On a hybrid fractional Caputo–Hadamard boundary value problem with hybrid Hadamard integral boundary value conditions

In the present research article, we find some important criteria on the existence of solutions for a class of the hybrid fractional Caputo–Hadamard differential equations and its corresponding inclusion problem supplemented with hybrid Hadamard integral boundary conditions. In this direction, we utilize some theorems due to Dhage’s fixed point results in our proofs. Finally, we demonstrate two numerical examples to confirm the validity of the main obtained results.


Introduction
One way mathematics helps economics is to become more powerful in modeling theory so that different types of processes with distinct parameters can be written in mathematical formulas. In this case, different software can be developed to allow for more cost-free testing and less material consumption. One of basic methods in this way is working with fractional calculus. Nowadays, many researchers are studying advanced fractional modelings and their related existence results and qualitative behaviors of solutions for distinct fractional problems (see, for example, [1][2][3][4][5]). In recent decades, fractional hybrid differential equations and inclusions with complicated boundary value conditions have achieved a great deal of interest and attention of many researchers (see, for example, [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). Also, there are many works on the fractional Hadamard derivative and its applications in different fields (see, for example, [22][23][24][25][26]).

Preliminaries
Prior to proceeding to reach the main purposes, we first recall some essential auxiliary concepts which are needed throughout the paper. Let γ ≥ 0 and assume that the realvalued function is integrable on (a, b). In this case, the Hadamard fractional integral of a continuous function : (a, b) → R of order γ is defined by H I 0 a + ( (t)) = (t) and provided that the RHS integral is finite-valued [31,32]. Note that, for each γ 1 , γ 2 ∈ R + , we have H I [32]. It is evident that for all t > a by letting γ 2 = 0 [32]. Now, let n = [γ ] + 1 or n -1 ≤ γ < n. The Hadamard fractional derivative of order γ for a function : (a, b) → R is defined by provided that the RHS integral has finite values [31,32]. The Caputo-Hadamard fractional derivative of order γ for an absolutely continuous function if the RHS integral exists [31,32]. Again, let ∈ AC n R ([a, b]) so that n -1 < γ ≤ n. In [31,32], it has been verified that the solution of the Caputo-Hadamard fractional differential equation CH D γ a + ( (t)) = 0 has general solutions of the form (t) = n-1 i=0 c i (ln t a ) i , and we have + · · · + c n-1 ln t a n-1 for any t > a.
Here, consider the normed space (X , · X ). Then all subsets of X , all closed subsets of X , all bounded subsets of X , all convex subsets of X , and all compact subsets of X are denoted by collections P(X ), P cls (X ), P bnd (X ), P cvx (X ), and P cmp (X ), respectively. A setvalued map Ψ is convex-valued if, for each ∈ X , the set Ψ ( ) is convex. The set-valued map Ψ has an upper semi-continuity property whenever, for every * ∈ X , Ψ (ρ * ) belongs to P cls (X ) and, for each open [33]. Moreover, * ∈ X is a fixed point for the set-valued map Ψ : X → P(X ) whenever * ∈ Ψ ( * ) [33]. The notation FIX(Ψ ) represents the set of all fixed points of Ψ [33]. Consider the metric space X furnished with the metric d X .
A set-valued operator Ψ has the complete continuity property if the set Ψ (W) has the relative compactness property for all W ∈ P bnd (X ). Let Ψ : X → P cls (Q) have the upper semi-continuity property. Then Graph(Ψ ) ⊆ X × Q is a closed set. On the other hand, assume that Ψ has a closed graph with the complete continuity property. Then Ψ has the upper semi-continuity property [33]. We say that Ψ : [1, e] × R → P(R) is a Caratheodory set-valued map if the mapping → Ψ (t, ) is upper semi-continuous for almost all t ∈ [1, e] and the mapping t → Ψ (t, ) is measurable for each ∈ R [33,34]. In addition, a Caratheodory set-valued map Ψ : [1,e] |q| : q ∈ Ψ (t, ) ≤ φ r (t) for almost all t ∈ [1, e] and for each | | ≤ r [33,34]. All selections of Ψ at ∈ C R ([1, e]) are defined by the following set: [33,34]. As it has been verified before in [33], we have (SEL) Ψ , = ∅ for all ∈ C X ([1, e]) whenever dim X < ∞. We need next results.

Theorem 2 ([36]) Consider the separable Banach space
, and the linear continuous map Ξ : is an operator which belongs to

Theorem 3 ([37])
Consider the Banach algebra X . Assume that there are a set-valued map Φ 2 : X → P cmp,cvx (X ) and a single-valued map Φ 1 : X → X satisfying:

Main results
In this part of the paper, we intend to state our main theoretical findings on the existence results. To reach this aim, we consider X = { (t) : (t) ∈ C R ([1, e])} equipped with the supremum norm X = sup t∈ [1,e] | (t)| and the multiplication action on the space X defined by ( · )(t) = (t) (t) for all , ∈ X . Then an ordered triple (X , · X , ·) is a Banach algebra. In this moment, we present an essential lemma which converts fractional BVP (1)-(2) into integral equation.

Lemma 4 Assume thatα belongs to X . Then 0 is a solution for the hybrid Caputo-Hadamard equation
furnished with hybrid Hadamard integral boundary value conditions iff the function 0 is a solution for the following Hadamard integral equation: Proof Let 0 be a solution for hybrid equation (5). Then the general solution of homogeneous equation (5) is obtained by the equality 0 (t) Now, we employ the following integro-derivative operators of arbitrary orders on both sides of equation (8), and we get Corresponding to the boundary value conditions, we obtain By inserting the valuesm * 0 ,m * 1 , andm * 2 into (8), we get This means that 0 is a solution for integral equation (7). On the contrary, it is easy to check that 0 satisfies fractional hybrid BVP (5)-(6) if 0 is a solution for the integral equation of fractional order (7). Now, we derive our first result about the existence of solutions of problem (1)- (2).
In what follows, we are going to provide another essential result for the fractional hybrid inclusion problem (3)-(4). Existence results herein are carried out in the light of the assumptions of Theorem 3. Here, we can formulate the desired theorem on the existence of a solution function of the above form.

Definition 6 We say that the function ∈ AC R ([1, e]) is a solution for the hybrid inclusion BVP (3)-(4) whenever there exists an integrable function
Theorem 7 Assume that the following statements are valid.

Conclusion
It is known that the most natural phenomena are modeled by different types of fractional differential equations and inclusions. This diversity in investigating complicate fractional differential equations and inclusions increases our ability for exact modelings of more phenomena. This is useful in making modern software which helps us to allow for more costfree testing and less material consumption. In this work, we investigate the existence of solutions for a hybrid fractional Caputo-Hadamard differential equation and its related inclusion problem with hybrid Hadamard integral boundary value conditions. In this way, we use some Dhage's fixed point results in our proofs. Eventually, we give two numerical examples to support the applicability of our findings.