Hilfer-type fractional differential switched inclusions with noninstantaneous impulsive and nonlocal conditions ∗

JinRong Wang, Ahmed Gamal Ibrahim, Donal O’Regan Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China sci.jrwang@gzu.edu.cn School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China Department of Mathematics, Faculty of Science, King Faisal University, Al-Ahasa 31982, Saudi Arabia agamal@kfu.edu.sa School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland donal.oregan@nuigalway.ie


Introduction
Fractional differential equations and fractional differential inclusions are important because of their applications in physics, mechanics, and engineering [6,25].For existence results for fractional differential equations and inclusions, we refer the reader to [1,2,10,25,26,38] and the references therein.Impulsive differential equations and impulsive differential inclusions have applications in physics, biology, engineering, medical type fractional differential switched inclusions inserting noninstantaneous impulsive and nonlocal conditions.The aim of the paper is to study the following nonlocal problem for Hilfer-type fractional noninstantaneous impulsive differential inclusions: D α,β s + i x(t) ∈ F t, x(t) , a.e.t ∈ (s i , t i+1 ], i = 0, 1, . . ., m, where 0 < α < 1, 0 is the Hilfer derivative with lower limit at s i of order α and type β, E is a real Banach space, 0 = s 0 < t 1 < s 1 < t 2 < • • • < t m < s m < t m+1 = b, x(t + i ), x(t − i ) are the right and left limits of x at the point t i , respectively, I 1−γ s + i is the Riemann-Liouville integral of order 1 − γ with lower limit at s i , and x(t).In addition, we set x(t − i ) = x(t i ).Moreover, F : J × E → 2 E − {φ} is a multifunction, g : P C 1−γ (J, E) → E, g i : [t i , s i ] × E → E, i = 1, 2, . . ., m, and x 0 is a fixed point of E. The space P C 1−γ (J, E) will be given in the next section.
The paper is organized as follows.In Section 2, we collect some background material concerning multifunctions and fractional calculus, and we introduce a measure of noncompactness on the space of piecewise weighted continuous functions.In Section 3, we establish the existence of mild solutions of (1), and in Section 4, we give an example to illustrate our theory.

Preliminaries
Denote L p (J, E) = {v : J → E: v is Bochner integrable} endowed with the norm v L p (J,E) = ( J v(t) p dt) 1/p , p ∈ [1, ∞), P b (E) = {B ⊆ E: B is nonempty and bounded}, P cl (E) = {B ⊆ E: B is nonempty, convex and closed}, P ck (E) = {B ⊆ E: B is nonempty, convex and compact}, conv(B) (respectively, conv(B)) be the convex hull (respectively, convex closed hull in E) of a subset B. Let C(J, E) be the Banach space of all E valued continuous functions from J to E with the norm x C(J,E) = sup t∈J x(t) .For a ∈ [0, b) and 0 γ 1, consider the weighted spaces of con- ), k = 0, 1, . . ., m}, and Let us recall some facts concerning multifunctions (see [5,20]).
Definition 1.Let X and Y be two topological spaces.A multifunction G : X → P (Y ) \ {∅} is said to be upper semicontinuous at x 0 ∈ X, u.s.c. for short, if for any open V containing G (x 0 ), there exists a neighborhood N (x 0 ) of x 0 such that G (x) ⊆ V for all x ∈ N (x 0 ).We say that G is upper semicontinuous if it is so at every x 0 ∈ X.
Lemma 1.Let X, Y be two Hausdorff topological spaces and G : , n 1, and (x n , y n ) → (x, y) with respect to the product topology on X × Y, then y ∈ G (x). (ii) If G is a closed and locally compact (i.e. for any x ∈ X, there is a neighborhood We recall some definitions and facts concerning fractional integral and derivatives [6,19,25]. Definition 2. The Riemann-Liouville fractional integral of order q > 0 with the lower limit at a for a function f ∈ L p ([a, b], E), p ∈ [1, ∞), is defined as follows: where the integration is in the sense of Bochner, Γ is the Euler gamma function defined by Γ(q) = ∞ 0 t q−1 e −t dt, 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 I 0 a + f (t) = f (t).
Definition 3. Let q > 0, m be the smallest integer greater than or equal to q, and The Riemann-Liouville fractional derivative of order q with the lower limit zero for f is defined by Definition 4. The Hilfer fractional derivative of order 0 < α < 1 and type 0 β 1 and with lower limit at a for a function f : a+ can also be written as ).Now we note some properties (the proofs are similar to the scalar case given in [14]).
, where c D α s + i denotes the Caputo fractional derivative of order α with lower limit at s + i , and γ = 1.In this case, (3) becomes and (4) becomes Based on Lemma 3, we give a concept of mild solutions of problem (1).
Then N has a fixed point.

