Some analytical and numerical results for a fractional q-differential inclusion problem with double integral boundary conditions

*Correspondence: mesamei@basu.ac.ir; sh.rezapour@azaruniv.ac.ir; rezapourshahram@yahoo.ca 2Department of Mathematics, Bu-Ali Sina University, 65178, Hamedan, Iran 3Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran Full list of author information is available at the end of the article Abstract In this work, we study a q-differential inclusion with doubled integral boundary conditions under the Caputo derivative. To achieve the desired result, we use the endpoint property introduced by Amini-Harandi and quantum calculus. Integral boundary conditions were considered on time scale Tt0 = {t0, t0q, t0q, . . .} ∪ {0}. To better evaluate the validity of our results, we provided an example, some graphs, and tables.

By combining the ideas mentioned above, we now intend to examine the following qinclusion: with introducing new integral boundary conditions: for all t ∈ J = [0, 1], and ð ∈ (0, 1), S = j=k j=0 u j , P = j=k j=0 w j with u j , w j ∈ R, and c D $ q denotes Caputo quantum fractional derivative of order $ ∈ (2, 3], also 1 , 2 ∈ (1, 2], where F : J × R 5 → P(R) is a compact-valued multifunction such that P(R) is a set of all subsets of R.

Preliminaries
In this section, we summarize what we need from quantum calculus to examine the subject of this research. Throughout this work we always apply quantum calculations to the time scale T t 0 = {t 0 , t 0 q, t 0 q 2 , . . .} ∪ {0} such that t 0 ∈ R and 0 < q < 1 [28]. ([28]) For every real number y, we define the q-analogue of y as [y] q = 1q y 1q = 1 + q + · · · + q y-1 .
Also, for the power function (ws) n q , its q-analogue for n ∈ N 0 is expressed as such that w, s ∈ R and N 0 = N ∪ {0}. The (3) can be expressed for any real number β as follows:
Here we present Algorithm 1 for calculating different values of the q-gamma function; also in Table 1 some numerical results for q = 1 5 , q = 1 2 , q = 8 9 are provided.

Algorithm 1
The proposed procedure to calculate q (w) function Gq = gamma-(w, q) h = 1; for k = 0 : as well as, (D q )(0) = lim w→0 (D q )(w). Moreover, we can extend the q-derivative of this function to any arbitrary order by means of (D n q )(w) = D q (D n-1 q )(w), such that n ∈ N, and (D n q )(w) = (w).
that the right-hand side absolutely converges. The q-antiderivative of can be extended to any arbitrary order by means of I n q (w) = I(I n-1 q (w)).
Remark 2.1 ( [53]) Let the function be continuous at w = 0, then we have Remark 2.2 ( [53]) According to the following relations, we can replace the order of double q-integral: Moreover, it can be written for the left
Here, to help visualize fractional calculations, we present graphs of two functions in Figs. 1 and 2.
Notation 2.8 Assume that (K, d) is a metric space. We denote the set of all subsets of K and the set of all nonempty subsets of K by P(K) and 2 K , respectively. Also assume that the symbols P bd (K), P cl (K), P cp (K), and P cv (K) represent the class of all bounded, closed, compact, and convex subsets of K, respectively. Definition 2.9 ([56]) Let F : K → 2 K be a mapping. It is called a multifunction on K, also an element p ∈ K is a fixed point of F whenever p ∈ F(p). Moreover, for multifunction  F , an element p ∈ K is called an endpoint of F whenever F(p) = {p}. Also, we say that F has an approximate property whenever inf p∈K sup r∈F (p) d(p, r) = 0. 1], and the multifunction F : Nadler's fixed point theorem states that: if F is a closed-valued contractive set-valued map on a complete metric space, then F has a fixed point [58].

Definition 2.12
For l = (l 1 , l 2 ) ∈ K, we define that is called the set of selection of S * . If dim K < ∞, then S * F ,l = φ for all l ∈ K [58].
To prove our main result, we use the endpoint technique presented in 2010 by Amini-Harandi [56].  (p, r)) for any p, r ∈ K. Then F has a unique endpoint iff F has an approximate endpoint property.

Main results
Now, after stating the above preparations, we can get our main results. First we start with a lemma.
such that and Proof With regard to Lemma 2.7 the solution of c D $ q l(t) = v(t) is such that y 0 , y 1 andy 2 ∈ R. Now, by taking derivative from l(t), we have and by exerting the boundary conditions (2) to (10) we have (1) . Now we can compute y 0 , y 1 , y 2 as follows: Now, by replacing y 0 , y 1 , y 2 in (9), we obtain (7).

Now, by applying quantum Caputo fractional derivative from order
from which it can be concluded The following conditions must be met to prove our main theorem. (C 1 ) Given the multivalued map F : J × R 5 → P cp (R) is integrable bounded, so that F (·, v, u, x, y, z) : [0.1] → P cp (R) is measurable.
(C 4 ) Let M : K → 2 K be given as follows: Proof We prove that the endpoint of M : K → 2 K is the solution to inclusion (1)- (2). For this, we first show that M(k) is a closed subset of K for all k ∈ K.
For ∀k ∈ K, the map t → F(t, l(t), l (t), l (t), c D 1 q l(t), c D 2 q l(t)) is measurable and closed value. So, it has measurable selection, and hence f ∈ S * F ,l = φ for all l ∈ K. Let k ∈ K, and {x n } n≥1 be a sequence in M(k) such that x n → x. ∀n ∈ N, choose f n ∈ S * F ,l , where for all t ∈ J.
As we know, F has compact values, then the sequence f n has a subsequence that converges to some f ∈ L 1 [0, 1]. We show this again with f n .
It is easy to check that f ∈ S * F ,l and for all t ∈ J. Indeed, this gives that x ∈ M(k), therefore K has closed values. Moreover, since F has compact values, then M(k) for all k ∈ K is a bounded set. Finally, we shall show that H d (M(u), M(v)) ≤ ψ( uv ). Let u, v ∈ K and p 1 ∈ M(v). Choose f 1 ∈ S * F ,l such that for almost all t ∈ J.

But since
thus ∃w ∈ F(t, l(t), l (t), l (t), c D 1 q l(t), c D 2 q l(t)) such that ∀t ∈ J: Regard the set-valued map N : J → P(R) by Now, for all t ∈ J, let p 2 ∈ M(k) by Afterwards, let sup t∈J | (t)| = , so Also, and p 1 (t)p 2 (t) ≤ 1 Moreover, for i = 1, 2, we have Finally, according to the above relations, it can be concluded that Using Lemma 2.13 and the endpoint property of M, there exists u * ∈ K such that M(u * ) = {u * }. Thereupon, u * is a solution for quantum inclusion problem (1)-(2).

Illustrative examples
To better understand our main result, we give an example in this section.

Conclusion
Understanding and interpreting physical phenomena have always been one of the topics of interest to researchers. Attempts to provide a better explanation of these phenomena have led to progress in various scientific fields and the connection between them. Quantum calculus, as an interdisciplinary subject in mathematics and physics, is one of the tools of modeling and approximation. In this paper, we investigated a quantum differential inclusion problem using the endpoint property technique with the new boundary conditions. One illustrative example and some numerical results have been provided to validate our results and to show their importance.