key: cord-0963258-pwhfuktj
authors: Samei, Mohammad Esmael; Ahmadi, Ahmad; Hajiseyedazizi, Sayyedeh Narges; Mishra, Shashi Kant; Ram, Bhagwat
title: The existence of nonnegative solutions for a nonlinear fractional q-differential problem via a different numerical approach
date: 2021-04-23
journal: J Inequal Appl
DOI: 10.1186/s13660-021-02612-z
sha: 99e207aa3c0ae0dbad349e34b3fbfbbfa027dc24
doc_id: 963258
cord_uid: pwhfuktj

This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q-differential equation [Formula: see text] with three-point conditions for [Formula: see text] on a time scale [Formula: see text] , where [Formula: see text] , [Formula: see text] , and [Formula: see text] , 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.

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.

In for t ∈ [0, 1], q ∈ (0, 1), 2 < α ≤ 3, 0 < β ≤ 3, and k(0) + k(1) = 0, D q k(0) + D q k(1) = 0, D 2 q k(0) + D 2 q k(1) = 0, where c D α q is the Caputo fractional q-derivative of order α, and F : [0, 1] × R × R × R → P(R) is a multivalued map with P(R) the class of all subsets of R.

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 c D α 0 [z](t) = φ(t, z(t), c D β 0 [z](t)) for t ∈ (0, 1) under the conditions z(0) = z (0) = 0 and z (τ ) = αz (1), where φ : [0, 1] × R 2 → R is a given function, α, β in (2, 3) and (0, 1), respectively, 0 < η < 1, and c D β 0 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 D α q [x](t) + w(t, x(t)) = 0 for each t ∈ [0, 1] with nonlocal boundary conditions x(0) = D α-2 q [x](0) = 0 and

where D α q is the standard Riemann-Liouville fractional q-derivative of order α such that 2 < α ≤ 3 and α -1β > 0, q ∈ (0, 1), φ ∈ L 1 [0, 1] is nonnegative, μ[x] is a linear functional given by μ[x] = 1 0 x(t) dN(t) involving the Stieltjes integral with respect to a nondecreasing function N : [0, 1] → R such that N(t) is right-continuous on [0, 1), leftcontinuous at t = 1, N(0) = 0, and dN 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 c D α q [u](t) equation in two modes and inclusions of order α ∈ (n -1, n], where the natural number n ≥ 5, 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 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.

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 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 = (1q a )/(1q) [8] . 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 [11] . Also, for α ∈ R and a = 0, we have

If y = 0, then it is clear that x (α) = x α [12] (Algorithm 1). The q-gamma function is given by

where z ∈ R\{0, -1, -2, . . .} [8] . Note that q (z + 1) = [z] q q (z). Algorithm 2 shows a pseudocode description of the technique for estimating the q-gamma 

, which is shown in Algorithm 3 [11] . Furthermore, the higher-order q-derivative of a function f is defined by [11] . The q-integral of a function f is defined on [0, b] by

provided that the series absolutely converges [11] . If x ∈ [0, T], then and [11] . The fractional Riemann-Liouville-type q-integral of a function h on J = (0, 1) for α ≥ 0 is defined by

for t ∈ J [15, 17] . We can use Algorithm 5 for calculating I α q [h](t) according to Eq. (2). Also, the Caputo fractional q-derivative of a function h is defined by

for t ∈ J and α > 0 [17] . 

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

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.

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.

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).

Assume that w(0, 0, 0) = 0 and there exist nonnegative functions g 1 , g 2 , g 3 ∈ C(J, R + ), nondecreasing functions φ 1 , φ 2 ∈ C(R + , [0, ∞]), and η > 0 such that

for almost all (t, k, k) ∈ J × R 2 , and

where (14). Then the fractional q-differential problem (1) has at least one nontrivial solution k * ∈ B.

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.

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.

(A1) w(t, k, l) = μ(t)γ (k, l), where μ ∈ C(J, (0, ∞)) and γ ∈ C(R + × R, R + ).

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.

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.

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.

We consider the nonlinear fractional q-differential equation

under the boundary conditions k(0) = k (0) = 0 and k ( 1 5 ) = 2 13 k (1) for t ∈ (0, 1). It is clear that σ = 11 4 ∈ (2, 3), ζ = 3 7 ∈ (0, 1), r = 1 5 ∈ (0, 1), and λ = 2 13 > 0. We define the function w :

