Using Krasnoselskii's theorem to investigate the Cauchy and neutral fractional q - integro di ﬀ erential equation via numerical technique

: This article discusses the stability results for solution of a fractional q - integro - di ﬀ erential problem via integral conditions. Utilizing the Krasnoselskii ’ s, Banach ﬁ xed point theorems, we demonstrate existence and uniqueness results. Based on the results obtained, conditions are provided to ensure the generalized Ulam and Ulam – Hyers – Rassias stabilities of the original system. The results are illustrated by two examples.


Introduction and formulation of the problem
The fractional derivative can be considered as a global operator that has greater degrees of flexibility as compared to integer-order derivative, because a classical derivative with integer-order could be a nearby operator. A few researchers have demonstrated that fractionalorder derivatives play a noteworthy part in electrochemical analysis to explain the mechanistic behavior of the concentration of a substrate at the electrode surface to the current [1][2][3][4][5][6][7][8][9][10][11][12][13]. Some interesting applications of fractional calculus in science and engineering have been discussed [14][15][16][17][18].
It is interesting to study solution to fractional q-integro-differential problem with integral conditions, which will allow a generalized stability. It is shown in [4] that, in a real k-dimensional Euclidean space, the local and global solutions exist for the following Cauchy problem: y h y ξ y ξ ξ y y , Θ , , d, 0 , , and σ C 0 + is the Caputo fractional operator. A class of abstract delayed fractional neutral integro-differential equations was introduced in [19], for σ 1, 2 ( ) ∈ , Using the Leray-Schauder alternative fixed point theorem, the existence results were obtained (for more details, see [3]). Recently, much attention has been paid to the study of differential equations with fractional derivatives [20][21][22][23][24][25]. Note that in [22], the authors introduced and studied a related problem. Shah et al. [8] investigated the following problem under delay differential equations involving Caputo fractional derivative and under nonlocal initial condition with non-monotone term as where , η α , j j ∈ and σ c 0 + is the of order σ, h h , 1 2 are given continuous functions. By using the classical tools of fixed point theory, the existence and uniqueness results are obtained. On an arbitrary domain, in [21], the authors studied an with non-conjugate Riemann-Stieltjes integromultipoint boundary conditions by using new tools on function analysis. For some more related works, refer to [26,27]. Shah [ ] × → and ϕ ϕ , : 1 2 → are continuous functions, where is complete norm space. The authors in [2], considered the problem for the system (1.9), and we generalized the system in the q which is not explicitly presented, and therefore it makes sense to consider for t ∈ , σ ν , ∈ , the problem for system as follows: ‖ ‖. Here, this study is focused on the question of existence and uniqueness in Section 3. In addition, Section 4 is devoted to show a generalized stability. Note that this representation also allows us to generalize the results obtained recently in the literature. The article is ended by two examples illustrating our results.
We recall some essential preliminaries that are used for the results of the subsequent sections. Let 0 t ∈ and q ∈ . The time scale t0 is defined by [31]). The q-factorial function y z q and y z 1 [32]). Also, we have Algorithms 1 and 2 simplify q-factorial functions y z q n ( ) ( ) − and y z q , respectively. In [33], the authors proved y z y z y q z sy sz s y z , and The q-gamma function is given by [31].
In fact, by using (2.2), we have Algorithm 3 shows the MATLAB lines for calculation of y Γ q ( ) which we tend n to infinity in it. Note that, Lemma 1]. For any positive numbers σ and ν, the q-beta function is defined by For a function w : → , the q-derivative of w, is [32]). Also, the higher order q-derivative of the function y is defined by (see [32]). In fact, (see ref. [27]).
≤ ≤ , provided the series absolutely converges (see [32]). By using Algorithm 6, we can obtain the numerical results of y It has been proved that whenever the function y is continuous at 0 t = (see [32]). The fractional Riemann-Liouville-type q-integral of the function y is defined by and σ 0 > (see refs [27,34] and σ 0 > (see [34,35]). It has been proved that 34]. Also, (see [34])  Algorithm 8 shows the MATLAB codes of numerical technique. One can find other algorithms in [36]. Now, we introduce some basic definitions, lemmas, and theorems, which are used in the subsequent sections.
. Then, one has (1) For σ, σ́0 > , Lemma 2.5. [37] Let n σ n n 1 , − < ≤ ∈ , and y C , . Then for all , , we have y y . Then for each y AC 0, 38], Banach fixed point theorem) Let B be a non-empty complete metric space and : B B → is contraction mapping. Then, there exists a unique point y B ∈ such that y y ( ) = .
Lemma 2.8. ( [38], Krasnoselskii fixed point theorem) Let E be bounded, closed, and convex subset in a Banach space B. If 1 , : 2 E E → are two applications satisfying the following conditions: (A1) y z

