Differential Inclusions of Arbitrary Fractional Order with Anti-periodic Conditions in Banach Spaces

In this paper, we establish various existence results of solutions for fractional differential equations and inclusions of arbitrary order q ∈ (m − 1, m), where m is an arbitrary natural number greater than or equal to two, in infinite dimensional Banach spaces, and involving the Caputo derivative in the generalized sense (via the Liouville–Riemann sense). We study the existence of solutions under both convexity and nonconvexity conditions on the multivalued side. Some examples of fractional differential inclusions on lattices are given to illustrate the obtained abstract results. in the generalized sense, measure of noncompactness, fractional lattice inclusions.


Introduction.
During the past two decades, fractional differential equations and fractional differential inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics and engineering.For some of these applications, one can see [18,23] and the references therein.For some recent developments on initial-value problems for differential equations and inclusions of fractional order, we refer the reader to the references [2,17,30,34,[38][39][40][41][42][43][44][45].
Some applied problems in physics require fractional differential equations and inclusions with boundary conditions.Recently, many authors have been studied differential inclusions with various boundary conditions.Some of these works have been done in finite dimensional spaces and of positive integer order, for example, Ibrahim et al. [25] and Gomaa [19,20].
Several results have studied fractional differential equations and inclusions with various boundary value conditions in finite dimensional spaces.We refer, for example, to Agarwal et al. [1] where conditions are established for the existence of solutions for various classes of initial and boundary value problems for fractional differential equations and inclusions involving the Caputo derivative.Next, Ouahab [34] studied a fractional differential inclusion with Dirichlet boundary conditions under both convexity and nonconvexity conditions on the multi-valued right-hand side and Ntouyas et al. [33] discussed the existence of solutions for fractional differential inclusions with three-point integral boundary conditions involving convex and non-convex multivalued maps.
For some recent works on boundary value problems for fractional differential equations and inclusions in infinite dimensional spaces, we refer to Benchohra et al. [7] where the existence of solutions are established for nonlinear fractional differential inclusions with two point boundary conditions.
Anti-periodic boundary value problems occur in the mathematical modeling of a variety of physical problems and have received a considerable attention.Examples include anti-periodic trigonometric polynomials in the study of interpolation problems, anti-periodic wavelets, antiperiodic conditions in physics, and so forth (for details, see [3]).
In this paper, we establish various existence results of solutions for fractional differential equations and inclusions of arbitrary order q ∈ (m − 1, m), where m is a natural number greater than or equal to two, in infinite dimensional Banach spaces, and involving the Caputo derivative in the generalized sense (via the Riemann-Liouville sense).More precisely, let J = [0, T], T > 0, E be a real separable Banach space with a norm • .We study the following fractional differential equations and inclusions with anti-periodic conditions: c D q g x(t) = f (t, x(t)) a.e. on J, x (k) (0) = −x (k) (T), k = 0, 1, 2, . . ., m − 1, (1.1) and c D q g x(t) ∈ F(t, x(t)) for a.e.t ∈ J, x (k) (0) = −x (k) (T), k = 0, 1, 2, . . ., m − 1, ( respectively, where c D q g x(t) is the generalized Caputo derivative which is defined via the Riemann-Liouville fractional derivative of order q with the lower limit zero for the function x at the point t, f : We would like to point out that Agarwal et al. [1] considered the problems (1.1) and (1.2) when m = 4 and the dimension of E is finite.Thereafter, Ahmad et al. [3] considered the problem (1.1) when m = 2, Ahmad [4] proved the existence of solutions for (1.1) when m = 3 and the dimension of E is finite, Cernea [12] proved existence theorems of solutions for (1.2) when m = 3 and the dimension of E is finite, and Ibrahim [26] established various existence achievements in infinite dimensional Banach spaces for (1.1) and (1.2) when m = 3. Alsaedi et al. [5] considered (1.1) in finite dimensional spaces in the case when m = 5.As a consequence, the obtained results in [1, 3-5, 11, 26] are particular cases of our derived results.We must mention that Kaslik and Sivasundaram [28] gave non-existence of periodic solutions of fractional order differential equations in the interval [0, ∞) by using the Mellin transform approach.However, we consider fractional differential equations and inclusions with antiperiodic conditions on [0, T] in this paper, not on the interval [0, ∞).In other words, we try to find solutions to fractional differential equations and inclusions with anti-periodic conditions on finite time intervals, not to find periodic solutions on the infinite interval.So our problem is much different from [28].
