On Caputo–Hadamard type coupled systems of nonconvex fractional differential inclusions

*Correspondence: fahd@cankaya.edu.tr; tabdeljawad@psu.edu.sa 2Department of Mathematics, Çankaya University, 06790 Etimesgut, Ankara, Turkey 4Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia Full list of author information is available at the end of the article Abstract This research article is mainly concerned with the existence of solutions for a coupled Caputo–Hadamard of nonconvex fractional differential inclusions equipped with boundary conditions. We derive our main result by applying Mizoguchi–Takahashi’s fixed point theorem with the help of P-function characterizations.


Introduction
In the previous two decades, fractional calculus has earned sizeable importance owing to diverse applications in scientific and engineering problems. Fractional-order boundary value problems, in particular, have became a rapidly growing area due to features of fractional derivatives which make the systems of fractional-order practical and realistic than the corresponding classical systems. For some current work, we suggest [1][2][3][4][5][6][7][8][9][10]. There are numerous definitions of fractional differentiation operators in the literature, the most common is the classical Riemann-Liouville type fractional derivative after which a beneficial alternative has been introduced to cope with disadvantages caused by the Riemann-Liouville expression, the so-called Caputo derivative. Fractional derivatives within the frame of Hadamard type differ from the Riemann-Liouville type and the Caputo type due to the appearance of a logarithmic function in the definition of the Hadamard derivatives. One can find manifold monographs and articles devoted exclusively to the theory of fractional derivatives, not merely on mathematical subjects but also physics, applied sciences, engineering, etc.; see [11][12][13][14].
This article involves the so-called Caputo-Hadamard fractional derivatives which modifies the Hadamard derivative into a more beneficial type using Caputo approach [15,16] Differential inclusions are found to be of great advantage as the field of study for these inclusions covers theoretical treatment, inequalities, and applications in a variety of disciplines in physical and industrial sciences. Examples cover optimal control systems [17], isothermal dynamics with stochastic velocities [18], control problems [19] and sweeping processes [20]. The study of fractional-order differential inclusions was first launched by Sayed and Ibrahim [21]. Since then the literature on fractional-order differential inclusions has found various qualitative results. We refer the reader to Ref. [22][23][24][25] for the recent advancement on the topic.
The study of coupled systems of fractional-order differential equations has also received great attention as such systems emerge in a diversity of problems of biological phenomena and environmental issues. For details and examples, the reader is referred to [26][27][28] and the references mentioned therein.
Recently, a class of coupled fractional-order differential inclusion was discussed in [29], of the form subject to the coupled boundary condition where c D z is the Caputo-Liouville fractional derivative of order 1 < z ≤ 2, z ∈ (γ , ζ ), W and Z are given multivalued maps. The authors investigated the existence criteria for solutions by applying standard fixed-point theorems for multivalued maps. Motivated by the above and inspired by the work in [26][27][28], in this paper, we study the following coupled fractional differential inclusions: where Hc D r is the Caputo-Hadamard fractional derivative of order 1 < r ≤ 2, r ∈ (γ , ζ ), δ = y d dy , and W , Z : is the family of all nonempty subsets.
The objective of the present paper is to establish new existence criteria of solutions for the problem (1.3)-(1.4) by applying Mizoguchi-Takahashi's fixed point theorem for multivalued maps. To the best of our knowledge, the application of fixed-point theorem due to Mizoguchi and Takahashi to the framework of the current problem is new and has not been investigated elsewhere.
The article is designed as follows. Some introductory materials that we need in the sequel are presented in the next section. The main results are derived in Sect. 3. An example is provided to illustrate the theory in Sect. 4.

Axillary results
Let J be a finite interval on R. We denote by = C( [1, T], R) the set of continuous functions on [1, T] supplied with the norm w = max θ∈ [1,T] |w(θ )|. The product set ( × , (w, z) ) is a Banach space endowed with the norm is the set of those Lebesgue measurable functions w : [1, T] → R with the norm Now we recall some essential outlines on multivalued maps [30]. For a normed space ( , · ), let is measurable. Next, we shall recall some known results concerning fractional operators.

Definition 2.1 ([31])
The fractional-order integral operator of Hadamard type of a function f ∈ L 1 ([1, T], R) is given as provided the integral exists.

Definition 2.2 ([15]) For a given function
, the Caputo-Hadamard fractional derivative of order r > 0 is defined as follows: where n = [r] + 1, [r] is the integer part of r and (·) is the Gamma function defined by (w) = ∞ 0 y w-1 e -y dy. If r = n ∈ N we have Hc D r f (y) = δ n f (y).

Lemma 2.3 ([15]) For a given function f
particularly, for 0 < r < 1, we obtain The following lemma is useful in the forthcoming analysis related to the problem (1.3)-(1.4).

Existence results
is an MT -function if and only if φ is a function of contractive factor, that is, for any strictly decreasing sequence (z n ) n≥1 ⊂ R + 0 we have 0 ≤ sup n≥1 α(z n ) < 1. • If we define α(y) = 2φ(y) for all y ∈ R + 0 then α is truly an MT -function. For more details about MT -functions see [35,36].
Hence the condition (H2) holds for w, z,z andw ∈ R a.e. 1 < γ , ζ ≤ 2. We see that ( n ) n∈N is a strictly decreasing sequence; then 0 ≤ sup n∈N φ 1 ( n ) < 1 2 and 0 ≤ sup It ensures that α is a function of a contractive factor, and thus verifies the Mizoguchi-Takahashi's condition. We showed that all the hypotheses of Theorem 3.6 are fulfilled, then the system (4.1) with W and Z provided by (4.2) and (4.3) has at least one solution on [1,2].

Conclusions
This paper was focused on the existence theory of solutions for coupled fractional differential inclusions involving Caputo-Hadamard type fractional derivative equipped with uncoupled boundary conditions. We make use of Mizoguchi-Takahashi's fixed point theorem for multivalued maps to reach the desired results, which are well illustrated with the aid of an example. The technique developed in the present work can also be used to give results for boundary value problems of coupled fractional differential inclusions consisting of different types of fractional derivatives along with a variety of boundary value conditions.