Positive Solvability for Conjugate Fractional Differential Inclusion of ( k , n − k ) Type without Continuity and Compactness

: The monotonicity of multi-valued operators serves as a guideline to prove the existence of the results in this article. This theory focuses on the existence of solutions without continuity and compactness conditions. We study these results for the ( k , n − k ) conjugate fractional differential inclusion type with λ > 0, 1 ≤ k ≤ n

The (k, n − k) conjugate differential equation of second order (n = 2) has been studied in [9]. After that, the main results with a high order were presented in [10][11][12][13] and the references given therein. The development, by adding the fractional derivative, is important and necessary to prove the strong extent of nonlinearity theory and its applications. The research into the conjugate fractional type of problem began with the case (n − 1, 1) in [14,15]. As far as we know, there are no papers exploring the existence of solutions for the (k, n − k) conjugate differential inclusion of fractional order.
Oscillation theory began with Sturm's work in 1836, and was further developed for the fifty years before 1996. At present, it is a full, self-contained, discipline, turning more towards nonlinear and functional differential equations. On one hand, oscillation theory has two research fields; one with linear operators and the other with nonlinear functional operators. On the other hand, it has two different fields under the conjugate and disconjugate operators topics. See [16] for a good overview. This theory strongly influences investigatations of strong solution results for (k, n − k) conjugate differential boundary value problems.
The aim of this paper is to take one more step with oscillation theory, to develop the previous results from another aspect, which is to study (n − k, k) fractional conjugate problems with multi-valued mappings instead of single-valued mappings. These results are devoted to the sufficient conditions for the existence of a positive solution to the problem where Θ is a monotone multi-valued map.
There are several contributions that generalize differential equations and inclusions and study their solvability. They depend on investigations into the properties of the solutions (existence, uniqueness, stability, controllability, . . . , etc.), see [17,18] and the references given therein.
It is worth mentioning that the literature on the existence and uniqueness of solutions to fractional differential equations is expanding at present, and this problem has drawn the attention of many contributors [19][20][21][22][23][24][25][26][27].
In the next section, we provide some basic definitions, properties, lemmas and theorems used to investigate the main upshots. The main theorems and results are included in Section 3. Consequently, Section 4 comes with some applications. Finally, Section 5 is formed by a brief overview of current and future works.

Fractional Calculus
In this subsection, we recall the definitions and some fundamental facts of Caputo-Hadamard fractional integral and the corresponding derivative [28,29].
Then, the Caputo-Hadamard integral CH I ρ b 0 of fractional order ρ is written by where δ := τ d dτ and

Monotone Multi-Valued Operators and Corresponding Fixed Point Theorems
We recall the following definitions and results from [30][31][32].
In addition to assumptions (1) and (2), the map A is a L 1 -Caratheodory map if for each A has a closed graph if, whenever v n → v * , z n → z * and z n ∈ A(v n ), it holds z * ∈ A(v * ).
Let (E, . ) be a real Banach space and P a normal cone of E. A partial ordering " " is induced by the cone P, namely, for any w, z ∈ E, w z if and only if z − w ∈ P.
Let X and Y be subsets of E. If, for all w ∈ X, there exists z ∈ Y such that w z, then we write X Y. For a nonempty subset D of X and A : D → 2 X /∅, we say that A is increasing (decreasing) upward if and only if, for all u, v ∈ D with u v, it is true that, for If A is increasing (decreasing) upward and downward, we say that A is increasing (decreasing).
Theorem 1. Let X be a real Banach space and P a normal cone of X. Suppose that T : X → 2 X /{∅} is an increasing multi-valued operator satisfying: (1) For any w ∈ X, T(w) is a nonempty and closed subset of X.
(2) There exists a linear operator L : X → X with a spectral radius r(L) < 1 and L(P) ⊂ P such that, for any w, z ∈ X with w z, (i) for any u ∈ T(w) there exists v ∈ T(z) satisfying Then T has a fixed point in X.
Theorem 2. Let X be a real Banach space and P a normal cone of X. Suppose that T : X → 2 X /{∅} is a decreasing multi-valued operator satisfying; (1) For any w ∈ X, T(w) a nonempty and a closed subset of X.
(2) There exists a constant c ∈ (0, 1) such that, for any w, z ∈ X with w z, Then T has a fixed point in X.

