Existence Results for Sequential Riemann–Liouville and Caputo Fractional Differential Inclusions with Generalized Fractional Integral Conditions

: Under different criteria, we prove the existence of solutions for sequential fractional differential inclusions containing Riemann–Liouville and Caputo type derivatives and supplemented with generalized fractional integral boundary conditions. Our existence results rely on the endpoint theory, the Krasnosel’ski˘i’s ﬁxed point theorem for multivalued maps and Wegrzyk’s ﬁxed point theorem for generalized contractions. We demonstrate the application of the obtained results with the help of examples. Wegrzyk’s ﬁxed point theorem

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Main Results
In this section, we present our main results in different subsections by applying a variety of fixed point theorems for multivalued maps.

Existence Result via Endpoint Theory
) and d(a, B) = inf b∈B d(a; b). Then (P cl,b (X), H d ) is a metric space (see [23]). Now we state the endpoint fixed point theorem that will be applied to prove our first result.

Lemma 2. ([24]
) Let (X, d) be a complete metric space and S : X → P cl,bd (X) be a multifunction such that H d (S x, Sy) ≤ ψ(d(x, y)) for all x, y ∈ X, where P cl,bd (X) is the collection of all nonempty closed and bounded subsets of X and ψ : [0, ∞) → [0, ∞) is an upper semi-continuous function such that ψ(t) < t and lim inf t→∞ (t − ψ(t)) > 0 for all t > 0. Then S has a unique endpoint if and only if S has an approximate endpoint property.
For more details about endpoint theory, we refer the reader to the article [25].
By Lemma 1, we define an operator G : Z → P (Z ) as follows: t q+r−1 RL I q+r v(T) for v ∈ S F,u , where S F,u denotes the set of selections of F defined by is a nondecreasing upper semi-continuous mapping such that lim inf t→∞ (t − ψ(t)) > 0 and ψ(t) < t for all t > 0. Moreover, assume that F : [0, T] × R → P cp (R) is an integrable bounded multifunction such that F(·, u) : [0, T] → P cp (R) is measurable for all u ∈ R; here P cp (R) denotes the collection of all nonempty compact subsets of R. In addition, we assume that there exists a function where Φ is defined by (6). If the multifunction G (defined by (7)) has the approximate endpoint property, then the inclusion problem (1)-(2) has a solution.
Proof. Our proof will be complete once it is shown that the multifunction G : Z → P (Z ) defined by (7) has an endpoint. To do this, we show that the operator G(u) is a closed subset of P (Z ) for all u ∈ Z. Note that the multivalued map t → F(t, u(t)) is measurable and has closed values for all u ∈ Z, and therefore has a measurable selection. So S F,u is nonempty for all u ∈ Z. Let {z n } n≥1 be a sequence in G(u) with z n → z for u ∈ Z. For every n ∈ N, choose v n ∈ S F,u n such that t q+r−1 RL I q+r v n (T) In view of the compactness of F, we deduce that the sequence {v n } n≥1 has a subsequence which converges to some v ∈ L 1 ([0, T]). Let this subsequence be denoted by {v n } n≥1 again. Clearly v ∈ S F,u and for all t ∈ [0, T], Thus z ∈ G(u) and consequently G is closed-valued. On the other hand, G(u) is a bounded set for all u ∈ Z as F is a compact multivalued map.
there exists z ∈ F(t, u(t)) provided that Let us consider the multivalued map V : [0, T] → P (R) given by We define the element h 2 ∈ G(u) as follows: Then, one can get Therefore there exists u * ∈ Z such that G(u * ) = {u * }, since the multifunction G (by the hypothesis) has an approximate endpoint property. Therefore we deduce that the problem (1)-(2) has a solution u * . The proof is complete.

Example 1. Consider the following inclusions problem containing Riemann-Liouville and Caputo derivative operators with generalized fractional integral boundary conditions
Here Let us take ψ(y) = y/3 and note that it is nondecreasing upper semi-continuous Observe that inf u∈Z sup s∈G(u) u − s = 0 in view of sup u∈G(0) u = 0. Therefore, the operator G has the approximate endpoint property. Clearly the hypothesis of Theorem 1 is satisfied. So the conclusion of Theorem 1 applies to the problem (8) with F(t, x) given by (9).

