Fractional hybrid systems involving ' -Caputo derivative

This paper considers a coupled hybrid thermostat system driven by the Ã -Caputo fractional derivative in a Banach algebra. We employ a version of Darbo’s fixed-point theorem combined with the measure of noncompactness (MNC) technique to establish certain existence results. Additionally, an example is provided to illustrate our findings


Introduction
It's commonly recognized that fractional calculus is beneficial in solving numerous real-world problems spanning various scientific and engineering fields, as evidenced by [1,16,20,25,28,31,39].The main advantage of using a non-integer order derivative instead of an integer order derivative is that the first one is non-local in nature, while the second is local in nature and doesn't have any memory terms in the system.So, recently, Numerous studies have appeared focused on fractional differential equations (FDEs), see [2, 3, 6, 23, 24, 32-35, 37, 38, 40] and the references cited therein.
In the literature, different types of integral and differential operators have been introduced by Kilbas et al. [20], including Caputo, Riemann-Liouville, Erdelyi-Kober, Riesz, and Hadamard operators.In the same regard, Almeida [5] generalized Caputo's fractional derivative (FD) to Ã -Caputo FD.This operator appears in various concrete models.For instance, several anomalous diffusions, including ultra-slow processes [22], financial crisis [27], random walks [18], Heston model [7] and Verhulst model [8].Moreover, the increase in global population via the prism of the Ã -Caputo boundary value problem has been investigated in [36].Therefore, considerable attention has been devoted to the quantitative and qualitative properties of solutions to various types of differential problems governed by Ã -Caputo FD [4,9,17,29,40,41].
Hybrid differential equations have attracted a considerable amount of interest and investigation by several researchers.This category of differential systems includes the perturbations of primitive differential equations in different manners.For example, the authors in [15] discussed the following coupled system
Baleanu et al. [11] examined the existence of solutions for the following hybrid thermostat differential model: and denotes the Caputo FD of order # and # -1, respectively, h C g C Î ( ) Motivated by the above papers and using the main idea of [12,19,26], this paper considers the following coupled hybrid system for the thermostat model involving Ã-Caputo FD: ( ) ( , ( ), ( )) ( , ( ), ( and where denotes the Caputo FD with respect to Ã of order # i and # i -1, respectively.Here is a given function fulfilling some hypothesis that will be indicated later and h( , , ) , z z 0 0 0 ¹ Î for all  Ã is increasing and positive monotone function and Ã '( ) .t > 0 Numerous physical systems undoubtedly merit an in-depth examination of ( 1)-( 2), for instance, binary mixture convection [30], geophysical morphodynamics [21] and so on.The above-mentioned motivational models present notable benefits, but the complexity of their associated mathematical models often escalates, making it challenging to establish the existence of results.Hence, the investigation of hybrid coupled systems involving Ã-Caputo type TFD in a Banach algebra has become important.
The contributions of our paper can be outlined as follows: • We consider a fractional hybrid system in a general configuration, enabling improvements over several earlier related papers [10,11,15,19].• By utilizing Darbo's fixed-point theorem and the MNC technique in a Banach algebra, we propose certain sufficient conditions to ensure the existence of solutions.
The paper is structured as follows: In Section 2, we collect the basic background necessary for subsequent discussions.Section 3 utilizes a version of Darbo's fixed point theorem to establish new existence criterion.Finally, our findings are illustrated through an example.

