On a fractional q-differential inclusion on a time scale via endpoints and numerical calculations

By using an endpoint result for set-valued maps, we study the existence of solutions for a fractional q-differential inclusion with sum and integral boundary value conditions on the time scale Tt0={t0q,t0q2,…}∪{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}_{t_{0}}= \{t_{0} q, t_{0} q^{2},\ldots \} \cup \{0\}$\end{document}, where t0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t_{0}$\end{document} is a real number and q∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q \in (0,1)$\end{document}. We provide an example involving some graphs and algorithms via numerical calculations to illustrate our main result.


Introduction
In 1910, Jackson started the subject of q-difference equations [1]. The fractional calculus provides a meaningful generalization for the classical integration and differentiation to any order. Also, quantum calculus is equivalent to traditional infinitesimal calculus without the notion of limits. Despite the long history of these two theories, the investigation of their properties has remained untouched until recent time. In last decades, some researchers investigated q-fractional difference equations [2][3][4][5]. Later, q-fractional boundary value problems considered by many researchers (see for examples, [6][7][8][9][10][11]).
Plasma is also known as the ionized state of the matter. In this state of matter, plasma contains components such as free electrons, ions neutrals, and dust. The multicomponent plasmas deal with the partially or fully ionized state of plasma. It also fulfils the condition of quasi neutrality. The multi-component plasmas play a crucial role in plasma discharge and other industrial processing. The multi-component plasma has more than two components while the general plasma has ions and electrons. The wide range of plasmas itself give an opportunity to analyze such plasmas on distinct scales. There are two main areas of multi-component plasma: dusty plasma and the negative ions plasma. Both areas of multi-component plasma have a wide range of emerging applications in science and engineering. The time fractional Kersten-Krasil'shchik coupled KdV-mKdV nonlinear system and homogeneous two component time fractional coupled third order KdV systems are very important fractional nonlinear systems for describing the behavior of waves in multi-component plasma and elaborate various nonlinear phenomena in plasma physics. In the past decades many researchers are used to various techniques for solving fractional nonlinear partial differential equation and find approximate and exact solutions of the fractional evolution equations (for more details, see [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]) and different applications of fractional calculus (see for example [27][28][29][30]).
This work is arranged as: In Sect. 2, we state some useful definitions and lemma on the fundamental concepts of q-fractional calculus and multifunctions. In Sect. 3, some main theorems on the solutions of fractional q-differential inclusion (1)-(2) are stated. Section 4 contains an illustrative example to show the validity and applicability of our results. The paper concludes with some interesting observations. In Sect. 5, conclusions are presented.

Essential preliminaries
Throughout this article, we shall apply the time scales calculus notation [37]. In fact, we consider the fractional q-calculus on the time scale T t 0 = {0} ∪ {t : t = t 0 q n }, where n ≥ 0, t 0 ∈ R and q ∈ (0, 1). Let a ∈ R. Define [a] q = 1-q a 1-q [1]. The power function (xy) n q with n ∈ N 0 is defined by (xy) (n) q = n-1 k=0 (xyq k ) for n ≥ 1 and (xy) (0) q = 1, where x and y are real numbers and N 0 := {0} ∪ N [2]. Also, for α ∈ R and a = 0, we have If y = 0, then it is clear that x (α) = x α [7] (see the Algorithm 1). The q-Gamma function is given by Γ q (z) = (1q) (z-1) /(1q) z-1 , where z ∈ R\{0, -1, -2, . . .} [1]. Note that, Γ q (z + 1) = [z] q Γ q (z). The 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 [2,3]. Furthermore, the higher order q-derivative of a function f is defined by provided the series is absolutely converges [2,3]. If x in [0, T], then  [38]. Actually the interchange of order is true, since In addition the left side can be written as The operator I n q is given by [2,3]. It has been proved that (D q [39]. Also, the Caputo fractional q-derivative of a function h is defined by where t ∈ J and σ > 0 [39]. It has been proved that I β [39]. The Algorithm 4 shows pesudo-code I α q [h](x). Let (E, ρ) be a metric space. Denote by P(E) and 2 E the class of all subsets and the class of all nonempty subsets of E, respectively. Thus, P cl (E), P bd (E), P cv (E) and P cp (E) denote the classes of all closed, bounded, convex and compact subsets of E, respectively. A mapping T : E → 2 E is called a multifunction on E and e ∈ E is called a fixed point of T whenever e ∈ T (e). An element e ∈ E is called an endpoint of a multifunction T : E → 2 E whenever T (e) = {e} [40]. Also, we say that T has an approximate endpoint property whenever inf e∈E sup f ∈T (e) ρ(e, f ) = 0 [40]. A function ψ : R → R is called upper semi-continuous whenever lim sup n→∞ ψ(r n ) ≤ ψ(r) for all sequence {r n } n≥1 with r n → r.