The present paper is organized as follows.In Section 2, we collect some background material and basic results about fractional calculus.Hence, we prove auxiliary lemmas which will be used later.In Section 3, we give existence results for (1.1).In Section 4, we prove various existence results for (1.2).We consider the case when the values of F are convex as well as nonconvex.In Section 5, we apply our abstract results to fractional differential inclusions on lattices continuing on our previous work [45].
The proofs rely on the methods and results for boundary value fractional differential inclusions, the properties of noncompact measure and fixed point techniques.

Preliminaries and notations
Let C (J, E) be the space of E-valued continuous functions on J with the norm x C(J,E) = max{ x(t) , t ∈ J}, AC n (J, E) be the space of E-valued functions f on J, which have continuous derivatives up to the order n − 1 on J such that f n−1 is absolute continuous on J, L 1 (J, E) be the space of all E-valued Bochner integrable functions on J with the norm Let G : J → 2 E be a multifunction.By S 1 G we denote the set of integrable selections of G, i.e., S 1 a.e.}.This set may be empty.For P cl (E)valued measurable multifunction, S 1 G is nonempty and bounded in L 1 (J, E) if and only if t → sup{ x : x ∈ G(t)} ∈ L 1 (J, R + ) (such a multifunction is said to be integrably bounded) (see [22,Theorem 3.2]).Note that S 1 G ⊆ L 1 (J, E) is closed if the values of G are closed and it is convex if and only if for almost all t ∈ J, G(t) is convex set in E. Definition 2.1.Let X and Y be two topological spaces.A multifunction G : For more information about multifunctions, see, [6,10,24,27].Now, let us recall the following definitions and facts about the integration and differentiation of fractional order.Definition 2.2.[29, p. 69] The Riemann-Liouville fractional integral of order q > 0 with the lower limit zero for a function f ∈ L p (J, E), p ∈ [1, ∞) is defined as follows: where the integration is in the sense of Bochner, Γ is the Euler gamma function, g q (t) = t q−1 Γ(q) , for t > 0, g q (t) = 0, for t ≤ 0 and * denotes the convolution of functions.For q = 0, we set It is known [29] that I q I β f (t) = I q+β f (t), β, q ≥ 0.Moreover, by applying Young's inequality, it follows that Then, I q maps L p (J, E) to L p (J, E).Let q be the least integer greater than or equal to the number q.The set of natural numbers is denoted by N. Definition 2.3.[29, p. 70] Let q > 0, m = q and f ∈ L 1 (J, E) be such that g m−q * f ∈ W m,1 (J, E).The Riemann-Liouville fractional derivative of order q for f is defined by where Note in the above definition that ϕ = f (m) and c k = f (k) (0), k = 0, 1, . . ., m − 1.In the following lemma, we mention some elementary properties for the Riemann-Liouville fractional integrals and derivatives.Lemma 2.4.Let q > 0 and m = q .
(iii) If p > 1 q and f ∈ L p (J, E) then I q f is continuous on J.
Proof.Proofs of these properties are exactly as in the scalar case [36,Chapter 1].
Definition 2.5.[29, p. 91] Let q > 0 and m = q .The Caputo derivative in the generalized sense (via Riemann-Liouville fractional derivative) of order q for a given function f is defined by provided that the right side is well defined.
Remark 2.6.Let q > 0, m = q and f : To proceed, we state the following lemma as a simple consequence of Lemma 2.4 and formula [36, (3.13)].
and hence, c D q g (I q f (t)) = f (t) a.e. on J.
The following lemma is essential to derive existence results of solutions for (1.1) and (1.2).
Lemma 2.9.Let m ∈ N and q ∈ (m ) ) Then, x (m−1) is continuous on J, c D q g x(t) exists a.e. for t ∈ J and c D q g x(t) = z(t) for a.e.t ∈ J, Proof.First we note, since for any k = 0, . . ., m − 1, it holds q − k ≥ q − (m − 1) > γ, by Lemma 2.8, the functions I q−k z, D k (I q z) are continuous on J. Hence b k+1 in (2.2), (2.3) and (2.4) are well-defined, and x (k) are continuous on J for all k = 0, . . ., m − 1.Now, in view of (ii) of Lemma 2.8, D q g z(t) exists for a.e.t ∈ J and c D q g z(t) = D q z(t) a.e.Thus, for a.e.
which implies (2.3).Furthermore, by differentiating both sides of (2.1), we get from (ii) of Lemma 2.4 for 1 which implies (2.4).Similarly, 2).The proof is finished.Now, by using Lemma 2.9 to establish formulae of solutions for (2.5), when m = 2, 3, 4, 5, we get known results mentioned in Introduction.We list them one by one as follows.

