Monotone Iterative Schemes for Positive Solutions of a Fractional q -Difference Equation with Integral Boundary Conditions on the Half-Line

In this paper, we study the boundary value problem of a fractional q -diﬀerence equation with nonlocal integral boundary conditions on the half-line. Using the properties of the Green function and monotone iterative method, the extremal solutions are obtained. Finally, an example is presented to illustrate our main results.


Introduction
In this paper, we are concerned with the following fractional q-differential equation with integral boundary value problem on the half-line: where t ∈ J � [0, +∞), D α q and D α− j q are the fractional q-derivative of Riemann-Liouville type of order α and α − j, D j q is the q-derivative of order j, 0 < q < 1, n ∈ N, n ≥ 3, f ∈ C(J × R n , J), and g is a given function and satisfies some conditions which will be given later. e q-difference calculus was initially developed by Jackson [1,2] and had proved to have important applications in many subjects, such as quantum mechanics, complex analysis, and hypergeometric series. e fractional q-difference calculus had its origin in the works by Al-Salam [3] and Agarwal [4]. Due to the development of the fractional differential equations, fractional q-differential equations, regarded as fractional analogue of q-difference equations, have been studied by several researchers, especially, about the existence of the solutions for the boundary value problems (see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and the references therein).
In [14], Wang et al. investigated the following nonlinear fractional q-difference equation with q-integral boundary condition: where α ∈ (n − 1, n], n ≥ 3, μ is a parameter with 0 < μ < [α] q , β > 0, and D α q is the fractional q-derivative of Riemann-Liouville type. By applying the hybrid monotone method, the existence and uniqueness of the positive solution of the q-integral boundary problem are obtained. In [17], the authors investigate the existence of solutions for the following boundary value problem of nonlinear fractional q-difference equations on the half-line D α q u(t) + f(t, u(t)) � 0, u(0) � 0, where 1 < α < 2, 0 ≤ m i�1 a i ξ α− 1 i < Γ q (α). D α q is the q-derivative of Riemann-Liouville type of order α. By means of Schauder fixed point theorem and Leggett-Williams fixed point theorem, some results on existence and multiplicity of solutions to the above boundary value problem are obtained.
To the best of our knowledge, there are few papers that consider the boundary value of nonlinear fractional q-difference equations with nonlocal conditions on the half-line, although the study of such problems is very important. In [17], the authors only proved the existence and multiplicity of solutions, for the uniqueness of positive solutions without being given. Moreover, how to seek the solutions? It is very important and helpful for computational purpose. is thought motivates the research of iterative schemes of positive solutions for problem (1). We should mention that the paper has some new features. Firstly, we consider the infinite interval problem for higher-order nonlinear fractional q-difference equation with integral boundary conditions. Secondly, the nonlinearity relies on the lower-order fractional q-derivative of unknown function. Our main purpose of this paper is by constructing a suitable Banach space, defining appropriate operators, and using the monotone iterative method, which is different from the method in [17] to obtain the existence and uniqueness of positive solutions.

Preliminaries on q-Calculus and Lemmas
Here we recall some definitions and fundamental results on fractional q-integral and fractional q-derivative. For more information regarding fractional q-calculus, see [23].
Let q ∈ (0, 1) and define If b � 0, then a (α) � a α . e q-gamma function is defined by and satisfies Γ q (s + 1) � [s] q Γ q (s). e q-derivative of a function u is defined by and q-derivative of higher order by e q-integral of a function f defined in the interval [0, b] is given by If a ∈ [0, b] and u is defined in the interval [0, b], its integral from a to b is defined by Similar to that for derivatives, an operator I n q can be defined, namely, e fundamental theorem of calculus applies to these operators I q and D q , i.e., and if u is continuous at s � 0, then e basic properties of the two operators can be found in the book [23]. e following definition was considered first in [4].
Definition 1 (see [4]). Let α ≥ 0 and u be a function defined on [0, 1]. e fractional q-integral of the Riemann-Liouville type is (I 0 q u)(s) � u(s) and Definition 2 (see [24]). e fractional q-derivative of the Riemann-Liouville type of order α ≥ 0 is defined by where m is the smallest integer greater than or equal to α.

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Equation (33) is multiplied by g(t) and q-integrating from 0 to +∞, and we obtain (34) Combining (33) and (35), we have e proof is completed.

Lemma 5. From Lemma 4, by direct calculation, we have
where Now, we shall prove a lemma which plays a key role in the next theorems.

Lemma 6. Function G(t, qs) defined in (24) satisfies the following conditions
Proof. From (24), it is obvious that, for all 0 ≤ t, s ≤ + ∞, e above two inequalities imply that Journal of Mathematics 5 qs) defined in Lemma 5 satisfies the following conditions: Proof. For 0 ≤ t, s ≤ + ∞ and i � 1, 2, . . . , n − 1, by Remark 2 and Lemma 5, it is easy to get is gives for, i � 2, . . . , n − 1, e proof is completed. In the following, we will construct a suitable Banach space. Let with the norm where Similar to the proof of Lemma 2.2 in [26], we can prove X is a Banach space. Note that the Arzela-Ascoli theorem fails to work in X. In order to proceed, we need the following compactness criterion. □ Lemma 8. Let B ⊂ X be a bounded set. en, B is relatively compact in X if the following conditions hold: , u ∈ B, the following inequalities hold: Proof. e method is similar to the proof of Lemma 2.3 in [26], so we omit the details. □ Lemma 9. If condition (H 2 ) holds, then for u ∈ X, we have Journal of Mathematics 7 Define the cone P ⊂ X by and define the operator T: P↦P as follows: We also define It is easy to see problem (1) has a solution, if and only if the operator T has a fixed point.

