Controllability for fractional evolution inclusions without compactness

In this paper, we study the existence and controllability for fractional evolution inclusions in Banach spaces. We use a new approach to obtain the existence of mild solutions and controllability results, avoiding hypotheses of compactness on the semigroup generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Finally, two examples are given to illustrate our theoretical results.

1. Introduction. Fractional differential equation is concerned with the notion and methods to solve differential equations involving fractional derivatives of the unknown function. It can be also considered as an alternative model to nonlinear differential equations. As a result, differential equation with fractional derivative can be considered as an excellent instrument for the description of memory and hereditary properties of various materials and processes. The fractional order models of real systems are always more adequate than the classical integer order models, since the description of some systems is more accurate when the fractional derivative is used. The advantages of fractional derivatives becomes evident in modeling mechanical and electrical properties of real materials, description of rheological properties of rocks and in various other fields. There has been a great deal of interest in the solutions of fractional differential equations in analytical and numerical senses. One can see the monographs of Kilbas et al. [14], Podlubny [20], Diethelm [6], Tarasov [23], Zhou [30,31] and the papers [25,21,15,8,29,32].
In this paper, we are interested in the fractional semilinear differential inclusions in Banach spaces of the type C D q t x(t) ∈ Ax(t) + F (t, x(t)), a.e. t ∈ [0, b], 0 < q ≤ 1, where C D q t is the Caputo fractional derivative of order q, b > 0 is a finite number, A is the infinitesimal generator of a strongly continuous semigroup {T (t)} t≥0 in X, the state x(·) takes values in a Banach space X, x 0 is an element of the Banach space X, F : [0, b] × X X is a multivalued map. Further, we investigate the following fractional control system where the control function u(·) takes its value in L 1 q 1 ([a, b]; U ) for q 1 ∈ (0, q), a Banach space of admissible control functions and U is a Banach space. B : U → X is a bounded linear operator.
In recent years, the existence of mild solutions and controllability problems for various types of nonlinear fractional evolution inclusions in infinite dimensional spaces by using different kinds of approaches have been considered in many recent publications (see, e.g., [24,11,16,26,27,28,1,17] and the references therein). In most of the existing articles, various fixed point theorems and measure of noncompactness are employed to obtain the fixed points of the solution operator of the Cauchy problems under the restrictive hypotheses of compactness on the semigroup generated by the linear part and on the nonlinear term. But, in infinite dimensional Banach spaces the compactness of the associated evolution operator is in contradiction with the controllability of a linear system while using locally L p −controls, for p > 1. As it was pointed out by [2], it is meaningful to introduce conditions assuring controllability for semilinear equations without requiring the compactness of the semigroup or evolution operator generated by the linear part. In this paper another approach is considered, it exploits the weak topology of the state space. This new tool was introduced to study semilinear differential inclusions associated to boundary value conditions, see [3]. We prove the existence of mild solutions of (1) and the controllability results of (2) by means of weak topology, avoiding hypotheses of compactness on the semigroup generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness.
The paper is organized as follows. In Section 2 we recall some notions and results that we use in the main part of the paper. In section 3 we study the existence of mild solutions for (1). In Section 4 we prove the controllability for the fractional control system (2) and in Section 5 two examples are given to illustrate the obtained theory.
2. Preliminaries. In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let A : D(A) → X be the infinitesimal generator of a strongly continuous semigroup {T (t)} t≥0 . There exists a constant M 1 > 0 such that sup t∈J T (t) ≤ M 1 . Let (X, · ) be a reflexive Banach space and X w denote the space X endowed with the weak topology. For a set D ⊂ X, the symbol D w denotes the weak closure of D. We recall that a bounded subset D of a reflexive Banach space X is weakly relatively compact. In the whole paper, without generating misunderstanding, we denote by · p both the L p ([0, b]; X)-norm and L p ([0, b]; R)-norm and by · 0 the C([0, b]; X)-norm. We recall (see [4,Theorem 4.3]) that a sequence {x n } ⊂ C([0, b]; X) weakly converges to an element x ∈ C([0, b]; X) if and only if (i) there exists N > 0 such that, for every n ∈ N and t ∈ For sake of completeness, we recall some results that we will need in the main section. Firstly we state the fixed point theorem by Donal O'Regan.
Theorem 2.1. [18] Let E be a metrizable locally convex linear topological space and let Q be a weakly compact, convex subset of E. Suppose G : Q → C(Q) has weakly sequentially closed graph. Then G has a fixed point; here C(Q) denoted the family of nonempty closed, convex subsets of Q.