Consequently, γ (j)
k is the coefficients of in b k for any j, k = 1, 2, . . ., m with k ≤ j.Next, by inserting (2.16) into (2.15), after elementary calculations, we derive Using (2.18), we can derive step by step, for instance γ (r) We can easily check So by uniqueness, ηj = η j , i.e., we get On the other hand, (2.16) implies Summarizing, we arrive at the following result.
Our main tools are the Schaefer fixed point theorem for single-valued mappings and the O'Regan-Precup fixed point theorem [35,Theorem 3.1] for multivalued mappings, which is a generalization of the Mönch fixed point theorem.Lemma 2.13.Let Z : E → E be continuous and map every bounded subset into relatively compact subset.If the set E(Z) = {x ∈ E : x = λZ(x), λ ∈ [0, 1)} is bounded, then Z has a fixed point.Lemma 2.14.Let D be a closed convex subset of E, and N : D → P c (D). Assume the graph of N is closed, N maps compact sets into relatively compact sets and that, for some x 0 ∈ U, one has (2.22) Then T has a fixed point.
For more about fixed point theorems see [16].Finally, we give the concept of solutions for (1.1) and (1.2).
Of course, f and F in the above definitions are specified below with appropriate properties.

Existence results for (1.1)
In the following, we give the first existence result for (1.1).Theorem 3.1.Let m ∈ N, m ≥ 2 and q ∈ (m − 1, m).Let f : J × E → E be a function such that the following conditions are satisfied.
(H 3 ) For a.e.s ∈ J, the function f (s, •) maps any bounded subset into relatively compact subset in E.
Then, the problem (1.1) has a solution provided that δ < 1 for Proof.Let us consider the operator N : C(J, E) → C(J, E) defined by where z(t) = f (t, x(t)), t ∈ J.Note that, since f is continuous, then for every x ∈ C(J, E), the function z is continuous.Therefore, the operator N is well defined.We shall prove that N satisfies the assumptions of the Schaefer fixed point theorem.We split the proof into several steps.
Step 1. N is continuous.Let {x n } n∈N be a sequence such that x n → x in C(J, E).For any t ∈ J, Since f is continuous, we get N(x n ) − N(x) → 0, as, n → ∞.
Step 2. N maps bounded sets into bounded sets.Let r > 0 and B r = {x ∈ C(J, E) : x C(J,E) ≤ r}.According to (H 2 ), for any Then, N(B r ) is bounded.
Step 3. N maps bounded sets into relatively compact subsets.
Step 4. The subset {x ∈ C(J, E) : As a consequence of steps 1 → 4 and the Schaefer fixed point theorem, the function N has a fixed point.In view of (H 2 ) and Corollary 2.10, this fixed point is a solution for the problem (1.1).The proof is completed.
In the following theorem, we give another existence result for (1.1).Theorem 3.2.Let m ∈ N, m ≥ 2 and q ∈ (m − 1, m).Let f : J × E → E be a function such that (H 1 ), (H 3 ) and the following condition is satisfied.
Then, the problem (1.1) has a solution.
Proof.Let us consider the operator N : C(J, E) → C(J, E) defined as (3.2).Arguing as in the proof of Theorem 3.1, we can show that N is continuous and maps any bounded subset to relatively compact subset.It remains to show that the set {x ∈ C(J, E) : Let λ ∈ [0, 1) and x ∈ C(J, E) be such that x = λN(x).Then, by (H 4 ) and arguments of (3.3), we have This proves that the subset {x ∈ C(J, E) :  (H 6 ) There exists a function ϕ ∈ L ) and a nondecreasing continuous function Ω : R + → R + such that for any x ∈ E F(t, x) ≤ ϕ(t)Ω( x ) for a.e.t ∈ J.
and for every bounded subset D ⊆ E, χ(F(t, D)) ≤ β(t)χ(D) for a.e.t ∈ J, where χ is the Hausdorff measure of noncompactness in E.
