Existence and uniqueness results to positive solutions of integral boundary value problem for fractional q-derivatives

In this paper,we are interested in the existence and uniqueness of positive solutions for integral boundary value problem with fractional q-derivative: Dqαu(t)+f(t,u(t),u(t))+g(t,u(t))=0,0<t<1,u(0)=Dqu(0)=0,u(1)=μ∫01u(s)dqs,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} &D_{q}^{\alpha}u(t)+f\bigl(t,u(t),u(t)\bigr)+g\bigl(t,u(t) \bigr)=0, \quad 0< t< 1, \\ & u(0)=D_{q}u(0)=0, \qquad u(1)=\mu \int_{0}^{1}u(s)\,d_{q}s, \end{aligned}$$ \end{document} where Dqα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D_{q}^{\alpha}$\end{document} is the fractional q-derivative of Riemann–Liouville type, 0<q<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< q<1$\end{document}, 2<α≤3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2<\alpha\leq3 $\end{document}, and μ is a parameter with 0<μ<[α]q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\mu<[\alpha]_{q}$\end{document}. By virtue of fixed point theorems for mixed monotone operators, we obtain some results on the existence and uniqueness of positive solutions.

(1.1) Ferreira [14] also considered the existence of positive solutions to the nonlinear qdifference boundary value problem ⎧ ⎨ ⎩ D α q u(t) + f (t, u(t)) = 0, 0 < t < 1, 1 < α ≤ 3, u(0) = D q u(0) = 0, D q u(1) = β ≥ 0. (1.2) EI-Shahed and AI-Askar [15] studied the existence of a positive solution to the fractional where γ , β ≤ 0, and c D α q is the fractional q-derivative of Caputo type. Darzi and Agheli [16] studied the existence of a positive solution to the fractional q- where D α q is the fractional q-derivative of Riemann-Liouville type, 0 < q < 1, 2 < α ≤ 3, 0 < μ < [α] q . Our results ensure the existence of a unique positive solution. Moreover, an iterative scheme is constructed for approximating the solution. As far as we know, there are still very few works utilizing the fixed point results for mixed monotone operators to study the existence and uniqueness of a positive solution for fractional q-derivative integral boundary value problems.
The plan of the paper is as follows. In Sect. 2, we give not only basic definitions of qfractional integral, but also some properties of certain Green's functions, which play a fundamental role in the process of proofs. In Sect. 3, in light of some sufficient conditions, we obtained some results on the existence and uniqueness of positive solutions to problem (1.5). At the closing part, two examples are given to demonstrate the serviceability of our main results in Sect. 4.

Preliminaries
For convenience of the reader, on one hand, we recall some well-known facts on q-calculus and, on the other hand, some notations and lemmas that will be used in the proofs of our theorems.
A nonempty closed convex set P ⊂ E is a cone if (1) x ∈ P, r ≥ 0 ⇒ rx ∈ P and (2) x ∈ P, -x ∈ P ⇒ x = θ (θ is the zero element of E), where (E, · ) is a real Banach space. For all x, y ∈ E, if there exist μ, ν > 0 such that μx ≤ y ≤ νx, then we write x ∼ y. Obviously, ∼ is an equivalence relation.
Let q ∈ (0, 1). Then the q-number is given by The q-analogue of the power function (ab) (n) with n ∈ N 0 is More generally, if α ∈ R, then Note that if b = 0, then a (α) = a α . The q-gamma function is defined by and satisfies q (x + 1) = [x] q q (x). The q-derivative of a function f is defined by and q-derivatives of higher order by The q-integral of a function f defined in the interval [0, b] is given by If a ∈ [0, b] and f is defined in the interval [0, b], then its integral from a to b is defined by Similarly to the derivatives, the operator I n q is given by The fundamental theorem of calculus applies to the operators I q and D q , that is, and if f is continuous at x = 0, then The following formulas will be used later ( t D q denotes the derivative with respect to variable t): Definition 2.1 (see [4]) Let α ≥ 0, and let f be a function defined on [0, 1]. The fractional q-integral of the Riemann-Liouville type is defined by and Definition 2.2 (see [10]) The fractional q-derivative of the Riemann-Liouville type is defined by where p is the smallest integer greater than or equal to α. Lemma 2.1 (see [10]) Let α, β ≥ 0, and let f be a function defined on [0, 1]. Then the following formulas hold: Lemma 2.2 (see [10]) Let α > 0, and let be p be a positive integer. Then the following equality holds:

2)
has a unique solution Lemma 2.4 (see [27]) The function G(t, qs) defined by (2.3) has the following properties:
Remark 2.1 When B = θ in Lemma 2.5, then the corresponding conclusion still holds.
Remark 2.2 When A = θ in Lemma 2.6, then the corresponding conclusion still holds.