Main results
In this section, we present existence results of mild solutions of (1).
Let p be a real number such that p > 1/α, S p F (.,x(.)) = {z ∈ L p (J, E): z(t) ∈ F (t, x(t)) a.e. for t ∈ J k , k = 0, 1, . . ., m}. and We introduce the following assumptions:  , x) is measurable for each x ∈ E (here E is separable) or alternatively F (., x) is strongly measurable for each x ∈ E (here E is not necessarily separable), then the multifunction F (., x) : J → P ck (E) has a measurable selection for every x ∈ E. If E is separable, then strongly measurable coincides with measurable.Also from [23, p. 29, Thm.1.3.5]note that if F (., x) : J → P ck (E) has a strongly measurable selection for every x ∈ E and if for a.e.t ∈ J, F (t, .): E → P ck (E) is upper semicontinuous, then for every strongly measurable function x : J → E, there exists a strongly measurable selection z : J → E with z(t) ∈ F (t, x(t)) a.e.(F 2 ) There exist a function ϕ ∈ L p (J, R + ) and a continuous nondecreasing function (F * 2 ) For any natural number n, there is a function ϕ n (t) for a.e.t ∈ J and https://www.mii.vu.lt/NA (F 3 ) There exists a function ς ∈ L p (J, R + ) such that for any bounded subset D ⊆ E and any k = 0, 1, 2, . . ., m, χ(F (t, D)) where η = b α−1/p ((p − 1)/(pα − 1)) (p−1)/p , and χ is the Hausdorff measure of noncompactness on E. (H g ) g : P C 1−γ (J, E) → E is continuous, completely continuous, and (H * g ) g : P C 1−γ (J, E) → E is Lipschitz continuous with the Lipschitz constant k and maps convergent sequences in P C(J, E) to strongly convergent sequences in E.
(H) For every i = 1, 2, . . ., m, g i : [t i , s i ] × E → E is uniformly continuous on bounded sets and for any t ∈ J, g i (t, .)maps bounded subsets of E to relatively compact subsets, and there exists a positive constant h i such that for any x ∈ E, We state the first existence result.
Theorem 1.Under assumptions (F 1 ), (F 2 ), (F 3 ), (H g ), and (H), problem (1) has a mild solution provided that where h = i=m i=0 h i .Proof.In view of (F 1 ), for every x ∈ P C 1−γ (J, E), S p F (.,x(.)) is non empty, and hence we can define a multifunction R : P C 1−γ (J, E) → 2 P C1−γ (J,E) as follows: let x ∈ P C 1−γ (J, E), a function y ∈ R(x) if and only if where f ∈ S p F (.,x(.)) .Our goal is to prove, using Lemma 6, that R has a fixed point.The proof will be given in several steps.It is easy to show that the values of R are convex.
Step 1.In this step, we claim that there is a natural number n such that R(B n ) ⊆ B n , where B n = {x ∈ P C 1−γ (J, E): x P C1−γ (J,E) n}.Suppose the contrary.
For I 2 , note that for almost s ∈ [0, t], Since ϕ ∈ L p (J, X) and ϕ(s) ds exists, then by the Lebesgue dominated convergence theorem we derive that lim δ→0 I 2 = 0 (independently of x).J,E) n 0 , it follows from the uniform continuity of g i on bounded sets that Then there is a x ∈ B n0 and f ∈ S p F (.,x(.)) such that for t ∈ (s k , t k+1 ], Next, let t, t + δ ∈ (s k , t k+1 ], δ > 0. Then we have Arguing as in Case 1, we conclude that Step 3. The graph of the multivalued function We need to show that y ∈ R(x).Recalling the definition of R, for any n 1, there is a f n ∈ S p F (.,xn(.))such that (15) holds.In view of ( 16), f n (t) n 0 ϕ(t) for every n 1 and for a.e.t ∈ J. Then E) is reflexive, and hence we can assume, without loss of generality, that (f n ) converges weakly to a function f ∈ L p (J, E).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 = k 0 (j), . . ., j, such that k0 k=j λ j,k = 1 and the sequence of convex combinations z j = k0 k=j λ j,k f k , j 1, converges strongly to f in L 1 (J, E) as j → ∞.
https://www.mii.vu.lt/NATake y n (t) = k0(n) k=n λ n,k y k .Then From Remark 1(ii), the continuity of g, the uniform continuity of g i on bounded sets, and the Lebesgue dominated convergence theorem we have that y n (t) → v(t) and Since y n → y, then y = v.Now for a.e.t, F (t, .) is upper semicontinuous with closed convex values, so from [5, Chap. 1, Sect.4] it follows that f ∈ S p F (.,x(.)) , so R is closed.
Step 5. R maps compact sets into relatively compact sets.Let B be a compact subset of B n0 .Let (y n ), n 1, be a sequence in R(B).Then there is a sequence (x n ), n 1, in B such that y n ∈ R(x n ) (so there exists f n ∈ S p F (.,xn(.))such that, for t ∈ J, (15) holds).We need to show that the set Z = {y n : n 1} is relatively compact in P C 1−γ (J, E).Note that, since B is compact in P C 1−γ (J, E), then from (F 3 ) we get for a.e.t ∈ J 0 , Arguing as in the previous step, we see that Z is relatively compact, and hence R(B) is relatively compact.Now apply Lemma 6.Then there is a x ∈ P C 1−γ (J, E) and f ∈ S p F (.,x(.)) such that Next, in view of (F 1 ), there is a δ , and from Lemma 3 the function x is a solution of (1).Remark 4. If, in (12), we assume lim sup x →∞ g(x) / x P C1−γ (J,E) = 0 is replaced by lim x →∞ g(x) / x P C1−γ (J,E) = 0, then, in (9), we could replace lim sup n→∞ Ω(n)/n = υ < ∞ with lim inf n→∞ Ω(n)/n = υ < ∞.
Next, we present the following affine results.Theorem 2. Under assumptions (F 1 ), (F * 2 ), (F 3 ), (H g ), and (H), problem (1) has a mild solution provided that Proof.The proof is similar to Theorem 1.The only difference is to show that there is a natural number n such that R(B n ) ⊆ B n under (F * 2 ).Suppose the contrary holds.Then, for any natural number n, there are x n , y n ∈ P C 1−γ (J, E) with y n ∈ R(x n ), x n P C1−γ (J,E) n and y n P C1−γ (J,E) > n, and y n is defined by (15).From Hölder's inequality we obtain that Similarly, we get for i = 1, 2, . . ., m that It follows from ( 18), (28), and (29) that By dividing both side by n and passing to the limit as n → ∞, we get 1 h + h/Γ(γ), which contradicts (27).This completes the proof.

Lemma 6 .
(See[29, Thm.3.1].)Let D be a closed convex subset of a Banach space X 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