(H 8 ) There is a positive number r such that for δ in (3.1).
Then the problem (1.2) has a solution.
Proof.In view of (H 5 ) and [27, Theorems 1.3.1 and 1.3.5]for every x ∈ C(J, E), the multifunction t → F(t, x(t)) has a measurable selection and by (H 6 ) this selection belongs to S 1 F(•,x(•)) .So, we can define a multifunction R : C(J, E) → 2 C(J,E) as follows: For any x ∈ C(J, E), a function y ∈ R(x) if and only if y = N ( f ), 2) with z = f .According to (H 6 ) and Corollary 2.10, any fixed point for R is a solutions for (1.2).So, our aim is to prove that the multivalued function R satisfies the assumptions of Lemma 2.14.Denote D = B r .It is clear since the values of F are convex, the values of R are convex also.
Step 1. R(D) ⊆ D. Let x ∈ D and y ∈ R(x).Then, by using the same arguments in Step 2 of the proof of Theorem 3.1, we can show that, for any t ∈ J y(t) C(J,E) ≤ δΩ(r).(4.3)This inequality with (4.2) imply ||y|| ∞ ≤ r.
Step 2. R(D) is a equicontinuous set in C(J, E).Let y ∈ R(D) and t 1 , t 2 ∈ J, (t 1 < t 2 ).Then there is x ∈ D with y ∈ R(x).By using the same arguments in Step 3 of the proof of Theorem 3.1 with (H 6 ), and recalling the definition of R, Then, the right hand side does not depend on x and uniformly tends to zero when t 2 → t 1 .
Step 3. The implication (2.22) holds with Then, for any n ≥ 1, there exists x n ∈ Z with y n ∈ R(x n ).This means that there is Observe that, by (H 6 ), for every natural number n, f n (s) ≤ ϕ(s) Ω(r) a.e.Then, by using (H 6 ) and the properties of the measure of noncompactness, one has for t ∈ J (see [9,21,32]) Observe that, since Z ⊆ C ⊆ conv ({x 0 } ∪ H), then from Step 2, Z is equicontinuous.Moreover, according to condition (H 7 ) for a.e.t ∈ J, This inequality with (4.1) imply that Z is relatively compact.
Step 4. R maps compact sets into relatively compact sets.Let G be a compact subset of D.
Arguing as above, one obtains from (4.5), for t ∈ J Now, since G is compact in C(J, E), then for t ∈ J, the subset G(t) is relatively compact in E. Therefore, χ({x n (t) : n ≥ 1}) = 0, t ∈ J. Hence, by (H 7 ) we get for a.e.t ∈ J, This with (4.6) imply χ({y n (t) : n ≥ 1}) = 0, for all t ∈ J.This together with the equicontinuity of {y n : n ≥ 1}, we conclude, from the Ascoli-Arzelà theorem, the set {y n : n ≥ 1} is relatively compact.Then, the sequence {y n } n∈N has a convergent subsequence.This shows that R(G) is relatively compact.
Step 5.The graph R is closed.Let {x n } n∈N be a sequence in D with x n → x in D and let y n ∈ R(x n ) with y n → y in C(J, E).We have to show that y ∈ R(x).Then, for any n ≥ 1, there is Hence, the sequence { f n } n∈N is semi compact, hence, it is weakly compact in L 1 (J, E) (see, [27,Proposition 4.2.1]).So, without loss of generality we can assume that f n converges weakly to a function f ∈ L 1 (J, R + ).From Mazur's lemma, for every natural number j there is a natural number k 0 (j) > j and a sequence of nonnegative real numbers λ j,k , k = j, . . ., k 0 (j) such that ∑ k 0 k=j λ j,k = 1 and the sequence of convex combinations z j = ∑ k 0 k=j λ j,k f k , j ≥ 1 converges strongly to f in L 1 (J, R + ) as j → ∞.Then we get for a.e.
As a result of the Steps 1 → 5 the multivalued R satisfies all assumptions of Lemma 2.14.Then, R has a fixed point which is a mild solution of (1.2).The proof is completed.Now we present our second result for the problem (1.2).Theorem 4.2.Let m ∈ N, m ≥ 2 and q ∈ (m − 1, m).Let F : J × E → P ck (E) be a multifunction.We suppose the following assumptions.
