Solutions of two fractional q-integro-differential equations under sum and integral boundary value conditions on a time scale

In this manuscript, by using the Caputo and Riemann–Liouville type fractional q-derivatives, we consider two fractional q-integro-differential equations of the forms Dqαc[x](t)+w1(t,x(t),φ(x(t)))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}^{c}\mathcal{D}_{q}^{\alpha }[x](t) + w_{1} (t, x(t), \varphi (x(t)) )=0$\end{document} and Dqαc[x](t)=w2(t,x(t),∫0tx(r)dr,cDqα[x](t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {}^{c}\mathcal{D}_{q}^{\alpha }[x](t) = w_{2} \biggl( t, x(t), \int _{0}^{t} x(r) \,\mathrm{d}r, {}^{c} \mathcal{D}_{q}^{\alpha }[x](t) \biggr) $$\end{document} for t∈[0,l]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t \in [0,l]$\end{document} under sum and integral boundary value conditions on a time scale Tt0={t:t=t0qn}∪{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}_{t_{0}}= \{ t: t =t_{0}q^{n}\}\cup \{0\}$\end{document} for n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\in \mathbb{N}$\end{document} where t0∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t_{0} \in \mathbb{R}$\end{document} and q in (0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0,1)$\end{document}. By employing the Banach contraction principle, sufficient conditions are established to ensure the existence of solutions for the addressed equations. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.


Introduction
It has been recognized that fractional calculus provides a meaningful generalization for the classical integration and differentiation to any order. On the other hand, quantum calculus is equivalent to traditional infinitesimal calculus without the notion of limits. It defines q-calculus where q stands for quantum. Despite the old history of these two theories, the investigation of their properties remains untouched until recent time. Fractional q-calculus, initially proposed by Jackson [1][2][3], was regarded as the fractional analogue of q-calculus. Soon afterwards, it was further promoted by Al-Salam in [4] and then continued by Agarwal in [5] where many outstanding theoretical results were given. Its emergence and development extended the application of interdisciplinarity and aroused widespread attention of scholars; see  and the references therein. The existence of solutions for q-fractional boundary value problems has been under consideration by many researchers; see for instance [29][30][31][32][33][34][35][36][37][38][39][40].
(P1) First we investigate the nonlinear fractional q-integro-differential equation for t ∈ J under sum and integral boundary value conditions for t ∈ J under the sum boundary conditions where 1 ≤ α < 2, 0 ≤ ζ < 1, 0 < b < 1, m ≥ 1, c i ≥ 0 for all i = 1, . . . , m and w 2 : J × B 3 → B is a continuous function. This paper is organized as follows: In Sect. 2, we state some useful definitions and lemmas on the fundamental concepts of q-fractional calculus and fixed point theory. In Sect. 3, some main theorems on the solutions of fractional q-integro-differential equations (1)- (2) and (3)-(4) are stated. Section 4 contains some illustrative examples to show the validity and applicability of our results. The paper concludes with some interesting observations.

Essential preliminaries
This section is devoted to some notations and essential preliminaries that are acting as necessary prerequisites for the results of the subsequent sections. Throughout this article, we apply the time scales calculus notation [9]. In fact, we consider the fractional q-calculus on the specific time scale T = R where T t 0 = {0} ∪ {t : t = t 0 q n } for nonnegative integer n, t 0 ∈ R and q ∈ (0, 1). Let a ∈ R. Define [a] q = 1-q a 1-q [2]. The power function (xy) n q with n ∈ N 0 is defined by for n ≥ 1 and (xy) (0) q = 1, where x and y are real numbers and N 0 := {0} ∪ N [6]. Also, for α ∈ R and a = 0, we have If y = 0, then it is clear that x (α) = x α [8] (Algorithm 1). The q-gamma function is given by [2]. Note that Γ q (z + 1) = [z] q Γ q (z). Algorithm 2 shows a pseudo-code description of the technique for estimating q-gamma function of order n. The q-derivative of function f is defined by , which is shown in Algorithm 3 [6,7]. Furthermore, the higher [6,7]. Tables 1, 2, and 3 show the values Γ q (z) for some z and q ∈ (0, 1). The q-integral of a function f is defined on [0, b] by   for t ∈ J [11,18]. One can use Algorithm 5 for calculating I α q [h](t) according to Eq. (5). Also, the Caputo fractional q-derivative of a function h is defined by where t ∈ J and α > 0 [18]. It has been proved that I β [18]. Algorithm 5 shows pseudo-code I α q [h](x). We use y = max t∈J |y(t)| as the norm of A = B = C 1 (J). Clearly, (A, . ) and (B, . ) are Banach spaces. Also, the product space (A × B, (y, z) ) is a Banach space where (y, z) = y + z . An operator O : A → A is called completely continuous if restricted to any bounded set in A is compact.

