Integro-differential equations implicated with Caputo-Hadamard derivatives under nonlocal boundary constraints

The goal of this work is to derive a new type of fractional system that arises from the combination of the Caputo-Hadamard derivative with the integro-differential equation. Also, the existence and uniqueness of solutions to this problem have been studied under nonlocal boundary conditions. Moreover, Hyer-Ulam stability has been studied for the considered problem. Finally, to reinforce the theoretical results and provide applications for our paper, two supporting examples have been emphasized.


Introduction
Recently, research on the analysis of differential equations of fractional order has become more prominent and fascinating.Fractional derivatives provide an incredible tool for investigating memory and the inherited characteristics of various materials and processes.Fractional derivatives are often used to simulate the dynamics of processes and phenomena in the actual world, where the behavior of the past determines the current state.Since the development of fractional differential equations (FDEs), significant efforts have been made to advance the theoretical and application aspects in a number of disciplines, such as physics, chemistry, biology, engineering sciences, etc.Researchers have made many contributions in these areas, and for more information, we recommend the reader study the works [1][2][3][4][5][6][7][8][9].
The boundary value problem (BVP) has been a pivotal component in the development of classical calculus in recent times.Furthermore, historical background has been included for a number of fractional calculus applications and analytical results; for instance, refer to [10][11][12][13][14][15][16][17][18][19][20].It has been noted that most of the studies conducted on the subject pertain to FDEs of the Caputo or Riemann-Liouville (RL) types.Apart from the previously mentioned fractional derivatives, another type of fractional derivative that has been documented in the literature is the Hadamard fractional (HF) derivative.This type of derivative differs from the previous ones in that a logarithmic function with any exponent makes up the kernel of the integral in the delineation of the (HF) derivative; further details can be found in [21][22][23][24][25][26][27][28].
The proof of solutions for fractional BVP with multi-point, HF integral, RL fractional integral, and Erdelyi-Kober fractional integral conditions has recently received a lot of attention.An inequality of the Lyapunov type with the HF derivative was studied by Ma et al [29].Similar research was conducted by Wang et al [30] on nonlocal HF BVP with discrete boundary conditions and the Hadamard integral.For more details, see [31][32][33][34].
The authors [35] discussed HF derivative-based FDEs with three-point border conditions.Fractional derivative Caputo-Hadamard (CH) type is the name given by Jarad et al [36] to a fractional derivative of Hadamard type that has been changed into a more appropriate one with physically interpretable initial conditions similar to the singles in the Caputo setting.
Dynamical problems usually call for the use of stability theory.For typical classical fractional calculus problems as well as exponential types of stabilities, Lyapunov and Mittag-Leffler stabilities have been developed with remarkable success.The required attention has recently been given to Hyer-Ulam (HU) [37] stability.Some stability and existence results from the fixed point (FP) approach were looked into in [38].Additionally, [39,40] has examined the stability analysis and existence theory for FDEs.
In this manuscript, we establish nonlocal integral boundary constraints on the CH fractional DE of the form: 1 where CH D θ is CH fractional derivative with order θ, h J I I , H H are HF integrals of orders η and ϑ, respectively, The rest of this manuscript is arranged as follows: section 2 is devoted to presenting some necessary preliminaries for the sequel.In section 3, we give an essential lemma and some fundamental hypotheses and conditions to prove our main results on the existence and uniqueness of solutions to Problem (1.1) (main contributions in sections 4 and 5, respectively).section 6 is concerned with the study of the Hyer-Ulam stability of such a problem.In the last section (section 7), we provide two illustrative examples making effective the required hypotheses given in section 3.

Preliminaries work
This section is devoted to presenting some definitions and previous works that help the reader understand our manuscript and help us overcome the difficulties that we face in proofs.These results are taken from [6,36,41].Definition 2.1.For the function ℓ, the left and right HF integrals of order θ are respectively described as Definition 2.2.Let q Î . For the function ℓ, the left and right HF derivatives of order θ are respectively given by ò , q Re 0 ( )  on (b, c) and q Re ( ) denotes the real part of q.
)), we get Definition 2.5.Let q Î . For the function ℓ, the left and right CH fractional derivatives of order θ are respectively defined by Lemma 2.6.Assume that q Î  with q > Re 0 ( ) and where )}is either unbounded or ¡ has a FP in Q.
Theorem 2.9.Let  be a non-empty, convex, closed and bounded subset of a BS Q.Also, let ¡ ¡   Q , : 1 2 be operators fulfill the statements below: • ¡ 2 is a contraction; • ¡ 1 is compact and continuous.