Existence Result via Krasnosel'skiȋ's Multi-Valued Fixed Point Theorem
This subsection is concerned with the second existence result for the problem (1)-(2) when the map F in (1) is L 1 −Carathéodory and convex valued. The proof of this result relies on Krasnosel'skiȋ's fixed point theorem for multivalued maps [26].
In the following we need the following assumptions: is the collection of all nonempty compact and convex subsets of R; (H 2 ) There exists a function p ∈ C([0, T], R + ) such that (H 3 ) There exists a function k ∈ Z satisfying H d (F(t, x), F(t, y)) ≤ k x − y , for a.e t ∈ [0, T] and all x, y ∈ Z and that k < 1/Λ, where

Proof. Define the multivalued operators
t q+r−1 RL I q+r v(T) Notice that G = A + B, where G is given by (7). In several steps, it will be shown that A and B satisfy the hypothesis of Krasnosel'skiȋ's multivalued fixed point theorem. Let B r = {x ∈ Z : x ≤ r} be a bounded set in Z. The operators A and B define the multivalued operators A, B : B r → P cp,c (Z ). Observe that the operator B is equivalent to the composition L • S F , where L is the continuous linear operator on For an arbitrary element x ∈ B r , let {v n } be a sequence in S F,x . Then v n (t) ∈ F(t, x(t)) for almost all t ∈ [0, T]. As F(t, x(t)) is compact for all t ∈ J, we can find a convergent subsequence of {v n (t)} (also labeled as {v n (t)}) converging in measure to some v(t) ∈ S F,x for almost all t ∈ [0, T]. On the other hand, continuity of L implies that L(v n )(t) → L(v)(t) pointwise on [0, T].
To ensure the uniform convergence, we have to establish that {L(v n )} is an equi-continuous sequence. Take τ 1 , τ 2 ∈ [0, T] with τ 1 < τ 2 and x ∈ B r . Then Obviously the right hand of the above inequality tends to zero as τ 2 → τ 1 independent of x ∈ B r . So {L(v n )} is an equi-continuous sequence. In consequence, the Arzelá-Ascoli theorem applies and hence there exists a uniformly convergent subsequence {v n } (labeled as {v n } again) such that L(v n ) → L(v).
Step 1: A is a multi-valued contraction on Z. Take x, y ∈ Z and h 1 ∈ Ax. Then, for each t ∈ [0, T], there exists v 1 (t) ∈ F(t, x(t)) such that Since H d (F(t, x), F(t, y)) ≤ k x − y , therefore, we can findŵ ∈ F(t, y) satisfying Define K(t) = {ŵ ∈ R : |v 1 (t) −ŵ| ≤ k x − y }. Then the multivalued operator U defined by U(t) = S F,y ∩ K(t), is measurable and nonempty. Let v 2 be a measurable selection for U, which exists by Kuratowski-Ryll-Nardzewski's selection theorem [27]. Then v 2 (t) ∈ F(t, y(t)) and for each t ∈ [0, T], we have |v For each t ∈ [0, T], let us define It follows that h 2 ∈ Ay and Consequently, Interchanging the roles of x and y, we obtain an analogous inequality: which, together with the condition k Λ < 1, implies that A is a multivalued contraction.
Step 2: We show that B is compact and upper semicontinuous through certain claims.
CLAIM I: B maps bounded sets into bounded sets in Z.
t q+r−1 RL I q+r v(T) + RL I q+r v(t).
CLAIM II: B maps bounded sets into equi-continuous sets.
q+r−1 RL I q+r |v(s)|(T) T q+r Γ(q + r + 1) independently of x ∈ B r . Then, by the Arzelá-Ascoli theorem, we deduce that B : Z → P (Z ) is completely continuous. Thus it follows by Claims I and II that B is completely continuous. Hence, by Proposition 1.2 in [28], it will be upper semicontinuous once it is shown to be closed graph. This will be shown in the next claim.
CLAIM III: B has a closed graph. Letting x n → x * , h n ∈ B(x n ) and h n → h * , we show that h * ∈ B(x * ). For h n ∈ B(x n ), we can find v n ∈ S F,x n such that, for each t ∈ [0, T], h n (t) = 1 Ω Ω 1 − Ω 2 Γ(q) Γ(q + r) t q+r−1 RL I q+r v n (T) + RL I q+r v n (t).
We will show that there exists v * ∈ S F,x * such that for each t ∈ [0, T], t q+r−1 RL I q+r v * (T) + RL I q+r v * (t).
If we consider the linear operator Θ : then we note that as n → ∞. Thus, it follows by a closed graph result [29] that Θ • S F is a closed graph operator. Further, let h n (t) ∈ Θ(S F,x n ). Since x n → x * , we have that t q+r−1 RL I q+r v * (T) + RL I q+r v * (t), for some v * ∈ S F,x * . Hence B has a closed graph.Thus, the operator B is compact and upper semicontinuous.
Step 3: Now, we establish that A(x) + B(x) ⊂ B r for all x ∈ B r . Take arbitrary elements x ∈ B r with r > p Φ (Φ defined by (6)) and h ∈ B. Then, selecting v ∈ S F,x , we have Then we have This shows that A(x) + B(x) ⊂ B r for all x ∈ B r . Thus, the operators A and B verify the hypothesis of Krasnosel'skiȋ's multivalued fixed point theorem and hence there exists a solution x ∈ A(x) + B(x) in B r . Therefore there exists a solution of the problem (1)-(2) in B r which completes the proof.