Main results
Now, we are ready to provide our main results.

Lemma 2 Let z(t) ∈ A and σ ∈ (2, 3]. Then the unique solution of the fractional problem
where and Proof It is known that the solution of the fractional q-differential where d 0 , d 1 are real constants and t ∈ J [48]. Thus, we have k (t) = I σ -1 q [z](t) + d 1 + 2d 2 t and k (t) = I σ -2 q [z](t) + 2d 2 . By using the boundary conditions, we obtain d 0 + 2d 2 Σ = 0, and Hence by an easy calculation, we get (1) , Now, substituting the values of d 0 , d 1 and d 2 in (7), we obtain (4).
From definition (6), we can see that Also, for t ∈ J and i = 1, . . . , m. From this we obtain Definition 3 A function (k 1 , k 2 , . . . , k m ) belongs to m i=1 AC 1 (J) is a solution for the fractional q-differential inclusion (1) whenever it satisfies the boundary value conditions and there exists a function (z 1 , z 2 , . . . , z m ) ∈ m i=1 L 1 (J) such that and for t ∈ J and 1 ≤ i ≤ m, where A 1 (t) and A 2 (t) are defined in (6).
is a nondecreasing upper semi-continuous mapping such that lim inf t→∞ (tψ(t)) > 0 and ψ(t) < t for all t > 0 and there exist continuous functions γ : for i = 1, 2, . . . , m. Define the operator Ω : E → 2 E by for z ∈ S T ,k . If the multifunction Ω has the approximate endpoint property, then the boundary value inclusion problem (1)-(2) has a solution.
Proof First, we prove that Ω(k) is a closed subset of P(E) for all k ∈ E. Since the multivalued map t → T (t, k(t), k (t), c D ζ 1 k(t), . . . , c D ζ m k(t)) is measurable and has closed values for all k ∈ E, has measurable selection and so S T ,k is nonempty for all k. Assume that k ∈ E and {v n } n≥1 be a sequence in Ω(k) with v n → v. For every n ≥ 1, choose z n ∈ S T ,k such that for all t ∈ J. By compactness of T , the sequence {z n } n≥1 has a subsequence which converges to some z ∈ L 1 (J). We denote this subsequence again by {z n } n≥1 . One can easily check that for all t ∈ J. This shows that v ∈ Ω(k) and so Ω is closed-valued. On the other hand, Ω(k) is a bounded set for all k ∈ E because T is a compact multivalued map. Finally, we show that P ρ (Ω(k), Ω(l)) ≤ ψ( kl ). Let k, l ∈ E and f 1 ∈ Ω(l). Choose z 1 ∈ S T ,l such that for almost all t ∈ J. Since for t ∈ J. Now, we consider the multivalued map K : J → P(R) which is given by Since z 1 and are measurable, the multifunction for all t ∈ J. Define the element f 2 ∈ Ω(k) by for all t ∈ J. Let sup t∈J |γ (t)| = γ . Thus, one can get On the other hand, and, for each i = 1, . . . , k, we have

An example by using the algorithms and some numerical calculations
Here, we give an example to illustrate our main result. In this way, we give a computational technique for checking the problem. 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 pseudocode description of the method for calculation of the q-Gamma function of order n in the Algorithms 2, 3, 4 and 5. The Algorithm 6 help us for numerical solving of the problem.
Also, we provide some figures and numerical tables.
The Table 7 shows the values of Ξ ≈ 0.2839, 0.3449, 0.3657 for q = 1 10 , 1 2 , 6 7 . One can easily check that inf k∈E sup l∈Ω(k) kl = 0. Thus, the operator Ω has the approximate  endpoint property. Now, by using Theorem 4, we get the fractional q-differential inclusion problem (11) has a solution.

Conclusion
Most natural phenomena could be modeled by different types of fractional differential equations and inclusions. Recently some physicists have been studying the role of fractional calculus in better describing of physical phenomena. They have found that by using the q-fractional they can provide a better description by some physical notions. Thus, we should investigate distinct fractional differential equations and inclusions to increase our