vu0 Ut, 0; uB hutdt − Ut, 0; vB hutdt 4 4 0 a a −1 −1 Ut, 0; vB4 hutdt − Ut, 0; vB4 hvtdt 0 0 0 8 Journal of Applied Mathematics t s Ut, s; u fs, B2 us g s, η, B3 u η dη 0 0 −Ut, s; v fs, B2 us s 0 g s, η, B3 u η dη s Ut, s; v fs, B2 us g s, η, B3 u η dη 0 −Ut, s; v fs, B2 vs s 0 g s, η, B3 v η dη ds ≤ Kau0 Y max uτ − vτ a2 Cα k1 K max uτ − vτ τ∈I τ∈I aM0 Cα k2 max uτ − vτ τ∈I s t a max uτ − vτ fs, B us g s, η, B u η dη K 2 3 0 0 M0 fs, B2 us − fs, B2 vs s Y τ∈I g s, η, B3 u η − g s, η, B3 v η dη ds 0 ≤ Kau0 Y a2 Cα k1 K aM0 Cα k2 max uτ − vτ τ∈I a{Kk3 k5 r k4 k6 a M0 k3 k5 } max uτ − vτ. τ∈I 4.4 Thus, P ut − P vt ≤ λ max uτ − vτ, τ∈I 4.5 which means that P is a strict contraction map from S into S, and therefore by the Banach contraction principle there exists a unique fixed point u ∈ S such that P u u. Hence, u is a unique mild solution of 1.1, 1.2 on I. Theorem 4.2. Assume the following. i Conditions (H1 )∼(H13 ) hold. ii The functions B2 tω and B3 tω are uniformly Hölder continuous in t ∈ I for every element ω in S2 ∩ S3 . iii There are numbers L1 , L2 > 0 and p, q ∈ 0, 1 such that Journal of Applied Mathematics ft1 , B2 u − ft2 , B2 v ≤ L1 |t1 − t2 |p B2 u − B2 v , g s1 , η, B3 u − g s2 , η, B3 v ≤ L2 |s1 − s2 |q 9 4.6 for all t1 , t2 ∈ I and all s1 , η, s2 , η ∈ Δ. Then, the problem 1.1, 1.2 has a unique strong solution on I. Proof. Applying Theorem 4.1, the problem 1.1, 1.2 has a mild solution u ∈ S. Now, we will show that u is a unique strong solution of the considered problem on I. According to ii, B2 u−B2 v is uniformly Hölder continuous in t ∈ I for every element t u in S2 ∩S3 , also iii implies that t → ft, B2 u and t → 0 gt, s, B3 uds are uniformly Hölder continuous on I 20, 26. Set V t ft, B2 u t gt, s, B3 uds. 4.7 0 Clearly V t is uniformly Hölder continuous in t ∈ I. Consider the following nonlocal Cauchy problem: dα vt At, B1 utut V t, dtα a hutdt. B4 u0 − u0 4.8 0 From Pazy, 4.8 has a unique solution v on I given by vt Ut, 0; uu0 Ut, 0; uB4−1 a hutdt 0 t Ut, s; uV sds. 4.9 0 Noting that 21, each term on the right hand side of 4.9 belongs to DA, thus vt ∈ DA, using the uniqueness of V t, we have that ut vt. Hence, u is the unique strong solution of 1.1, 1.2 on I. In next section, some results are obtained from Sakthivel et al. 29, 30. 5. Exactly Null Controllability Results Consider fractional nonlocal evolution integrodiﬀerential control system of the form t dα ut At, B1 utut Φμ t Ψ t, ft, B2 ut, gt, s, B3 usds , dtα 0 a hutdt, B4 u0 − u0 0 5.1 10 Journal of Applied Mathematics where the unknown u· takes values in the Banach space X, the control function μ belongs to the spaces L2 I, H, a Banach space of admissible control functions with H, a Banach space. Further, Φ is a bounded linear operator from H into X, the function Ψ : I × X × X → X is given, and