Main results
In this section, we give and prove our main results by applying Lemmas 2.5 and 2.6. We consider the Banach space X = C[0, 1] endowed with standard norm x = sup{|x(t)| : t ∈ [0, 1]}. Clearly, this space can be equipped with a partial order given by We define the cone P = {x ∈ X : x(t) ≥ 0, t ∈ [0, 1]}. Notice that P is a normal cone in C[0, 1] and the normality constant is 1. (1) there exist x 0 , y 0 ∈ P h and r ∈ (0, 1) such that ry 0 ≤ x 0 < y 0 and we have x nu * → 0 and y nu * → 0 as n → ∞.
Proof We note that if u is a solution of boundary value problem (1.5), then

(3.2)
We transform the boundary value problem (1.5) into a fixed point problem u = T 1 (u, u) + T 2 u. From (F 1 ), (F 2 ), and Lemma 2.4 it is easy to see that T 1 : P × P → P and T 2 : P → P. Next, we want to prove that T 1 and T 2 satisfy the conditions of Lemma 2.5.

G(t, qs)f s, h(s), h(s) d q s
and

G(t, qs)f s, h(s), h(s) d q s
From (F 2 ) and (F 4 ) we have the inequality Since g(t, 0) ≡ 0, we also obtain Next, we prove that the operators T 1 and T 2 satisfy condition (ii) of Lemma 2.5. In fact, for u, v ∈ P and any t ∈ [0, 1], by (F 4 ) we have (3.9) Then we have T 1 (u, v) ≥ δ 0 T 2 u for u, v ∈ P. By Lemma 2.5 we can deduce: there exist u 0 , v 0 ∈ P h and r ∈ (0, 1) such that rv the operator equation T 1 (u, u) + T 2 u = u has a unique solution u * ∈ P h ; and for any initial values x 0 , y 0 ∈ P h , constructing successively the sequences x n = T 1 (x n-1 , y n-1 ) + T 2 x n-1 , y n = T 1 (y n-1 , x n-1 ) + T 2 y n-1 , n = 1, 2, . . . , we get x n → u * and y n → u * as n → ∞. We have the following two inequalities: we have x nu * → 0 and y nu * → 0 as n → ∞. (ii) the BVP has a unique positive solution u * in P h ; (iii) for any x 0 , y 0 ∈ P h , the sequences Theorem 3.2 Suppose that (F 1 )-(F 2 ) hold. In addition, suppose that f , g satisfy the following conditions: (1) there exist u 0 , v 0 ∈ P h and r ∈ (0, 1) such that rv 0 ≤ u 0 < v 0 and u 0 ≤ Proof Similarly to the proof of Theorem 3.1, T 1 and T 2 are given in (3.2). From (F 1 ) and (F 2 ) we know that T 1 : P × P → P is a mixed monotone operator and T 2 : P → P is increasing. By (F 5 ) we obtain According to (F 2 ) and (F 6 ), we have From f (t, 0, 1) ≡ 0 we get and the following inequalities hold: Hence we can easily check that T 1 (h, h) ∈ P, T 2 h ∈ P, t ∈ [0, 1], and, by using (F 6 ), we have (3.13) Then we have T 1 (u, v) ≤ δ 0 T 2 u for u, v ∈ P. Thus, from Lemma 2.6 we get that there ex- the operator equation T 1 (u, u) + T 2 u = u has a unique solution u * ∈ P h ; and for any initial values x 0 , y 0 ∈ P h , the sequences x n = T 1 (x n-1 , y n-1 ) + T 2 x n-1 , y n = T 1 (y n-1 , x n-1 ) + T 2 y n-1 , n = 1, 2, . . . , satisfy x n → u * and y n → u * as n → ∞. That is,

Example
Now, we give two examples to illustrate our results.