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

This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q-differential equation D q σ c [ k ] ( t ) = w ( t , k ( t ) , c D q ζ [ k ] ( t ) ) with three-point conditions for t ∈ ( 0 , 1 ) on a time scale T t 0 = { t : t = t 0 q n } ∪ { 0 } , where n ∈ N , t 0 ∈ R , and 0 < q < 1 , based on the Leray-Schauder nonlinear alternative and Guo-Krasnoselskii theorem. Moreover, we discuss the existence of nonnegative solutions. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

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 [1-7]). 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 q-calculus. 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 [11-23] and references therein.

In 2012, Zhoujin et al. considered the fractional differential equation for and under the boundary conditions and for , where denotes the Caputo fractional derivative, , and . The existence results are derived by means of Schauder’s fixed-point theorem. Then Liang and Zhang [24] studied the existence and uniqueness of positive solutions by properties of the Green function, the lower and upper solution method, and the fixed point theorem for the fractional equation for under the boundary conditions and , where , and is the Riemann–Liouville fractional derivative. In 2015, Zhang et al. [25] through the spectral analysis and fixed point index theorem obtained the existence of positive solutions of the singular nonlinear fractional differential equation for almost all with integral boundary value conditions and where , , may be singular at both , 1 and , denotes the Riemann–Stieltjes integral with signed measure, in which is a function of bounded variation. In 2016, Ahmad et al. [16] investigated the existence of solutions for a q-antiperiodic boundary value problem of fractional q-difference inclusions for , , , , and , , , where is the Caputo fractional q-derivative of order α, and is a multivalued map with the class of all subsets of .

In 2018, Guezane-Lakoud and Belakroum [26] considered the existence and uniqueness of nonnegative solutions of the boundary value problem for nonlinear fractional differential equation for under the conditions and , where is a given function, α, β in and , respectively, , and denotes the Caputo fractional derivative. In 2019, Ren and Zhai [27] discussed the existence of a unique solution and multiple positive solutions for the fractional q-differential equation for each with nonlocal boundary conditions and where is the standard Riemann–Liouville fractional q-derivative of order α such that and , , is nonnegative, is a linear functional given by involving the Stieltjes integral with respect to a nondecreasing function such that is right-continuous on , left-continuous at , , and is a positive Stieltjes measure. Rehman et al. [28] developed Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. They introduced the Green–Haar approach for a family of generalized fractional boundary value problems and compared the method with the classical Haar wavelets technique. The existence of solutions for the multiterm nonlinear fractional q-integro-differential equation in two modes and inclusions of order , where the natural number , with nonseparated boundary and initial boundary conditions was considered in [29]. In [30] the investigation is centered around the quantum estimates by utilizing the quantum Hahn integral operator via the quantum shift operator. In [20] the q-fractional integral inequalities of Henry–Gronwall type are presented.

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 and for and , where is a given function with , , , ,and , and 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].

In fact, we consider the fractional q-calculus on the specific time scale , where for nonnegative integer n, , and . Let . Define  [8]. The power function with is defined by for and , where x and y are real numbers, and  [11]. Also, for and , we have If , then it is clear that  [12] (Algorithm 1). The q-gamma function is given by , where  [8]. Note that . Algorithm 2 shows a pseudocode description of the technique for estimating the q-gamma function of order n. The q-derivative of a function f is defined by and , which is shown in Algorithm 3 [11]. Furthermore, the higher-order q-derivative of a function f is defined by for , where  [11]. The q-integral of a function f is defined on by for , provided that the series absolutely converges [11]. If , then , which is equal to whenever the series exists. The operator is given by and for and  [11]. It has been proved that and whenever h is continuous at  [11]. The fractional Riemann–Liouville-type q-integral of a function h on for is defined by and for  [15, 17]. We can use Algorithm 5 for calculating according to Eq. (2). Also, the Caputo fractional q-derivative of a function h is defined by for and  [17]. It has been proved that and for  [17]. Algorithm 5 gives a pseudocode for .

The proposed method for calculating

The proposed method for calculating

The proposed method for calculating

Lemma 2.1

([17])

Let α, and . Then and for all .

Lemma 2.2

Let . Then almost everywhere on for , and it is valid at any point if .

Lemma 2.3

([22])

Let and . Then we have for .

To prove the theorems, we further apply the Leray–Schauder nonlinear alternative.

Lemma 2.4

([32])

Let be a Banach space, let be a bounded open subset of , and let be a completely continuous operator. Then either there exist and such that , or there exists a fixed point .