Lemma 11. If conditions (H 1 ) and (H 2 ) hold, then the operator T: P↦P is completely continuous.
Proof. It is easy to see T: P↦P. Since G(t, qs) ≥ 0 and f ≥ 0, we know T(u)(t) ≥ 0, for u ∈ P, t ∈ J. In the following, we divide the proof into four steps: Step 1: we show that T is uniformly bounded. Let Ω be any bounded subset of P, i.e., there exists M > 0 such that ‖u‖ ≤ M for each u ∈ Ω. We only need to show that Tu is bounded in P. For u ∈ Ω, by Lemmas 5 and 8, we have By Lemmas 5, 7, and 9, we have From (54)-(56), we obtain 8 Journal of Mathematics which means that Tu is uniformly bounded.
Note that, from (53) and Lemma 5, we have where which means G * 1 (t, qs) does not rely on t. Hence, D α− 1 q Tu is equicontinuous on any compact intervals of J.
Step 3: we show that T is equiconvergent at ∞. We need to prove that, for ∀ε > 0, ∃T(ε) > 0, such that, for any t 1 , t 2 > T(ε), the following inequalities hold: (66) Note that, since the function G * 1 (t, s) does not rely on t, it is easy to infer that i.e., D α− 1 q Tu(t) is equiconvergent at +∞.
By (H 1 ) and Lemma 9, for any ε > 0, there exists a > 0 such that (68) On the other hand, since ere exists sufficiently large T 1 > 0 such that for any t 1 , t 2 > T 1 , we have . . , n − 1, there exists sufficiently large T 2 > a such that, for any t 1 , t 2 > T 3 and 0 ≤ qs < s ≤ a, we have Choose T > max T 1 , T 2 , for any t 1 , t 2 > T, and let t 1 ⟶ t 2 . For convenience, we denote (72)

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By (68) and (72), we obtain Similar to (72) and (73), we obtain From (67), (73), and (74), we obtain T is equiconvergent at ∞. e above three steps and Lemma 8 imply that the operator T: P ⟶ P is relatively compact.
Step 4: we show that the operator T: P ⟶ P is continuous. For u m ∈ P, such that u m ⟶ u as m ⟶ ∞, and we need to show that ‖Tu m − Tu‖ ⟶ 0 as m ⟶ ∞. By Lemma 6, the continuity of function f, and the Lebesgue dominated convergence theorem, we obtain Similar to (75), by Lemmas 7 and 9, we get for all i � 2, 3, . . . , n − 1. From (75)-(77), we have which means that the operator T is continuous.
In view of all the above arguments, we deduce that the operator T: P ⟶ P is completely continuous. is completes the proof.

Main Results
In this section, we give the main results of this paper. For convenience, we denote Let We have the following two theorems.
(90) By Lemmas 5 and 7, we have By the induction, we have In view of the complete continuity of the operator T and u m+1 (t) � Tu m (t), u m (t) ∞ m�0 is relatively compact. at is, u m (t) ∞ m�0 has a convergent subsequence u m k (t) ∞ k�1 and there exists a u * ∈ P R such that u m k ⟶ u * as k ⟶ ∞. is together with (90) implies lim m⟶∞ u m � u * . Since T is continuous and u m+1 (t) � Tu m (t), then we have u * � Tu * , that is, u * is a fixed point of operator T.
For the scheme v m ∞ m�0 , we use a similar discussion. For t ∈ J, by Lemma 5 and f ∈ C(J × R n , J), we have Applying the monotonicity of function f, we have By the induction, we have v m+1 (t) � Tv m (t) and v m (t) ≤ v m+1 (t), m � 0, 1, 2, . . . .

(95)
By Remark 2 and Lemma 6, we get By the induction, we have Using the complete continuity of the operator T and the iterative scheme, it is easy to get v m ⟶ v * and Tv * � v * .
Finally, we show that u * and v * are the maximal and minimal positive solutions of problem (1). Let w(t) be any positive solution of (1), then v 0 (t) � 0 < w(t) ≤ Rt α− 1 � u 0 (t), Noting that T is increasing, then we have Similarly, for t ∈ J and i � 1, 2, . . . , n − 1, we can get v m (t) ≤ w(t) ≤ u m (t), Since v * � lim m⟶∞ v m and u * � u m m⟶∞ lim, it follows that, for t ∈ J and i � 1, 2, . . . , n − 1, we have is implies that v * (t) and u * (t) are the minimal and maximal solutions of problem (1), and (53) and (54) hold.
On the other hand, we infer that v * (t) and u * (t) are two positive solutions of (1). In fact, by the condition (H 4 ), we get 0 function is not the solution of problem (1), which means v * (t) > 0, u * (t) > 0, and they can be constructed by means of two monotone iterative schemes in (81) and (82).

Theorem 2. Suppose conditions (H 1 ) and (H 3 ) are satisfied. If
hold. L is defined in (80). In addition, there is an error estimate for the approximation scheme Proof.
In the following, we show that T is a contraction. For any u 1 , u 2 ∈ P r , by condition (H 3 ) and Lemma 10, we obtain

Conclusions
In this paper, we study the boundary value problem of a fractional q-difference equation with nonlocal integral boundary conditions on the half-line. We obtain some new results as follows: (1) the existence and uniqueness of positive solutions for a higher-order nonlinear fractional q-difference equation with integral boundary conditions on the half-line are obtained, and (2) the monotone iterative schemes of positive solutions for the problem which considered in our paper are obtained.

Data Availability
All the data included in this study are available upon request by contact with the corresponding author.

Conflicts of Interest
e authors declare no conflicts of interest.