The existence of nonnegative solutions for a nonlinear fractional q-differential problem via a different numerical approach

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Introduction
It is recognized that fractional calculus provides a meaningful generalization for the classical integration and differentiation to any order. They can describe many phenomena in various fields of science and engineering such as control, porous media, electro chemistry, HIV-immune system with memory, epidemic model for COVID-19, chaotic synchronization, dynamical networks, continuum mechanics, financial economics, impulsive phenomena, complex dynamic networks, and so on (for more details, see [1][2][3][4][5][6][7]). It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of linear initial value fractional differential equation in terms of special functions.
The study of q-difference equations has gained intensive interest in the last years. It has been shown that these equations have numerous applications in diverse fields and thus have evolved into multidisciplinary subjects. On the other hand, quantum calculus is equivalent to traditional infinitesimal calculus without the notion of limits. Fractional q-calculus, initially proposed by Jackson [8], is regarded as the fractional analogue of qcalculus. Soon afterward, it is further promoted by Al-Salam and Agarwal [9,10], where many outstanding theoretical results are given. Its emergence and development extended the application of interdisciplinary to be further and aroused widespread attention of the scholars; see [11][12][13][14][15][16][17][18][19][20][21][22][23] and references therein.
Inspired by all the works mentioned, in this research, we investigate the existence and uniqueness of nonnegative solutions of the nonlinear fractional q-differential equation under the boundary conditions k(0) = k (0) = 0 and k (r) = λk (1) for t ∈ J := (0, 1) and 0 < q < 1, where w : J × R 2 → R is a given function with J := [0, 1], 2 < σ < 3, ζ ∈ J, r ∈ J,and λ > 0, and c D σ q denotes the Caputo fractional q-derivative. The rest of the paper is organized as follows. In Sect. 2, we cite some definitions and lemmas needed in our proofs. Section 3 treats the existence and uniqueness of solutions by using the Banach contraction principle and Leray-Schauder nonlinear alternative. Also, Sect. 3 is devoted to prove the existence of nonnegative solutions with the help of the Guo-Krasnoselskii theorem. Finally, Sect. 4 contains some illustrative examples showing the validity and applicability of our results. The paper concludes with some interesting observations.

Preliminaries and lemmas
In this section, we recall some basic notions and definitions, which are necessary for the next goals. This section is devoted to state some notations and essential preliminaries acting as necessary prerequisites for the results of the subsequent sections. Throughout this paper, we will apply the time-scale calculus notation [31].
To prove the theorems, we further apply the Leray-Schauder nonlinear alternative.

Main results
To facilitate exposition, we will provide our analysis in two separate folds. Now we give a solution of an auxiliary problem. Denote by L = L 1 (J, R) the Banach space of Lebesgueintegrable functions with the norm k = 1 0 |k(ξ )| dξ .
for t ∈ J is given by Proof First, by Lemma 2.1 and equation (4) we get Differentiating both sides of (7) and using Lemma 2.2, we get The first condition in equation (4) implies d 1 = d 3 = 0, and the second one gives Substituting d 2 into equation (7), we obtain which can be written as Indeed, where 1 G ζ q (t, ξ ) is defined by (6). The proof is complete.

Existence and uniqueness results
In this section, we prove the existence and uniqueness of nonnegative solutions in the Banach space B of all functions k ∈ C(J) into R with the norm Throughout this section, we suppose that w ∈ C(J ×R 2 , R). We define the integral operator : Then we have the following lemma.

Lemma 3.2 The function k ∈ B is a solution of problem (1) if and only if [k](t) = k(t) for t ∈ J.
Theorem 3. 3 The nonlinear fractional q-differential equation (1) has a unique solution k ∈ B whenever there exist nonnegative functions g 1 , for r ∈ J and λ > 1.
Proof We transform the fractional q-differential equation to a fixed point problem. By Lemma 3.2 the fractional q-differential problem (1) has a solution if and only if the operator has a fixed point in B. First, we will prove that is a contraction. Let k, l ∈ B. Then By inequality (13) we obtain On the other hand, Lemma 2.3 implies In view of (13), it yields for t ∈ J. Also, we have where Therefore Applying inequality (13), we get Now let us estimate the term We have and, consequently, (22) becomes By (15) this yields Taking into account (18) From here the contraction principle ensures the uniqueness of solution for the fractional qdifferential problem (1), which finishes the proof.
We now give an existence result for the fractional q-differential problem (1).
Proof First, let us prove that is completely continuous. It is clear that is continuous since w and 1 G ζ q are continuous. Let B η = {k ∈ B : k ≤ η} be a bounded subset in B. We will prove that (B η ) is relatively compact.
(i) For k ∈ B η , using inequality (26), we get Since φ 1 and φ 2 are nondecreasing, inequality (28) implies Using similar techniques to get (18), this yields Hence Moreover, we have and On the other hand, by (23) and (24) we obtain and from (31) and (32) we get Then (B η ) is uniformly bounded. (ii) (B η ) is equicontinuous. Indeed, for all k ∈ B η and t 1 , t 2 ∈ J with t 1 < t 2 , denoting Also, we have Using (23), (24), and (32), this yields and As t 1 → t 2 in (36) and (39), | [k](t 1 ) -[k](t 2 )| and tend to 0. Consequently, (B η ) is equicontinuous. By the Arzelá-Ascoli theorem we deduce that is a completely continuous operator. Now we apply the Leray-Schauder nonlinear alternative to prove that has at least one nontrivial solution in B. Letting O = {k ∈ B : k < η}, for any k ∈ ∂O such that k = τ [k](t), 0 < τ < 1, by (31) we get Taking into account (34), we obtain From (40) and (41) we deduce that which contradicts the fact that k ∈ ∂O. In this stage, Lemma 2.4 allows us to conclude that the operator has a fixed point k * ∈ O, and thus the fractional q-differential problem (1) has a nontrivial solution k * ∈ O. The proof is completed.