Preliminaries
Let  : [ , ], = a b be a finite interval.Consider C( )  the space of continuous functions f :  ®  with the supremum (uniform) norm:  the space of Lebesgue integrable real-valued functions on J equipped with the norm ),   and z z s s we pose and Definition 1. [5,20] The Ã-Riemann-Liouville fractional integral of a function f L Î 1 ( )  of order #>0 is given by and G the gamma function.
Definition 2. [5,20] Let 0 1 . Then, the following equality holds Now, let us assume that ( , . )    is a real Banach space and the zero element 0. If V X Ì is non-empty, then Conv  and denote the convex hull and the closure of V, respectively.If V X Ì is a bounded, diamV represents the diameter of V and We denote by M  is the family of all bounded subsets of X and by N  its subfamily comprising of the relatively compact subsets.Moreover, uv denotes the product of elements Definition 3. [13] We say that L : [ , ) M  ® 0 ¥ is a MNC in X if all the assumptions below are satisfied: , , ,  and lim ( ) [14] We say that the MNC Λ in the Banach algebra X satisfies condition (m) if: We need to introduce the class E of functions: where k n denotes the n-iteration of k.
Theorem 1. [15,Theorem 13] Let Λ be an MNC in the Banach space X and V X Ì be a nonempty, closed, bounded, and convex.X. Assume that  :    ´® is a continuous mapping satisfying for any nonempty subsets V 1 and V 2 of V, where k 2 E. Then S possesses a fixed point in V.
Then, for e z £ £ 2 b, we get z |.

Main Results
Before stating our result, we need to present the auxiliary lemma. and satisfies the following integral equation Proof.Applying the operator  a + J 1 ,Ã to both sides of (7), by Lemma 2, one gets . Thus, the solution of ( 7) is Then, by using Lemmas 1, we have which, together with the boundary condition Substituting the value of c a in (10) we get (9).Let us introduce the following assertions: for any z Î  , and v v u u , , Î  and are continuous.
(H3) There exists a constant K > 0 such that Notice that hypothesis (H1) gives us the existence of two constants g*, h* > 0 such that and for any z Î  , .
Theorem 2. Assume that (H1)-(H3) hold.Then the system (1)-( 2) admits at least one solution defined We equip the Banach algebra  = for all ( , ) .Z Z 1 2 Î  Now, from Lemma 5, we introduce an operator  :   ® as follows: where, U J i C i : ( ), ,  ® = 1 2 are defined by: Clearly, the fixed points of the operator U coincide with the solutions of problem ( 1)-( 2).We introduce the operators and as follows (for i = 1, 2): and for all ( , ) .Z Z 1 2 Î  and any z Î  , .Then, Clearly, the fixed point of operator U coincides with the solution of the problem (1)-(2).For K > 0, we define the ball   It's clear that B K is closed, bounded and convex in P.
Step 1.The operator U maps P into C( )  .
To demonstrate that U J To do so, let z Î  , be fixed . Without loss of generality, we may suppose ³n>³.Thus, we obtain:

Z Z
In view of hypothesis (H1), g is bounded on the compact set From the last estimate, we obtain , we infer that From the above inequality, we conclude that   On one hand, by using hypothesis (H1) and (H2), we get Similarly, we obtain On the other hand, Therefore, from ( 17), ( 18) and ( 19) and by G( we have Hence, by (H3), we get We deduce that the operator U maps B K into itself, Moreover, from the last estimates, it follows that Step 3. The continuity of the operators H and L on B K .
Firstly, we demonstrate that the operator H is continuous on B K .To do so, fix V > 0 and take ( , ),( , ) Z Z Then, for z Î  , , one has where we have used Remark 1. Then The above inequality entails the continuity of the operator H on B K .Now, the operator L is continuous on B K .In fact, '( ) ( , ( ), s s s ds g s y s y s ds ( )) ( , ( ), ( )) .s g s y s y s ds -By (H2), we get Thus, the preceding inequality demonstrates that the operator L is a continuous operator on B K .Consequently, we conclude that U is a continuous operator on B K .
Step 4. all subsets V 1 and V 2 of B K .We must solely examine the condition (6).
´ and Z Z V V we can suppose that z z 1 2 < ; Then, according to hypothesis (H2), we obtain Using Lemma 4, one has After that, by Lemma 3, and using ( 20), (21), and ( 22), with i = 1, 2, we have:  ) ) Thus, k z s z s z k which is verified by K 0 = 1.Therefore, all hypotheses of Theorem 2 are met, and as a result, the problem ( 23)-( 24) admits at least one solution ( , ) .Z Z