the others terms are defined as above. For all u0 ∈ X and admissible control μ ∈ L2 I, H, the problem 5.1 admits a mild solution given by uμ t Ut, 0; uu0 t Ut, 0; uB4−1 a hutdt 0 s Ut, s; u Φμs Ψ s, fs, B2 us, gs, τ, B3 uτdτ ds. 0 5.2 0 Definition 5.1. We will say that system 5.1 is exactly null controllable on the interval I if for a11 u0 ∈ X, there exists a control μ ∈ L2 I, H, such that the mild solution ut of 5.1 a corresponding to μ verifies u0 − B4−1 0 hutdt u0 and ua 0. In order to prove the controllability result, in addition, we consider the following conditions. H14 Ψ : I × X × X → X is continuous, and there exist constants N1 and N2 such that for all xi , yi ∈ X, i 1, 2, we have Ψ t, x1 , y1 − Ψ t, x2 , y2 ≤ N1 x1 − x2 y1 − y2 , N2 max Ψt, 0, 0. 5.3 t∈I H15 Let ρ aM0 M1 M2 ρ σ σ ≤ r, 5.4 where σ aM0 {N1 k3 k5 r k4 k6 N2 } and ρ M0 u0 aM0 Cα k1 , and let λ Kau0 a2 Cα k1 K aM0 Cα k2 2a M0 M1 M2 ρ ap a Kqa p , 5.5 where p M0 N1 k3 k5 , q N1 k3 k5 r k4 k6 N2 , and 0 ≤ λ < 1. H16 The bounded linear operator W : L2 I, H → X defined by Wμ a Ua, s; uΦμds 5.6 0 has an induced inverse operator W −1 which takes values in L2 I, H/ker W and there exist positive constants M1 , M2 , such that Φ ≤ M1 and W −1 ≤ M2 . Journal of Applied Mathematics 11 Theorem 5.2. If hypotheses (H1 )∼(H16 ) are satisfied, then the control nonlocal fractional integrodifferential system 5.1 is exactly null controllable on I. a Proof. Let Sr {u : u ∈ CI : X, u0 − B4−1 0 hutdt u0 , u ≤ r, t ∈ I}. We define an operator Q : Sr → Sr by a hutdt Quμ t Ut, 0; uu0 Ut, 0; uB4−1 t 0 U t, η; u ΦW −1 −Ua, 0; uu0 − 0 − a a hutdt 0 Ua, s; uΨ s, fs, B2 us, 0 t Ua, 0; uB4−1 s Ut, s; uΨ s, fs, B2 us, 0 gs, τ, B3 uτdτ ds η dη 0 s 5.7 gs, τ, B3 uτdτ ds. 0 Using the hypothesis H14 , for an arbitrary function u·, we define the control a −1 −1 μt W −Ua, 0; uu0 − Ua, 0; uB4 hutdt − a 0 s Ua, s; uΨ s, fs, B2 us, 0 gs, τ, B3 uτdτ ds t. 5.8 0 Using this controller, we will show that the operator Q has a fixed point. This fixed point is then a solution of 5.2. Clearly, Qμ ua 0, which means that the control μ steers system 5.1 from the initial state u0 to origin in time a, provided we can obtain a fixed point of the nonlinear operator Q. Now, we show that Q maps Sr into itself. We have Quμ t a −1 ≤ Ut, 0; uu0 Ut, 0; uB4 hutdt t 0 0 a −1 U t, η; u Ua, 0; uu Ua, 0; uB hutdt ΦW −1 0 4 a 0 0 s Ψ s, fs, B us, gs, τ, B uτdτ Ua, s; u 2 3 0 − Ψs, 0, 0 Ψs, 0, 0 ds dη 12 Journal of Applied Mathematics t Ut, s; u 0 s × Ψ s, fs, B2 us, gs, τ, B3 uτdτ − Ψs, 0, 0 Ψs, 0, 0 ds. 0 ≤ M0 u0 aM0 Cα k1 aM0 M1 M2 M0 u0 aM0 Cα k1 aM0 {N1 k5 r k6 k3 r k4 N2 } aM0 {N1 k5 r k6 k3 r k4 N2 } ≤ r. 5.9 Thus, Q maps Sr into itself. Now, for u, v ∈ Sr , we have Quμ t − Qvμ t a a −1 −1 ≤ Ut, 0; uu0 − Ut, 0; vu0 Ut, 0; uB4 hutdt − Ut, 0; vB4 hvtdt 0 