We mention also two results that are contained in the so called Eberlein-Smulian theory. Let us recall the following known definitions. For more details, see [14].
Definition 2.5. The fractional integral of order γ with the lower limit zero for a function f is defined as provided the right side is point-wise defined on [0, ∞), where Γ(·) is the gamma function.
Definition 2.6. The Riemann-Liouville derivative of order γ with the lower limit zero for a function f : [0, ∞) → R can be written as Definition 2.7. The Caputo derivative of order γ for a function f : [0, ∞) → R can be written as (ii) the Caputo derivative of a constant is equal to zero; (iii) if f is an abstract function with values in X, then integrals which appear in Definitions 2.5, 2.6 and 2.7 are taken in Bochner's sense. 3. Existence of mild solutions. We study the fractional semilinear differential inclusion (1) under the following assumptions: (H A ) the operator A generates a strongly continuous semigroup {T (t)} t≥0 in X, and there exists a constant M 1 ≥ 1 such that sup t∈J T (t) ≤ M 1 . We assume that the multivalued nonlinearity F : [0, b] × X X has nonempty convex and weakly compact values and: X has a measurable selection for every x ∈ X; (H 2 ) the multimap F (t, ·) : X X is weakly sequentially closed for a.e. t ∈ [0, b], i.e., it has a weakly sequentially closed graph; (H 3 ) there exists a constant q 1 ∈ (0, q) and for every r > 0, there exists a function µ r ∈ L 1 q 1 ([0, b]; R + ) such that for each c ∈ X, c ≤ r: Now, we define the mild solution of fractional evolution inclusion (1).
] and x satisfies the following integral equation and ξ q is a probability density function defined on (0, ∞), that is, The following results will be used in the proof of our main results.
Lemma 3.2. [30,33] The operators T and S have the following properties: (i) for any fixed t ≥ 0, T (t) and S (t) are linear and bounded operators, i.e., for any x ∈ X, The set Λ p is always nonempty as Proposition 3.1 below shows.
Proof. Let p ∈ C([0, b]; X). By the uniform continuity of p there exists a sequence {p n } of step functions, p n : Hence, by (H 1 ), there exists a sequence of functions {f n } such that f n (t) ∈ F (t, p n (t)) for a.e. t ∈ [0, b] and f n : [0, b] → X is measurable for any n ∈ N. From (3) there exists a bounded set E ⊂ X such that p n (t), p(t) ∈ E, for any t ∈ [0, b] and n ∈ N and by ( According to the reflexivity of the space X and by the Dunford-Pettis theorem (see [7, p. 294]), we have the existence of a subsequence, denoted as the sequence, such that By Mazur's convexity theorem we obtain a sequence By (H 3 ), the multimap F (t, ·) is locally weakly compact for a.e. t ∈ [0, b], i.e., for a.e. t and every x ∈ X there is a neighbourhood V of x such that the restriction of F (t, ·) to V is weakly compact. Hence by (H 2 ) and the locally weak compactness, we To conclude we have only to prove that Fix t 0 / ∈ N 0 and assume, by contradiction, that p(t 0 ) as m → ∞ then implies the existence of n 0 ∈ N such that p n (t 0 ) ∈ U for all n > n 0 . Therefore g 0 (t 0 ) ∈ F (t 0 , p n (t 0 )) ⊂ V for all n > n 0 . Since V is convex we also have that f n (t 0 ) ∈ V for all n > n 0 and, by the convergence, we arrive to the contradictory conclusion that g(t 0 ) ∈ V . We obtain that We define the solution multioperator Γ : where We first prove that the operator S is continuous.
On the other hand, the function Thus So the operator S is continuous.
It is easy to verify that the fixed points of the multioperator Γ are mild solutions of fractional differential system (1).