Let k 1 , k 2 , l 1 , l 2 ∈ R. Then we have

Therefore g 1 (t) = 1 2 √ 3 sin 2 t and g 2 (t) = 1 5 (1t) 3 , and by using equality (2) we obtain for q = 1 5 , 1 2 , 7 8 , respectively. Table 1 shows these results. Figures 2a and 2b show the curves of A and B . Also, Figs. 1a and 1b show the curves of I σ -1 q [g 1 ] L 1 and I σ -1 q [g 2 ] L 1 , respectively. Thus Theorem 3.3 implies that the nonlinear fractional q-differential equation (59) has a unique solution in B. under the boundary conditions k(0) = k (0) = 0 and k ( 1 4 ) = 5 3 k (1) for t ∈ (0, 1), has at least one nontrivial solution. It is obvious that σ = 8 3 ∈ (2, 3), ζ = 6 11 ∈ (0, 1), r = 1 4 ∈ (0, 1), and λ = 5 3 > 0. We define function the w : J × R 2 → R by w t, k(t), l(t) = 1 -1 t + 1 2 (k(t)) 2 6 + (k(t)) 4 + ln 1 + l(t) 2 + 1 . 

Now from inequality (26) we can consider g i (t) = (1 -1 t+1 ) 2 for i = 1, 2, 3 and φ 1 k(t) = (k(t)) 2 6 + (k(t)) 4 , φ 2 k(t) = ln 1 + k(t) 2 . Let us find η such that inequality (27) holds. In this case, by (14) .   Tables 2 and 3 show these results. Also, Fig. 4 shows the curve of the p base on Table 2 for q = 1 5 , 1 2 , 7 8 . Now we see that inequality (27) is equivalent to φ 1 (η) + φ 2 (η) + 1 pη = η 2 6 + η 4 + ln 1 + η 2 + 1 (1.5075)η < 0, 

φ 1 (η) + φ 2 (η) + 1 pη = η 2 6 + η 4 + ln 1 + η 2 + 1 (0.9073)η < 0, for q = 1 5 , 1 2 , 7 8 , 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 are η = 4, 5, 8 for q = 1 5 , 1 2 , 7 8 , respectively. Note that (η) defined by (η) = φ 1 (η) + φ 2 (η)

η w t, k(t), l(t) = 1t 2 1 + t 2 3π k(t) + l(t) + 6π + exp -π k(t) + l(t) . Figure 5 shows the curve of the base on Table 5 for q = 1 5 , 1 2 , 7 8 . If we define the functions γ : R 2 → R and μ : J → R + by γ k(t), l(t) = 3π k(t) + l(t) + 6π + exp -π k(t) + l(t) 

So assumption (A2) holds. Table 6 showsthese results. For this, we use Algorithm 8. (1 -( t * q^k )^2 ) / ( 1 + ( t * q^k )^2 ) ; 13 fun2= varrho_q ( q , sigma , k , e , lambda , t * q^k ) ; 14 p=p + q^k * s1 * fun1 * fun2 / s2 ; Thus by Theorem 3.7 we get that problem (62) has at least one nonnegative solution.

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.

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The first, second, and third authors were supported by Bu-Ali Sina University. The fourth author was supported by the 

Not applicable.

 Table 5 Numerical results of q (σ -2), q (ζ ), and = 1+(σ -2) q (ζ ) q (σ -2) q (ζ ) in equation (62) for q = 1 5 , 1 2 , 7 8 in Example 4.3 n q= 1 5 q = 1 2 q = 7 8 q (σ -2) q (ζ ) q (σ -2) q (ζ ) q (σ -2) q (ζ ) 1 1 is negative for values of η. Thus Theorem 3.4 implies that the nonlinear fractional qdifferential equation (60) has at least one nontrivial solution in B.

In this example, we consider the fractional q-differential equationunder boundary conditions k(0) = k (0) = 0 and k ( 3 8 ) = 15 4 k (1) for t ∈ (0, 1) such that the assumptions of Lemma 3.6 hold. Clearly, σ = 18 7 ∈ (2, 3), ζ = 5 6 ∈ (0, 1), r = 3 8 ∈ (0, 1), and λ = 15 4 > 0. Also, λ(σ -2) = 15 7 > 1. Table 5 shows that ≈ 1.1887, 1.0505, 0.9579 for q = 1 5 , 1 2 , 7 8 , respectively, which we calculated by Algorithm 7. In the algorithm, we define the matrix for saving the results for q = 1 5 , 1 2 , 7 8 . We define the function w : J × R 2 → R by Assumption ( i , column ) = i ; 10 Assumption ( i , column +1) = qGamma( q ( j ) , sigma -2 , i ) ; 11 Assumption ( i , column +2) = qGamma( q ( j ) , z e t a , i ) ; 12 Assumption ( i , column +3) = (1+ ( sigma -2) * Assumption ( i , column +2) ) / ( Assumption ( i , column +1) * Assumption ( i , column +2) ) ; and μ(t) = 1-t 2 1+t 2 , then assumption (A1) holds. Now we verify assumption (A2). Let q μ(ξ ) q (ξ ) (1)Availability of data and materials Data sharing not applicable to this paper as no datasets were generated or analyzed during the current study.

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The authors declare that they have no competing interests.

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The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.