0 t a U t, η; u ΦW −1 −Ua, 0; uu0 − Ua, 0; uB−1 hutdt 4 0 − a 0 s Ua, s; uΨ s, fs, B2 us, 0 − U t, η; v ΦW −1 −Ua, 0; vu0 − Ua, 0; vB4−1 − a a gs, τ, B3 uτdτ ds 0 hvtdt 0 Ua, s; vΨ s, fs, B2 vs, 0 s 0 gs, τ, B3 vτdτ ds dη t s Ut, s; uΨ s, fs, B us, gs, τ, B uτdτ 2 3 0 0 −Ut, s; vΨ s, fs, B2 vs, s 0 gs, τ, B3 vτdτ ds. a a −1 −1 ≤ Ut, 0; uu0 − Ut, 0; vu0 Ut, 0; uB4 hutdt − Ut, 0; vB4 hutdt 0 a a −1 −1 Ut, 0; vB hutdt − Ut, 0; vB hvtdt 4 4 0 0 t a −1 −1 hutdt U t, η; u ΦW −Ua, 0; uu0 − Ua, 0; uB4 0 0 0 Journal of Applied Mathematics 13 − a s Ua, s; uΨ s, fs, B2 us, gs, τ, B3 uτdτ ds 0 − U t, η; v ΦW −1 −Ua, 0; vu0 − Ua, 0; vB4−1 − a a 0 hvtdt 0 Ua, s; vΨ s, fs, B2 vs, 0 s 0 gs, τ, B3 vτdτ ds dη t s Ut, s; uΨ s, fs, B2 us, gs, τ, B3 uτdτ 0 0 −Ut, s; vΨ s, fs, B2 us, s 0 gs, τ, B3 uτdτ s Ut, s; vΨ s, fs, B2 us, gs, τ, B3 uτdτ 0 s −Ut, s; vΨ s, fs, B2 vs, gs, τ, B3 vτdτ ds. 0 ≤ Kau0 a2 Cα k1 K aM0 Cα k2 max uτ − vτ τ∈I 2a{M0 M1 M2 M0 u0 aM0 Cα k1 aM0 N1 k5 k3 } max uτ − vτ τ∈I a{aKN1 k5 r k6 k3 r k4 N2 M0 N1 k5 k3 } max uτ − vτ τ∈I ≤ λ max uτ − vτ. τ∈I 5.10 Therefore, Q is a contraction mapping, and hence there exists a unique fixed point u ∈ X, such that Qut ut. Any fixed point of Q is a mild solution of 5.1 on I which satisfies ua 0. Thus, system 5.1 is exactly null controllable on I. 6. Examples To illustrate the abstract results, we give the following examples. Example 6.1. Consider the nonlinear integropartial diﬀerential equation of fractional order t q ∂α ux, t aq x, t; bq ux, t Dx ux, t Fx, t, w1 Gx, t, s, w2 sds, ∂tα 0 |q|≤2m 6.1 14 Journal of Applied Mathematics with nonlocal condition ux, 0 p ck ux, tk gx, 6.2 k1 q q q where 0 < α ≤ 1, 0 ≤ t1 < · · · < tp ≤ a, x ∈ Rn , Dx Dx11 · · · Dxnn , Dxi ∂/∂xi , q q1 , . . . , qn is an n-dimensional multi-index, |q| q1 · · · qn , and wi , i 1, 2, is given by wi x, t q bqi x, tDx ux, t |q|≤2m−1 q cqi x, tDy u y, t dy. Ω q ≤2m−1 || 6.3 Let L2 Rn be the set of all square integrable functions on Rn . We denote by Cm Rn the set of all continuous real-valued functions defined on Rn which have continuous partial derivatives of order less than or equal to m. By C0m Rn , we denote the set of all functions f ∈ Cm Rn with compact supports. Let H m Rn be the completion of C0m Rn with respect to the norm !2 !! q 2 ! f !Dx fx! dx. m n |q|≤m R 6.4 It is supposed that the following hold. " q i The operator A − |q|≤2m aq x, t; bq ux, tDx is uniformly elliptic on Rn . In other words, all the coeﬃcients aq , |q| 2m, are continuous and bounded on Rn , and there is a positive number c such that −1m1 eq x, t; bq ξq ≥ c|ξ|2m , |q|2m q 6.5 q for all x ∈ Rn and all ξ / 0, ξ ∈ Rn , ξq ξ1 1 · · · ξnn , and |ξ|2 ξ12 · · · ξn2 . ii All the coeﬃcients aq , |q| 2m, satisfy a uniform Hölder condition on Rn . Under these conditions, the operator A with domain