Existence results
Before presenting our main results, we need the following auxiliary lemma.
, then y is solution of (1.9) iff y satisfies the IE be a solution of system (1.9). First, we show that y is a solution of integral Eq.
From integral boundary condition of our problem with using Fubini's theorem and after some computations, we obtain Finally, by substituting (3.5) in (3.4), we find (3.1). Conversely, from Lemma 3.1 and by applying the operator q σ ν C + on both sides of (3.1), we find This means that y satisfies the equation in problem (1.9). Furthermore, by substituting t by 0 in integral Eq. (3.1), we are clear that the integral boundary condition in (1.9) holds. Therefore, y is solution of problem (1.9), which completes the proof.
In order to prove the existence and uniqueness of solution for problem (1.9) in C , H ( ), we use two fixed point theorem. First, we transform the system (1.9) into fixed point problem as 3.
Proof. For any function y C , H ( ) ∈ , we define the norm and consider the closed ball y C y , : .
Next, let us define the operators ,          Let for any y ∈ ℓ and for each , The RHS of the last inequality is independent of y and tends to zero when 0 → , which implies that 1 U ℓ is equicontinuous, then 1 U is relatively compact on ℓ . Hence, by the Arzelá-Ascoli theorem, 1 U is compact on ℓ . Now, all hypotheses of Theorem 3.2 hold; therefore, the operator 1 2 U U + has a fixed point on ℓ . So problem (1.9) has at least one solution on . This proves the theorem. :

Existence and uniqueness result
.
Since μλ 1 < , it follows that U is a contraction. All assumptions of Lemma (3.1) are satisfied, then there exists such that y y U = , which is the unique solution of problem (1.1) in C , H ( ).

Generalized Ulam stabilities
The aim is to discuss the Ulam stability for problem (1.9), by using the integration  of problem (1.9) such that Then problem (1.9) is said to be generalized Ulam-Hyers stable. For each ε 0 > and for each solution z of problem (1.9), problem (1.9) is called Ulam-Hyers-Rassias stable with respect to C ϱ , and there exist a real number γ 0 > and a solution where ε * is a positive real number depending on ε. < . Therefore, we deduce by the fixed-point property of the operator that and γ 1 = , we obtain the Ulam-Hyers stability condition. In addition, the generalized Ulam-Hyers stability follows by taking ε ε μλ 1 Theorem 4.2. Assume that H 1 ( ) holds with μ λ1 < , and there exists a function C ϱ , ( ) ∈ + satisfying the condition 4.2. Then problem (1.9) is Ulam-Hyers-Rassias stable with respect to ϱ.
Proof. We have from the proof of Theorem 4.1, where ε ε μλ 1 = * − , and so the proof is completed.

Illustrative of our outcome
First we present Example 5.1, for illustrative our main result.   , we can see the results of λ and λ * in Table 1. These results are plotted in Figure 1.
which it satisfies in assumption of Theorem 4.2.
In the next example, we review and check Theorem 3.3 numerically.
Example 5.2. Consider the following fractional integrodifferential problem:   = , we can see the results of λ and λ * in Table 2. These results are plotted in Figure 2

Conclusion
Determining the answer of differential equations from the order of fractions in the discrete state simplifies many problems. The q-integro-differential boundary equations and their applications have attracted several researchers' interests in the field of fractional q-calculus and its applications in various phenomena from science and technology. q-Integro-differential boundary value problems occur in the mathematical modeling of a variety of physical operations. Using the Krasnoselskii's, Banach fixed point theorems, we prove existence and uniqueness results. Funding information: There is no funding to declare for this research study.  ⌊ totalout = totalout + q n * eval(subs(funx, totalout q n * )); 4 H = s * (1-q) * totalout;    1 : paramsmatrix n, column n ; C1 qGamma q s ,sigma 1,n ; C2 qGamma q s ,sigma nu,n ; C3 qGamma q s ,sigma nu 1,n ; C4 qGamma q s ,2 sigma nu 1,n ; C5 qGamma q s ,sigma nu 2,n ; C6 qGamma q s ,2 sigma nu 2,n ;