Pivotal lemma and hypotheses
In this section, we present the necessary and sufficient hypotheses to study the existence and uniqueness of the solution to Problem (1.1).
Clearly, the pair , .( )    is a BS.We begin this section with the proof of the following Lemma: 1 if and only if Proof.By influencing both sides of (3.2) by q I H , we have where D 0 , D 1 and D 2 are arbitrary real constants.Applying the conditions where λ is defined in (3.4).Substituting the values of D 0 , D 1 and D 2 in (3.5), we obtain (3.3)., In light of Lemma 3.1, the operator ϒ :J → J is interpreted as .
Clearly, trying to find a solution to the proposed problem (1.1) is sufficient to find a FP for the operator (3.6).Now, we assume that Assume also the hypotheses below hold: (H 1 ) There are a function P ä C([1, U], R + ) and a nondecreasing function f

Existence theorems
In this part, we will apply the fixed point technique to obtain the existence of the solution to Problem (1.1) under the constraints formulated in the above section.This approach is simple because it does not require more complex assumptions.We discuss the existence of the solution to Problem (1.1) in two different ways.The first method is to use the alternative Leray-Schauder type, which was presented in Theorem 2.8.The second way is to apply Theorem 2.9 for the existence of FPs.Proof.We split the proof onto the following steps: (I) Prove that ϒ maps bounded sets into bounded sets in .
q h q J h q h q h q J h q h q h q J h q h q J h .
, that is, ϒ is a bounded mapping.
(II) Show that ϒ maps bounded sets into equicontinuous sets of  , that is, ϒ is CC.For this, assume that t t Î U , 1, . Hence, the operator ¡ ℓ ( ) is equicontinuous.Thanks to Arzela-Ascoli theorem, the operator ϒ is CC.

,
The existence of a solution to Problem (1.1) with another method according to the following theorem: Theorem 4.2.According to H 3 ( ) and H 4 ( ), the BVP (1.1) has at least one solution on U 1, [ ] provided that X < K 1, where Ξ is given by (3.7).

Proof. Let us interpret
The following three steps completes the proof: Step 1: Step 2: Show that ¡ 2 is a contraction.Assume that Î   , 1 2 From the hypothesis H 3 ( ), we get Step 3: Claim that ¡ 1 is compact and continuous.Since ψ is continuous, the mapping ¡ 1 is continuous.Further, for Î W M ℓ , we get . Thus, ¡ 1 is relatively compact on.Thanks to Arzela-Ascoli theorem, the operator ϒ is compact on W M .Therefore, all assumptions of Theorem 2.9 are fulfilled.Hence, ϒ owns at least one FP, which is a solution to BVP (1.1).,

Uniqueness theorem
To test whether the solution to Problem (1.1) is unique, we present in this section the following theorem, which uses the Banach contraction principle (every contraction mapping defined on a BS has a unique FP) in the test.] provided that X < K 1, where Ξ is given by (3.7). .

Proof. Assume that
.
Since KΞ < 1, ϒ is a contraction.Based on Banach contraction principle, ϒ possesses a unique FP, which is the unique solution to the BVP (1.1).

Hyer-Ulam stability
In physical problems, stability of solution is crucial because the mathematical equations describing the problem will not be able to predict the future with any degree of accuracy if small departures from the mathematical model brought about by inevitable measurement errors do not also have a correspondingly small effect on the solution.The mentioned aspects are very important from a numerical and optimization point of view.This is due to the fact that most of the nonlinear problems in fractional calculus and applied analysis are quite difficult to solve for an actual solution.In such a situation, one needs approximate solutions that are near the actual solution of the corresponding problem.In the mentioned situation, stability of the solutions is necessary.In such circumstances, one needs approximations that are close to the real solution of the related problem.The stability of the solutions is required in the given scenario.In this part, we discuss the HU stability of the solution to the BVP (1.1).Firstly, the HU stability of the considered problem is described as follows: Definition 6.1.We say that the solution of the BVP (1.1) is HU stable if there exists j > 0 such that for every >  0 and Î ℓ  as a solution to the problem 1 there is a unique solution Î ℓ   to the problem (1.1) such that 2. An element ℓ ä J is a solution to the BVP (1.1) if and only if there exists U ä J such that ( ) Lemma 6.3.According to Remark 6.2, the solution of the system Proof.It comes directly using Lemma 6.3., Theorem 6.4.Suppose that the condition H 3 ( ) is true, then the solution to BVP (1.1) is HU-stable if where Ξ is described as (3.6).
Proof.Using Lemma 6.3, if ℓ is a solution to (6.1), and ℓ  is a solution to (1.1), then, we have . This illustrates that the solution of the problem (1.1) is HU stable.,

Illustrative examples
There is no doubt that giving numerical examples as special cases of the problem under study increases the sobriety of the model and clarifies each parameter and its importance.It also strengthens the theoretical results and tests the validity of the hypotheses used to achieve the desired goal.Therefore, in this section, we present two numerical examples in which the wave functions are involved under fractional order derivatives as special models of the problem (1.1).
Thus, X » < K 0.0002 1.Therefore, the requirements of Theorem 5.1 are fulfilled, and so the BVP (7.1) has a unique solution on e 1, [ ].Moreover, X » ¹ K 0.0002 1, then by Theorem 6.4, the solution of (7.1) is UH stable.

Conclusion
The study of differential equations with inclusions and various boundary conditions involving fractional derivatives has seen a rapid advancement in recent years.The related operators of a fractional derivative are nonlocal, which allow an increased modeling of many problems in many scientific disciplines.In this work, we were able to confirm that there is only one solution to an integro-differential equation with nonlocal boundary conditions.In order to apply specific FP approaches, certain technological requirements have been set.We also looked at the associated HU stability, and in this regard, a related result was found.Ultimately, two examples were provided to illustrate our results.

Theorem 4 . 1 . 1
Under the hypotheses H 1 ( )and H 2 ( ), the proposed problem (1.1) admits at least one solution on U