of definition DA H 2m Rn generates an evolution operator defined on L2 Rn , and it is well known that H 2m Rn is dense in X L2 Rn and the initial function gx is an element in Hilbert space H 2m Rn , see 26, page 438. Applying Theorem 4.1, this achieves the proof of the existence of mild solutions of the problem 6.1, 6.2. In addition, iii If the coeﬃcients bq , cq , |q| ≤ 2m − 1 satisfy a uniform Hölder condition on Rn and the operators F and G satisfy. There are numbers L1 , L2 ≥ 0 and λ1 , λ2 ∈ 0, 1 such that |q|≤2m−1 Rn ! !2 ! !2 q q ! ! !F x, t, Dx w1 − F x, s, Dx w1∗ ! dx ≤ L1 |t − s|λ1 !w1 − w1∗ ! dx , |q|≤2m−1 Rn ! !2 q q ! ! !G x, t, η, Dx w2 − G x, s, η, Dx w2 ! dx ≤ L2 |t − s|λ2 , 6.6 Journal of Applied Mathematics 15 for all t, s ∈ I, t, η, s, η ∈ Δ, and all x ∈ Rn . Applying Theorem 4.2, we deduce that 6.1, 6.2 has a unique strong solution. The second example is concerned with the controllability result. Example 6.2. Consider the fractional nonlocal evolution integropartial diﬀerential control system of the form t ∂2 ux, t ∂α ux, t ax, t, bux, t ζx, t Υ t, μ1 t, ux, t, μ2 t, s, ux, tds , ∂tα ∂x2 0 ux, 0 p ck ux, tk gx, x ∈ 0, π, k1 u0, t uπ, t 0, t ∈ I, 6.7 where 0 < α ≤ 1, 0 ≤ t1 < · · · < tp ≤ a, and the functions ax, t, ·, bx, t are continuous. Let us take X L2 0, π, Sr y ∈ L2 0, π : y ≤ r . 6.8 Put Φμxt ζx, t, x ∈ 0, π where μt ζ·, t and ζ : 0, π×I → 0, π is continuous. We define At, · : X → X by At, ·wx ax, t, ·w with domain DA {w ∈ X : w, w are absolutely continuous, w ∈ X, w0 wπ 0}. Assume that −At, · generates an evolution system Ut, s, · such that for every positive numbers n1 and n2 , Ut, s, · ≤ n1 and Ut, s, ·As, · ≤ n2 . Also, define B1 t : DB1 S1 → X by B1 tzx bx, tz, for all z ∈ X and x ∈ 0, π. Assume that the linear operator W that is given by Wμx a Ua, s; uζx, sds, x ∈ 0, π, 6.9 0 has a bounded invertible operator W −1 in L2 I, H/kerW. Let us assume that the nonlinear functions Υ, μ1 , and μ2 satisfy the following Lipschitz conditions Υ t, z1 , y1 − Υ t, z2 , y2 ≤ c1 z1 − z2 y1 − y2 , μ1 t, x1 − μ1 t, x2 ≤ c2 x1 − x2 , t μ2 t, s, v1 − μ2 t, s, v2 ds ≤ c3 v1 − v2 , 0 where cj > 0, j 1, 2, 3, xi , yi , zi , vi ∈ X, i 1, 2, and s, t ∈ I. 6.10 16 Journal of Applied Mathematics All the conditions stated in Theorem 5.2 are satisfied. Hence, system 6.7 is exactly null controllable on I. References 1 D. Baleanu, R. P. Agarwal, O. G. Mustafa, and M. C. Sulschi, “A symptotic integration of some nonlinear diﬀerential equations with fractional time derivative,” Journal of Physics A, vol. 44, pp. 1–9, 2011. 2 D. Baleanu and A. K. Golmankhaneh, “On electromagnetic field in fractional space,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 288–292, 2010. 3 A. A. Kilbas, H. M. Srivastava, and J. J. 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