Fix n ∈ N, consider Q n the closed ball of radius n in C([0, b]; X) centered at the origin and denote by Γ n = Γ| Qn : Q n C([0, b]; X) the restriction of the multioperator Γ on the set Q n . We describe some properties of Γ n . ; X); we will prove that x ∈ Γ n (p). Since p m ∈ Q n for all m and p m (t) p(t) for every t ∈ [0, b], it follows that p(t) ≤ lim inf m→∞ p m (t) ≤ n for all t (see [5,Proposition III.5]). The fact that We observe that, according to (H 3 ), f m (t) ≤ µ n (t) for a.e. t and every m, i.e., {f m } is bounded and uniformly integrable and {f m (t)} is bounded in X for a.e. t ∈ [0, b]. Hence, by the reflexivity of the space X and by the Dunford-Pettis theorem (see [7, p. 294]), we have the existence of a subsequence, denoted as the sequence, and a function g such that f m g in L implying, for the uniqueness of the weak limit in X, that x 0 (t) = x(t) for all t ∈ [0, b]. Finally, as the reason for the fourth part of Proposition 3.1, it is possible to show that g(t) ∈ F (t, p(t)) for a.e. t ∈ [a, b]. Proposition 3.3. The multioperator Γ n is weakly compact.
Proof. We first prove that Γ n (Q n ) is relatively weakly sequentially compact.
Let {p m } ⊂ Q n and {x m } ⊂ C([0, b]; X) satisfying x m ∈ Γ n (p m ) for all m. By the definition of the multioperator Γ n , there exists a sequence {f m }, f m ∈ Λ pm such that Further, as the reason for Proposition 3.2, we have that there exists a subsequence, denoted as the sequence, and a function g such that f m g in L 1 q 1 ([0, b]; X). Therefore, Furthermore, by the weak convergence of {f m }, by (H A ), we have for all m ∈ N and t ∈ [0, b]. As the reason for Proposition 3.2, it is then easy to prove that x m l in C([0, b]; X). Thus Γ n (Q n ) is relatively weakly compact by Theorem 2.2. Proof. Fix p ∈ Q n , since F is convex valued, from the linearity of the integral and of the operators T (t) and S (t), it follows that the set Γ n (p) is convex. The weak compactness of Γ n (p) follows by Propositions 3.2 and 3.3.
We are able now to state the main results of this section.
Then inclusion (1) has at least a mild solution.
Proof. We show that there exists n ∈ N such that the operator Γ n maps the ball Q n into itself. Assume, to the contrary, that there exist sequences {z n }, {y n } such that z n ∈ Q n , y n ∈ Γ n (z n ) and y n / ∈ Q n , ∀ n ∈ N. Then there exists a sequence {f n } ⊂ L 1 q 1 ([0, b]; X), f n (s) ∈ F (s, z n (s)) such that As the reason for Proposition 3.3, we have n < y n 0 ≤M 1 x 0 + M 1 q Γ(1 + q) which contracts (5). Fix n ∈ N such that Γ n (Q n ) ⊆ Q n . By Proposition 3.3 the set V n = Γ n (Q n ) w is weakly compact. Let now W n = co(V n ), where co(V n ) denotes the closed convex hull of V n . By Theorem 2.3, W n is a weakly compact set. Moreover, from the fact that Γ n (Q n ) ⊂ Q n and that Q n is a convex closed set, we have that W n ⊂ Q n and hence In view of Proposition 3.2, Γ n has a weakly sequentially closed graph. Thus from Theorem 2.1, inclusion (1) has a solution. The proof is now completed.
then inclusion (1) has at least a mild solution.
Proof. As the reason for Theorem 3.3. Assume that there exist {z n }, {y n } such that z n ∈ Q n , y n ∈ Γ n (z n ) and y n / ∈ Q n , ∀ n ∈ N, we get n < y n 0 which contracts (6). The conclusion then follows by Theorem 2.1, like Theorem 3.3.
Furthermore we are able to consider also superlinear growth condition, as next theorem shows.
then inclusion (1) has at least a mild solution.
Proof. It is sufficient to prove that the operator Γ maps the ball Q L into itself. In fact, given any z ∈ Q L and y ∈ Γ(z), we have The conclusion then follows by Theorem 2.1, like Theorem 3.3.
4. Controllability results. In this section, we deals with the controllability for fractional semilinear differential inclusions (2) in a reflexive Banach space. We assume that  ([0, b]; X) such that f (t) ∈ F (t, x(t)) on t ∈ [0, b] and x satisfies the following integral equation We will consider the controllability problem for system (2), i.e., we will study conditions which guarantee the existence of a mild solution to problem (2) satisfying where x 1 ∈ X is a given point. A pair (x, u) consisting of a mild solution x(·) to (2) satisfying (9) and of the corresponding control u(·) ∈ L 1 q 1 ([0, b]; U ) is called a solution of the controllability problem.