Theorem 2.5

Let be a Banach space, and let be a cone. Let and be open subsets of with , , and let be a completely continuous operator such that Then Θ has a fixed point in .

for and for ,

for and for .

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 the Banach space of Lebesgue-integrable functions with the norm .

Lemma 3.1

Let and . The unique solution of the q-fractional problem for is given by where

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 , and the second one gives Substituting into equation (7), we obtain which can be written as Indeed, where 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 of all functions into with the norm Note that if . Denote Throughout this section, we suppose that . We define the integral operator by Then we have the following lemma.

Lemma 3.2

The function is a solution of problem (1) if and only if for .

Theorem 3.3

The nonlinear fractional q-differential equation (1) has a unique solution whenever there exist nonnegative functions , such that

for all with and , we have

and , where for and .

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 Θ has a fixed point in . First, we will prove that Θ is a contraction. Let . Then By inequality (13) we obtain On the other hand, Lemma 2.3 implies In view of (13), it yields for . Also, we have where and 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)–(25), we obtain for . From here the contraction principle ensures the uniqueness of solution for the fractional q-differential problem (1), which finishes the proof. □

We now give an existence result for the fractional q-differential problem (1).

Theorem 3.4

Assume that and there exist nonnegative functions , nondecreasing functions , and such that for almost all , and where  and with and defined as in Theorem 3.3by (14). Then the fractional q-differential problem (1) has at least one nontrivial solution .

First, let us prove that Θ is completely continuous. It is clear that Θ is continuous since w and are continuous. Let be a bounded subset in . We will prove that is relatively compact. By the Arzelá–Ascoli theorem we deduce that Θ is a completely continuous operator. Now we apply the Leray–Schauder nonlinear alternative to prove that Θ has at least one nontrivial solution in . Letting , for any such that , , by (31) we get Taking into account (34), we obtain From (40) and (41) we deduce that which contradicts the fact that . In this stage, Lemma 2.4 allows us to conclude that the operator Θ has a fixed point , and thus the fractional q-differential problem (1) has a nontrivial solution . The proof is completed. □

For , using inequality (26), we get Since and 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 is uniformly bounded.

is equicontinuous. Indeed, for all and with , denoting we have Also, we have Using (23), (24), and (32), this yields and As in (36) and (39), and tend to 0. Consequently, is equicontinuous.

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 where Hence where Now we give the properties of the Green function .

, where and .

, where

Lemma 3.5

If , then and belong to with and for all . Furthermore, if , , then for each , we have and

It is obvious that . Moreover, we have which is positive if . Hence is nonnegative for all . Let . It is easy to see that . Then we have whenever , whenever , , whenever , and whenever , . Thus in all the cases. Since is nonnegative, we obtain Similarly, we can prove that 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 for all .

Lemma 3.6

If and , then the solution of the fractional q-differential problem (1) is nonnegative and satisfies

First, let us remark that under the assumptions on k and w, the function is nonnegative. Applying the right-hand side of inequality (45), we get Also, inequality (45) implies that where . Combining (47) and (48) yields which is equivalent to Indeed, In view of the left-hand side of (45), we obtain that for all , On the other hand, we have From (50) and (51) we get and by (49) we deduce that This completes the proof. □

Define the quantities and by The case of and is called the superlinear case, and the case of and 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.

First, we define the cone We can easily check that is a nonempty closed convex subset of , and hence it is a cone. Using (3.6), we see that . Also, from the proof of Theorem (3.4) we know that Θ is completely continuous in . Let us prove the superlinear case. The first part of Theorem (2.5) implies that Θ has a fixed point in such that . To prove the sublinear case, we apply similar techniques. The proof is complete. □

Since , for any , there exists such that for . Letting , for any , this yields 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 for each .

Second, in view of , for any , there exists such that for . Take and denote by the open set . If , then 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 Then we get . So, for each .

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.

The proposed method for calculating

The proposed method for calculating

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.

Example 4.1

We consider the nonlinear fractional q-differential equation under the boundary conditions and for . It is clear that , , , and . We define the function by Let , , . Then we have Therefore and , and by using equality (2) we obtain for , , , respectively, for , , , respectively, and which are less than one, for , , , respectively, for , , , respectively. Table 1 shows these results. Figures 2a and 2b show the curves of and . Also, Figs. 1a and 1b show the curves of and , respectively. Thus Theorem 3.3 implies that the nonlinear fractional q-differential equation (59) has a unique solution in .