Main results
The main results are presented in this section. To facilitate exposition, we will provide our analysis in two separate folds.

The nonlinear sum and integral boundary value problem (1)-(2)
First, we provide our key lemma.

Lemma 2
The function x 0 ∈ A is a solution for problem (1) under the sum and integral boundary value conditions (2) if and only if x 0 is a solution for the fractional q-integral Thus, we obtain x 0 (t) = -I α-1 q [v 0 ](t) + d 1 . At present, by using the boundary conditions (2), we conclude that d 1 Hence, by substituting d 0 in Eq. (7), we get Put δ = 1 0 x 0 (r) dr. By computing the value of δ and substituting it in (8), we get Thus, x 0 is a solution for the fractional q-integral equation (7). It is obvious that x 0 is a solution for the fractional q-integro-differential equation (1) whenever x 0 is a solution for the fractional q-integral equation. This completes the proof.

Theorem 3
Let g ∈ C(J, R) be a bounded function with upper bound L > 0. Assume that for each t ∈ J there exist positive continuous functions m 1 (t) and m 2 (t) such that and for s = 1, s = a, s = b, and If < 1, then the nonlinear fractional q-integro-differential equation (1)-(2) has a unique solution.
Proof We define the operator Θ : Take = sup t∈J |w 1 (t, 0, 0)| and choose r 0 > 0 such that Hence, Θ(B r 0 ) ⊂ B r 0 . On the other hand, one can write Since < 1, Θ is a contraction. Thus, by using the Banach contraction principle, Θ has a unique fixed point x 0 in A. At present, by using Lemma 2, one can get that c D α q [x 0 ] ∈ A and x 0 is the unique solution for the fractional q-integro-differential equation (1)-(2).

The nonlinear boundary value problem (3)-(4)
Lemma 4 Let w 2 : J × B 3 → B be a continuous function. An element x 0 ∈ B is a solution for the fractional q-integro-differential equation (3) under the sum boundary conditions (4) if and only if x 0 is a solution for the fractional integral equation Let x 0 be a solution for the fractional q-integro-differential equation (3).
. By using the sum boundary conditions (4), we get d 0 = 0 and d 1 = . By substituting d 0 and d 1 , we obtain Thus, x 0 is a solution for the fractional q-integral equation. It is obvious that x 0 is a solution for the fractional q-integro-differential equation (3) whenever x 0 is a solution for the fractional q-integral equation. This completes the proof.
for all x, y ∈ B and t ∈ J. Let and If < 1, then the nonlinear fractional q-integro-differential equation (3)-(4) has a unique solution.
Proof Define the operator Θ : Choose r > 0 such that where = sup t∈J |w 2 (t, 0, 0, 0)|. We show that ΘB r 0 ⊂ B r 0 , where On the other hand, we have Hence, On the other hand, Hence, Since < 1, Θ is a contraction and so, by using the Banach contraction principle, Θ has a unique fixed point. By using Lemma 4, it is clear that the unique fixed point of Θ is the unique solution for the nonlinear fractional integro-differential problem (3)-(4).

Examples, numerical results, and algorithms
Herein, we give an example to show the validity of the main results. In this way, we give a computational technique for checking problems (1)- (2) and (3)-(4). We need to present a simplified analysis that is able to execute the values of the q-gamma function. For this purpose, we provided a pseudo-code description of the method for calculation of the q-gamma function of order n in Algorithms 2, 3, 4, and 5; for more details, follow these addresses https://en.wikipedia.org/wiki/Q-gamma_function and https://www.dm.uniba.it/members/garrappa/software. Tables 1, 2, and 3 show the values Γ q (z) for some z and q ∈ (0, 1). For problems for which the analytical solution is not known, we will use, as reference solution, the numerical approximation obtained with a tiny step h by the implicit trapezoidal PI rule, which, as we will see, usually shows an excellent accuracy [49]. All the experiments are carried out in MATLAB Ver. 8.5.0.197613 (R2015a) on a computer equipped with a CPU AMD Athlon(tm) II X2 245 at 2.90 GHz running under the operating system Windows 7.

Conclusion
The q-integro-differential boundary equations and their applications represent a matter of high interest in the area of fractional q-calculus and its applications in various areas Algorithm 7 (Continued) of science and technology. q-integro-differential boundary value problems occur in the mathematical modeling of a variety of physical operations. The end of this article is to investigate a complicated case by utilizing an appropriate basic theory. In this manner, we prove the existence of a solution for two new q-integro-differential equations under sum and integral boundary conditions (1)-(2) and (3)-(4) on a time scale and show the perfect numerical effects for the problem which confirmed our results.