We assume the standard assumption that the corresponding linear problem (i.e., when F (t, c) ≡ 0) has a solution. More precisely, we suppose that (H W ) The controllability operator W : U → X given by has a bounded inverse which takes value in U /ker(W ) and there exists a positive constant M 3 > 0 such that Let q 1 ∈ (0, q). We denote with S 1 : and we define the solution multioperator Π : It is easy to verify that the fixed points of the multioperator Π are mild solutions of fractional differential system (2) and (9). Proof. The linearity follows from the linearity of the integral operator and of the operators B, W −1 , we have Moreover, by (8) and (10), we obtain Fix n ∈ N, we denote by Π n = Π| Qn : Q n C([0, b]; X) the restriction of the multioperator Π on the set Q n . We describe some properties of Π n . ; X); we will prove that x ∈ Π n (p). Since p m ∈ Q n for all m and p m (t) p(t) for every t ∈ [0, b], it follows that p(t) ≤ lim inf m→∞ p m (t) ≤ n for all t (see [5,Proposition III.5]). The fact that x m ∈ Π n (p m ) means that there exists a sequence {f m }, f m ∈ Λ pm such that for every t ∈ [0, b], We observe that, according to (H 3 ), f m (t) ≤ µ n (t) for a.e. t and every m, i.e., {f m } is bounded and uniformly integrable and {f m (t)} is bounded in X for a.e. t ∈ [0, b]. Hence, by the reflexivity of the space X and by the Dunford-Pettis theorem (see [7, p. 294]), we have the existence of a subsequence, denoted as the sequence, and a function g such that f m g in L 1 q 1 ([0, b]; X). In view of the linearity and continuity for S i , we have S i f m S i g for i = 1, 2. Thus implying, for the uniqueness of the weak limit in X, that x 0 (t) = x(t) for all t ∈ [0, b].
Similar to the proof of Proposition 3.2, we can prove that g(t) ∈ F (t, p(t)) for a.e. t ∈ [0, b]. Proposition 4.3. The multioperator Π n is weakly compact.
Proof. We first prove that Π n (Q n ) is weakly relatively sequentially compact.
Let {p m } ⊂ Q n and {x m } ⊂ C([0, b]; X) satisfying x m ∈ Π n (p m ) for all m. By the definition of the multioperator Π n , there exist a sequence {f m }, f m ∈ Λ pm such that Further, as the reason for Proposition 4.2, we have that there exists a subsequence, denoted as the sequence, and a function g such that f m g in L 1 q 1 ([0, b]; X). Therefore Furthermore, by the weak convergence of {f m }, by (H A ), (8), (10), and the continuity of the operators S 1 and S 2 we have , for all m ∈ N and t ∈ [0, b]. As the reason for Proposition 4.2, it is then easy to prove that x m l in C([0, b]; X). Thus Π n (Q n ) is relatively weakly compact by Theorem 2.2. Proof. Fix p ∈ Q n , since F is convex valued, from the linearity of the integral and of the operators T (t), S (t), B and W −1 , it follows that the set Π n (p) is convex. The weak compactness of Π n (p) follows by Propositions 4.2 and 4.3.
We are able now to state the main results of this section.
Proof. We show that there exists n ∈ N such that the operator Π n maps the ball Q n into itself. Assume, to the contrary, that there exist sequences {z n }, {y n } such that z n ∈ Q n , y n ∈ Π n (z n ) and y n / ∈ Q n , ∀ n ∈ N. Then there exists a sequence {f n } ⊂ L 1 q 1 ([0, b]; X), f n (s) ∈ F (s, z n (s)) such that As the reason for Proposition 4.3, we have which contracts (13). Fix n ∈ N such that Π n (Q n ) ⊆ Q n . By Proposition 4.3 the set V n = Π n (Q n ) w is weakly compact. Let now W n = co(V n ), where co(V n ) denotes the closed convex hull of V n . By Theorem 2.3, W n is a weakly compact set. Moreover from the fact that Π n (Q n ) ⊂ Q n and that Q n is a convex closed set we have that W n ⊂ Q n and hence In view of Proposition 4.2, Π n has a weakly sequentially closed graph. Thus from Theorem 2.1, the system (1) has a solution. The proof is now completed.
We are able to prove the controllability results also under less restrictive growth assumptions, for instance sublinearity.
Furthermore we are able to consider also superlinear growth condition, as next theorem shows. where C 1 and C 2 are the positive constants defined in (14) and (15), then the controllability problem (2) and (9) has a solution.