Table 1

Numerical results of problem (59) for , , and (1) and (2)  in Example 4.1

n	g_1(t)			g_2(t)			Σ_A	Σ_B	Γ_q(1−ζ)(1−ζ)^−1
	(1)	A_1	B_1	(2)	A_2	B_2
	q=15
1	0.2413	0.2941	0.1930	0.1672	0.2516	0.0267	0.4716	0.2196	0.7280
2	0.2507	0.3036	0.1930	0.1737	0.2630	0.0344	0.4896	0.2274	0.7280
3	0.2526	0.3055	0.1930	0.1750	0.2653	0.0361	0.4932	0.2291	0.7280
4	0.2530	0.3058	0.1930	0.1753	0.2658	0.0364	0.4939	0.2294	0.7280
5	0.2531	0.3059	0.1930	0.1753	0.2659	0.0365	0.4941	0.2295	0.7280
6	0.2531	0.3059	0.1930	0.1753	0.2659	0.0365	0.4941	0.2295	0.7280
7	0.2531	0.3059	0.1930	0.1754	0.2659	0.0365	0.4941	0.2295	0.7280
8	0.2531	0.3059	0.1930	0.1754	0.2659	0.0365	0.4941	0.2295	0.7280
9	0.2531	0.3059	0.1930	0.1754	0.2659	0.0365	0.4941	0.2295	0.7280
	q=12
1	0.1461	0.1876	0.1161	0.1012	0.1067	0.0107	0.2943	0.1268	0.8077
2	0.1804	0.2225	0.1188	0.1250	0.1345	0.0248	0.3569	0.1436	0.8077
3	0.1985	0.2406	0.1192	0.1375	0.1499	0.0361	0.3906	0.1552	0.8077
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
9	0.2169	0.2591	0.1192	0.1503	0.1663	0.0509	0.4254	0.1701	0.8077
10	0.2171	0.2593	0.1192	0.1504	0.1664	0.0510	0.4257	0.1702	0.8077
11	0.2171	0.2593	0.1192	0.1504	0.1665	0.0511	0.4258	0.1703	0.8077
12	0.2172	0.2594	0.1192	0.1505	0.1665	0.0511	0.4259	0.1703	0.8077
13	0.2172	0.2594	0.1192	0.1505	0.1665	0.0511	0.4259	0.1704	0.8077
14	0.2172	0.2594	0.1192	0.1505	0.1665	0.0511	0.4259	0.1704	0.8077
15	0.2172	0.2594	0.1192	0.1505	0.1665	0.0511	0.4259	0.1704	0.8077
	q=78
1	0.0187	0.0387	0.0320	0.0129	0.0134	0.0004	0.0521	0.0324	0.4938
2	0.0313	0.0560	0.0426	0.0217	0.0226	0.0010	0.0785	0.0436	0.5608
3	0.0446	0.0722	0.0507	0.0309	0.0323	0.0018	0.1045	0.0526	0.6119
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
21	0.1710	0.2046	0.0713	0.1185	0.1338	0.0485	0.3383	0.1199	0.8535
22	0.1730	0.2065	0.0714	0.1199	0.1355	0.0501	0.3421	0.1214	0.8560
23	0.1748	0.2083	0.0714	0.1211	0.1371	0.0514	0.3454	0.1228	0.8581
24	0.1763	0.2098	0.0714	0.1222	0.1384	0.0527	0.3483	0.1240	0.8600
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
50	0.1869	0.2204	0.0714	0.1295	0.1478	0.0616	0.3682	0.1330	0.8725
51	0.1870	0.2205	0.0714	0.1295	0.1478	0.0617	0.3683	0.1330	0.8725
52	0.1870	0.2205	0.0714	0.1296	0.1479	0.0617	0.3684	0.1331	0.8726
53	0.1870	0.2206	0.0714	0.1296	0.1479	0.0617	0.3685	0.1331	0.8726
54	0.1871	0.2206	0.0714	0.1296	0.1479	0.0618	0.3685	0.1331	0.8726
55	0.1871	0.2206	0.0714	0.1296	0.1480	0.0618	0.3686	0.1331	0.8727
56	0.1871	0.2206	0.0714	0.1296	0.1480	0.0618	0.3686	0.1332	0.8727
57	0.1871	0.2206	0.0714	0.1296	0.1480	0.0618	0.3686	0.1332	0.8727
58	0.1871	0.2207	0.0714	0.1297	0.1480	0.0618	0.3687	0.1332	0.8727
59	0.1872	0.2207	0.0714	0.1297	0.1480	0.0618	0.3687	0.1332	0.8728
60	0.1872	0.2207	0.0714	0.1297	0.1480	0.0619	0.3687	0.1332	0.8728