Main Results
To show the main results, we need to explain some basic facts. Let Σ = R, J = [b 0 , b] and define the set-valued map S Θ (w) by Then we have below Lemmas:

Some Auxiliary Results
Lemma 3. Let η(τ) ∈ L 1 ([b 0 , b], R) and consider the following problem then, the unique solution is given by where Proof. To get (8) and (9) we apply the integral operator CH I ρ b 0 to the both sides of (5). We obtain Using the conditions in (6), we find c r = 0, ∀r = 1, . . . , n − 1 and then Under the effect of the condition (7) we have which completes the proof.
Define the normal cone P ⊂ E by the set of all non-negative functions P = {w(τ)| w(τ) ≥ 0}. Consider the multi-valued map Let η k λ (τ) ∈ H λ k (τ, w(τ)) and consider the problem which has a solution We fix the set K + with values of (τ, s) such that Depending on (14), we define the green functions G K + (τ, s) by , then the problem (10)-(12) admits a unique solution with respect to (14) given by where G K + (τ, s) is defined by (15). (14) we obtain the result.
Proof. The proof is divided in two cases: then, by using the fact that log w < w and log w ≤ log z if w ≤ z, we have From both cases, we get (17).

Now, define the linear operator
Define the set Λ by Consequently, define the multi-valued operator N Λ K + (w)(τ) by the relation

Main Results
Consider X = L 1 ([b 0 , b], R) and P = {w ∈ X| w ≥ 0} as a normal cone in X. Then we can study two different cases.

Theorem 3 (Increasing Map). Suppose that
is a L 1 -Caratheodory multi-valued map subject to the following conditions: θ w ∈ S Θ (w), θ z ∈ S Θ (z) and w, z ∈ P with w z, it holds Then, the problem (1)-(3) has at least one positive solution.
Proof. Here, bearing in mind Theorem 1, the proof is shown in the following steps: Step1: We claim that N Λ K + has a closed graph. Indeed, let us consider u n (τ) ∈ N Λ K + (w n ), where u n → u * and w n → w * . It follows that there exists θ w n ∈ S Θ (w n ) such that u n (τ) = ∆ λ K + θ w n (τ). Since the operator ∆ λ K + is a closed linear operator (Lemma 2) and u n → u * ; then, there exists θ w * ∈ S Θ (w * ) such that Take u * (τ) = ∆ λ K + θ w * (τ): then u * (τ) ∈ N Λ K + (w * ) concludes the proof of the claim.
Step2: Define the linear operator Then, L(P) ⊆ P and Since S Θ is increasing upward, then there exists θ z ∈ S Θ (z) with θ w θ z . Consequently, By Theorem 1, the previous steps imply that the problem (1)-(3) admits at least a solution in P, i.e, a positive solution.

Theorem 4. (Decreasing Map)
Suppose that Θ is a L 1 − −Caratheodory multi-valued map subject to the following conditions: (N 6 ) For the function M(s) defined as in Lemma 5 we have the condition where Λ is defined in (19) and Υ * b = Υ b ∞ . Then, the problem (1)-(3) has at least one positive solution.

Applications
Here, we present some examples related to the main results. To obtain the desired conditions, we make use of the Poincaré inequality in L 1 (J, R).
Then we have the followings (2) It is known that the function cos w is decreasing in the compact interval [0, π]. Therefore, Θ is decreasing since 0 < πw 2 n (π+w) < π 2 .
(3) For all 0 ≤ w z we have which tends to take Υ b (τ) = 1

Conclusions
For monotone-type multi-valued operator, we investigate the existence results and provide some applications for them. Our analysis relies on nonlinear monotone fixed point theorems and is connected with oscillation theory in the sense of (k, n − k) conjugate-type differential operator. It is worth generalizing the results on fractional differential equation by multi-valued maps in order to get new extents for phenomena modeling. Data Availability Statement: This study did not report any data.