Existence Result via Wegrzyk's Fixed Point Theorem
In this subsection we apply Wegrzyk's fixed point theorem [30] to prove an existence result for problem (1)-(2) when the right hand side of the inclusions (1) is not necessarily nonconvex valued.
Thus the set S F,x is nonempty for each x ∈ Z. Now we verify that the operator G satisfies the hypothesis of Lemma 3. Let us first show that G(x) ∈ P cl (Z ) for each x ∈ Z. Let {u n } n≥0 ∈ G(x) be such that u n → u in Z as n → ∞. Then u ∈ Z and we can find v n ∈ S F,x n such that, for each t ∈ [0, T], u n (t) = 1 Ω Ω 1 − Ω 2 Γ(q) Γ(q + r) t q+r−1 RL I q+r v n (T) Since F is compact valued, we pass onto a subsequence (if necessary) to get that v n converges to v in L 1 ([0, T], R). So v ∈ S F,x and for each t ∈ [0, T], we have t q+r−1 RL I q+r v(T) Therefore, u ∈ G(x). Next we establish that Let x,x ∈ Z and h 1 ∈ G(x). Then there exists v 1 (t) ∈ F(t, x(t)) such that, for each t ∈ [0, T], In consequence, we can find w ∈ F(t,x(t)) satisfying |v 1 (t) − w(t)| ≤ µ(t)ν(|x(t) −x(t)|), t ∈ [0, T].
while Krasnosel'skiȋ's fixed point theorem for multivalued maps is applied to derive the second result (Theorem 2). In the third result (Theorem 3), we used Wegrzyk's fixed point theorem for generalized contractions to establish the existence of solutions for the given problem. It is imperative to note that Wegrzyk's fixed point theorem is a generalization of Covit and Nadler's fixed point theorem [33] in the sense that it deals with the generalized contractions. Thus Theorem 3 holds for several values of the function ν (defined in (A 2 )); for example, ν(t) = ln(1 + t) 3 , ν(t) = t (contraction case), etc. Some interesting special cases of our results follow by fixing the parameters involved in the boundary conditions (2). For instance, our results correspond to Dirichlet boundary conditions if we take γ i = 0 for all i = 1, 2, . . . , m and σ j = 0 for all j = 1, 2, . . . , n. Fixing γ i = 0 for all i = 1, 2, . . . , m and σ j = 0 for some j = 1, 2, . . . , n, we obtain the existence results for the boundary conditions of the form: x(0) = 0, x(T) = n ∑ j=1 σ j ρ j I α j ,β j η j ,κ j x(δ j ). On the other hand, we obtain the existence results for (1) associated with the boundary conditions: γ iρ i Iᾱ i ,β ī η i ,κ i x(ξ i ), x(T) = 0, by letting γ i = 0 for some i = 1, 2, . . . , m, and σ j = 0 for all j = 1, 2, . . . , n. We emphasize that the existence results indicated for special forms of the boundary conditions are new.