Figure 2

Graphical representation of and for , , in Example 4.1

Figure 1

Graphical representation of for , , in Example 4.1

Example 4.2

In this example, we apply Theorem 3.4 to prove that the fractional q-differential equation under the boundary conditions and for , has at least one nontrivial solution. It is obvious that , , , and . We define function the by Figures 3a and 3b show the curves of and . Let k, . Then we have Now from inequality (26) we can consider for and Let us find η such that inequality (27) holds. In this case, by (14) we calculate and for . We obtain for , , , respectively, and so Tables 2 and 3 show these results. Also, Fig. 4 shows the curve of the p base on Table 2 for . Now we see that inequality (27) is equivalent to for , , , respectively. Now by using Algorithm 6 we try to find a suitable value for η in inequalities (61). The algorithm is created for the same problems. On the other hand, the results show that it works exactly. According to Table 4, the suitable values of η in (61) are for , , , respectively. Note that defined by is negative for values of η. Thus Theorem 3.4 implies that the nonlinear fractional q-differential equation (60) has at least one nontrivial solution in .

Figure 3

Graphical representation of and for , , in Example 4.2

Table 2

Numerical results of problem (59) for , , and (1) , (2) , and (3) in Example 4.2

n	(1)	A_1	B_1	(2)	A_2	B_2	(3)	A_3	B_3
	g_1(t)			g_2(t)			g_3(t)
	q=15
1	0.2125	0.5809	0.5452	0.2125	0.5809	0.5452	0.2125	0.5809	0.5452
2	0.2207	0.5892	0.5453	0.2207	0.5892	0.5453	0.2207	0.5892	0.5453
3	0.2224	0.5909	0.5453	0.2224	0.5909	0.5453	0.2224	0.5909	0.5453
4	0.2227	0.5912	0.5453	0.2227	0.5912	0.5453	0.2227	0.5912	0.5453
5	0.2228	0.5913	0.5453	0.2228	0.5913	0.5453	0.2228	0.5913	0.5453
6	0.2228	0.5913	0.5453	0.2228	0.5913	0.5453	0.2228	0.5913	0.5453
7	0.2228	0.5913	0.5453	0.2228	0.5913	0.5453	0.2228	0.5913	0.5453
	q=12
1	0.1327	0.4399	0.4099	0.1327	0.4399	0.4099	0.1327	0.4399	0.4099
2	0.1630	0.4773	0.4219	0.1630	0.4773	0.4219	0.1630	0.4773	0.4219
3	0.1789	0.4943	0.4237	0.1789	0.4943	0.4237	0.1789	0.4943	0.4237
4	0.1871	0.5026	0.4240	0.1871	0.5026	0.4240	0.1871	0.5026	0.4240
5	0.1912	0.5068	0.4240	0.1912	0.5068	0.4240	0.1912	0.5068	0.4240
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
10	0.1953	0.5108	0.4240	0.1953	0.5108	0.4240	0.1953	0.5108	0.4240
11	0.1953	0.5109	0.4240	0.1953	0.5109	0.4240	0.1953	0.5109	0.4240
12	0.1954	0.5109	0.4240	0.1954	0.5109	0.4240	0.1954	0.5109	0.4240
13	0.1954	0.5109	0.4240	0.1954	0.5109	0.4240	0.1954	0.5109	0.4240
14	0.1954	0.5109	0.4240	0.1954	0.5109	0.4240	0.1954	0.5109	0.4240
	q=78
1	0.0187	0.1773	0.1759	0.0187	0.1773	0.1759	0.0187	0.1773	0.1759
2	0.0309	0.2244	0.2199	0.0309	0.2244	0.2199	0.0309	0.2244	0.2199
3	0.0435	0.2607	0.2518	0.0435	0.2607	0.2518	0.0435	0.2607	0.2518
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
24	0.1626	0.4350	0.3352	0.1626	0.4350	0.3352	0.1626	0.4350	0.3352
25	0.1638	0.4362	0.3352	0.1638	0.4362	0.3352	0.1638	0.4362	0.3352
26	0.1648	0.4373	0.3353	0.1648	0.4373	0.3353	0.1648	0.4373	0.3353
27	0.1658	0.4382	0.3353	0.1658	0.4382	0.3353	0.1658	0.4382	0.3353
28	0.1666	0.4390	0.3353	0.1666	0.4390	0.3353	0.1666	0.4390	0.3353
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
50	0.1719	0.4444	0.3353	0.1719	0.4444	0.3353	0.1719	0.4444	0.3353
51	0.1720	0.4444	0.3353	0.1720	0.4444	0.3353	0.1720	0.4444	0.3353
52	0.1720	0.4444	0.3353	0.1720	0.4444	0.3353	0.1720	0.4444	0.3353
53	0.1720	0.4445	0.3353	0.1720	0.4445	0.3353	0.1720	0.4445	0.3353
54	0.1721	0.4445	0.3353	0.1721	0.4445	0.3353	0.1721	0.4445	0.3353
55	0.1721	0.4445	0.3353	0.1721	0.4445	0.3353	0.1721	0.4445	0.3353
56	0.1721	0.4445	0.3353	0.1721	0.4445	0.3353	0.1721	0.4445	0.3353