(H 10 ) There is a function ς ∈ L 1 σ (J, E), σ ∈ (0, q − (m − 1)) such that for every x, y ∈ E h (F(t, x), F(t, y)) ≤ ς(t) x − y for a.e.t ∈ J, where h is the Hausdorff metric, and Then the problem (1.2) has a solution provided that (4.10) Proof.According to (H 9 ), (H 10 ) and [27, Theorems 1.1.9,1.3.1 and 1.3.5]for every x ∈ C(J, E), the multifunction t → F(t, x(t)) has an integrable selection.So, we can define a multifunction R : C(J, E) → 2 C(J,E) as in Theorem 4.1.Now, we show that R satisfies the assumptions of the theorem of Covitz and Nadler [13].The proof will be given in two steps.
Step 1.The values of R are nonempty and closed.Since S 1 F(•,x(•)) is nonempty, the values of R are nonempty.In order to prove the values of R are closed, let x ∈ C(J, E) and {u n } n∈N be a sequence in R(x) such that u n → u in C(J, E).Then, according to the definition of R there is a sequence { f n } n∈N in S 1 F(•,x(•)) such that u n = N ( f n ).Now, let t ∈ J be a fixed.In view of (H 10 ), for every n ≥ 1, and for a.e.t ∈ J Then, for every n ≥ 1, and for a.e.t ∈ J, f n (t) ≤ ς(t)(1 + x C(J,E) ).This show that the set { f n : n ≥ 1} is integrably bounded.Arguing as in Step 5 in the proof of Theorem 4.1, we can show that u ∈ R(x).
Step 2. R is a contraction.Let z 1 , z 2 ∈ C(J, E) and y 1 ∈ R(z 1 ).Then, there is f ∈ S 1 F(•,z 1 (•)) such that This proves that u λ is closed.By applying Theorem of Bressan-Colombo [8, Theorem 3], there is a continuous map Z : C(J, E) → L 1 (J, E) such that Z(x) ∈ N(x), for every x ∈ C(J, E).Then, Z(x)(s) ∈ F(s, x(s)) a.e. for s ∈ J. Now consider a map π : C(J, E) → C(J, E) defined by π(x) = N (Z(x)). (4.17) Our aim now is to prove that the function π satisfies the assumptions of Mönch's fixed point theorem.Arguing as in (3.3) we can show that, by using (H 11 )(ii), for any x ∈ C(J, E), we have π(x) C(J,E) ≤ Υ for .
Let D = B Υ .Then π(D) ⊆ D. Also, arguing as in the proof of Theorem 4.1, we can show that π is continuous and satisfies (2.22).So, as result of the Mönch fixed point theorem, there is x ∈ C(J, E) such that x = π(x).Since p ∈ L 1 σ (J, R + ) and 0 < σ < q − (m − 1), by Corollary 2.10, the function x is a solution for (1.2).The proof is completed.

Applications to fractional lattice inclusions
In this section, we apply the above abstract results to a model of fractional differential inclusions on lattices with local neighborhood interactions represented by c D q 0,t x n (t) ∈ f n (t, x n−1 (t), x n (t), x n+1 (t)), n ∈ Z, a.e. on J, ( where q ∈ (m − 1, m), m ∈ N, m ≥ 2, f n : J × R 3 → 2 R be specified latter and x n ∈ R with lim n→±∞ x n = 0. We are looking for forced localized solutions of (5.1).We are motivated by [14,15,31,37], where countable systems of ODEs are studied.Here we study countable systems of fractional differential inclusions.Set E = c 0 = {x = {x n } n∈Z : x n ∈ R, lim n→±∞ x n = 0} with the norm x = sup n∈Z |x n |.Then E is a separable Banach space.The following two examples will illustrate the feasibility of our assumptions in the above theorems.
nonempty, closed and convex}, conv(B) (respectively, conv(B)) be the convex hull (respectively, convex and closed hull in E) of a subset B.

4 Existence results for (1.2) 4.1 Convex case
At first, we consider the case when values of F are convex.In the following, we give the first existence result for (1.2).Let m ∈ N, m ≥ 2 and q ∈ (m − 1, m).Let F : J × E → P ck (E) be a multifunction.Assume the following conditions.(H5 ) For every x ∈ E, t → F(t, x) is measurable and for a.e.t ∈ J, x → F(t, x) is upper semicontinuous.