Existence of nonnegative solutions
In this section, we investigate the positivity of nonnegative solutions for the fractional q-differential problem (1). To do this, we introduce the following assumptions.
Let us rewrite the function k as Hence Now we give the properties of the Green function 2 H ζ q (t, ξ ).
Proof It is obvious that 2 G ζ q (t, ξ ) ∈ C(J 2 ). Moreover, we have which is positive if λ(σ -2) ≥ 1. Hence 2 G ζ q (t, ξ ) is nonnegative for all t, ξ ∈ J. Let t ∈ [τ , 1]. It is easy to see that q (ξ ) = 0. Then we have in all the cases. Since q (ξ ) is nonnegative, we obtain Similarly, we can prove that 2 H ζ q (t, ξ ) has the stated properties. The proof is completed.
We recall the definition of a positive solution. A function k is called a positive solution of the fractional q-differential problem (1) if k(t) ≥ 0 for all t ∈ J.
Proof First, let us remark that under the assumptions on k and w, the function c D ζ q [k] is nonnegative. Applying the right-hand side of inequality (45), we get Also, inequality (45) implies that where = 1+(σ -2) q (ζ ) q (σ -2) q (ζ ) . Combining (47) and (48) yields which is equivalent to In view of the left-hand side of (45), we obtain that for all t ∈ [τ , l], On the other hand, we have From (50) and (51) and by (49) we deduce that This completes the proof.
Define the quantities L 0 and L ∞ by The case of L 0 = 0 and L ∞ = ∞ is called the superlinear case, and the case of L 0 = ∞ and L ∞ = 0 is called the sublinear case. To prove the main result of this section, we apply the well-known Guo-Krasnoselkii fixed point Theorem 2.5 on a cone.

Theorem 3.7
Under the assumptions of Lemma 3.6, the fractional q-differential problem (1) has at least one nonnegative solution in the both superlinear and sublinear cases.
Proof First, we define the cone We can easily check that C is a nonempty closed convex subset of B, and hence it is a cone. Using (3.6), we see that [C] ⊂ C. Also, from the proof of Theorem (3.4) we know that is completely continuous in B. Let us prove the superlinear case.
(1) Since L 0 = 0, for any ε > 0, there exists δ 1 > 0 such that γ (k, l) ≤ ε(|k| + |l|) for Moreover, we have From (53) and (54) we conclude In view of assumption (A2), we can choose ε such that Inequalities (55) and (56) imply that Using the left-hand side of (45) and Lemma (3.6), we obtain Moreover, by inequality (51) we get In view of inequalities (57) and (58), we can write Let us choose M such that The first part of Theorem (2.5) implies that has a fixed point in C ∩ (O 2 \ O 1 ) such that δ 2 ≤ k ≤ δ. To prove the sublinear case, we apply similar techniques. The proof is complete.

Some illustrative examples
Herein, we give some examples to show the validity of the main results. In this way, we give a computational technique for checking problem (1). 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 Algorithms 2, 3, 4, and 5; for more detail, follow these address https://www.dm.uniba.it/members/garrappa/software.
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 [33]. 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-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 of science and technology. q-differential boundary value problems occur in the mathematical modeling of a variety of physical operations. In the end of this paper, we investigated a complicated case by utilizing an appropriate basic theory. An interesting feature of the proposed method is replacing the classical derivative with q-derivative to prove the existence of nonnegative solutions for a familiar problem for q-differential equations on a time scale, and under suitable assumptions, we have presented the global convergence of the proposed method with the line searches. The results of numerical experiments demonstrated the effectiveness of the proposed algorithm.