Table 3

Numerical results of , , and for , , in Example 4.2

n	q=15			q=12			q=78
	M_1	M_2	p	M_1	M_2	p	M_1	M_2	p
1	0.5809	0.5452	1.5070	0.4399	0.4099	1.1219	0.1773	0.1759	0.6224
2	0.5892	0.5453	1.5075	0.4773	0.4219	1.1421	0.2244	0.2199	0.7104
3	0.5909	0.5453	1.5075	0.4943	0.4237	1.1455	0.2607	0.2518	0.7677
4	0.5912	0.5453	1.5075	0.5026	0.4240	1.1465	0.2894	0.2750	0.8067
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
9	0.5913	0.5453	1.5075	0.5107	0.4240	1.1473	0.3722	0.3243	0.8834
10	0.5913	0.5453	1.5075	0.5108	0.4240	1.1473	0.3817	0.3276	0.8884
11	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.3899	0.3299	0.8921
12	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.3969	0.3316	0.8949
13	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.4029	0.3327	0.8970
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
40	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.4435	0.3353	0.9071
41	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.4437	0.3353	0.9072
42	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.4438	0.3353	0.9072
43	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.4439	0.3353	0.9072
44	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.4440	0.3353	0.9072
45	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.4441	0.3353	0.9073
46	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.4442	0.3353	0.9073
47	0.5913	0.5453	1.5075	0.5109	0.4240	1.1474	0.4442	0.3353	0.9073

Figure 4

2D graphs of for , , in Example 4.2

Table 4

Numerical results for finding suitable values of η in equation (27) for , , in Example 4.2, where

n	η	Ω(η)<0
		q=15	q=12	q=78
1	3.0000	2.1347	0.9079	0.0906
2	3.1000	2.1154	0.8693	0.0392
3	3.2000	2.0943	0.8294
4	3.3000	2.0715	0.7882	−0.0668
⋮	⋮	⋮	⋮	⋮
9	3.8000	1.9351	0.5649	−0.3480
10	3.9000	1.9036	0.5170	−0.4068
11	4.0000	1.8708	0.4681	−0.4663
12	4.1000	1.8367	0.4183	−0.5266
13	4.2000	1.8014	0.3676	−0.5877
⋮	⋮	⋮	⋮	⋮
18	4.7000	1.6077	0.1007	−0.9033
19	4.8000	1.5658	0.0449	−0.9684
20	4.9000	1.5229		−1.0340
21	5.0000	1.4790	−0.0690	−1.1002
22	5.1000	1.4342	−0.1270	−1.1670
23	5.2000	1.3884	−0.1857	−1.2344
⋮	⋮	⋮	⋮	⋮
47	7.6000	0.0745	−1.7591	−2.9806
48	7.7000	0.0126	−1.8301	−3.0577
49	7.8000		−1.9015	−3.1351
50	7.9000	−0.1126	−1.9732	−3.2127
51	8.0000	−0.1759	−2.0452	−3.2906
52	8.1000	−0.2395	−2.1176	−3.3687
53	8.2000	−0.3037	−2.1903	−3.4471

MATLAB lines for finding suitable values of η in Eq. (27) for q variable in Example 4.2

Example 4.3

In this example, we consider the fractional q-differential equation under boundary conditions and for such that the assumptions of Lemma 3.6 hold. Clearly, , , , and . Also, . Table 5 shows that, 1.0505, 0.9579 for , , , respectively, which we calculated by Algorithm 7. In the algorithm, we define the matrix for saving the results for . We define the function by Figure 5 shows the curve of the Λ base on Table 5 for . If we define the functions and by and , then assumption (A1) holds. Now we verify assumption (A2). Let Therefore So assumption (A2) holds. Table 6 shows these results. For this, we use Algorithm 8. Figure 6 shows the results of equation (63). On the other hand, Thus by Theorem 3.7 we get that problem (62) has at least one nonnegative solution.

Table 5

Numerical results of , , and in equation (62) for , , in Example 4.3

n	q=15			q=12			q=78
	Γ_q(σ−2)	Γ_q(ζ)	Λ	Γ_q(σ−2)	Γ_q(ζ)	Λ	Γ_q(σ−2)	Γ_q(ζ)	Λ
1	1.2612	1.0573	1.2030	1.2839	1.0584	1.1809	0.8642	0.9059	1.9386
2	1.2714	1.0600	1.1915	1.3507	1.0773	1.1103	0.9815	0.9493	1.6555
3	1.2734	1.0605	1.1892	1.3826	1.0861	1.0792	1.0708	0.9805	1.4860
4	1.2738	1.0606	1.1888	1.3982	1.0905	1.0646	1.1416	1.0043	1.3727
5	1.2739	1.0606		1.4059	1.0926	1.0575	1.1991	1.0231	1.2917
6	1.2739	1.0606	1.1887	1.4097	1.0936	1.0540	1.2466	1.0382	1.2311
7	1.2739	1.0606	1.1887	1.4116	1.0942	1.0522	1.2863	1.0506	1.1843
8	1.2739	1.0606	1.1887	1.4126	1.0944	1.0514	1.3197	1.0608	1.1473
9	1.2739	1.0606	1.1887	1.4131	1.0945	1.0509	1.3480	1.0694	1.1176
10	1.2739	1.0606	1.1887	1.4133	1.0946	1.0507	1.3722	1.0767	1.0933
11	1.2739	1.0606	1.1887	1.4134	1.0946	1.0506	1.3929	1.0828	1.0733
12	1.2739	1.0606	1.1887	1.4135	1.0947		1.4106	1.0881	1.0566
13	1.2739	1.0606	1.1887	1.4135	1.0947	1.0505	1.4260	1.0925	1.0426
14	1.2739	1.0606	1.1887	1.4135	1.0947	1.0505	1.4392	1.0964	1.0308
15	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.4506	1.0997	1.0208
16	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.4605	1.1026	1.0122
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
51	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5269	1.1214	0.9582
52	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5270	1.1215	0.9582
53	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5271	1.1215	0.9581
54	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5271	1.1215	0.9581
55	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5272	1.1215	0.9580
56	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5272	1.1215	0.9580
57	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5273	1.1215	0.9580
58	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5273	1.1215
59	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5273	1.1216	0.9579
60	1.2739	1.0606	1.1887	1.4136	1.0947	1.0505	1.5273	1.1216	0.9579

Figure 5

2D graphs of for , , in Example 4.3

Table 6

Numerical results of for in Assumption (A2) and for , , in Example 4.3

n	0<∫01(1−qξ)^(σ−3)μ(ξ)ϱ_q(ξ)d_qξ<1
	q=15	q=12	q=78
1	0.28515	0.31485	0.00482
2	0.34871	0.47839	0.01824
3	0.36161	0.57866	0.04195
4	0.36420	0.63251	0.07572
5	0.36472	0.66022	0.11804
6	0.36482	0.67425	0.16674
7	0.36484	0.68130	0.21946
8	0.36485	0.68484	0.25289
9	0.36485	0.68661	0.28682
10	0.36485	0.68750	0.31984
11	0.36485	0.68794	0.35125
⋮	⋮	⋮	⋮
15	0.36485	0.68836	0.45528
16	0.36485	0.68837	0.47565
17	0.36485	0.68838	0.49394
18	0.36485	0.68838	0.51030
19	0.36485	0.68838	0.52486
⋮	⋮	⋮	⋮
77	0.36485	0.68838	0.63313
78	0.36485	0.68838	0.63314
79	0.36485	0.68838	0.63314
80	0.36485	0.68838	0.63315
81	0.36485	0.68838	0.63315
82	0.36485	0.68838	0.63315
83	0.36485	0.68838	0.63316
84	0.36485	0.68838	0.63316
85	0.36485	0.68838	0.63316

Figure 6

2D graphs of for , , in Example 4.3

MATLAB lines for calculating values of  in Theorem 3.7 for q variable in Example 4.3

MATLAB lines for calculating in Assumption (A2) for q variable in Example 4.3

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.