Recently, many mathematicians, physicists and engineers have extensively studied various families of fractional differential equations and their applications. The development of the theory of fractional calculus stems from the applications in many widespread disciplines such as engineering, economics and other fields. Jackson [1] introduced the qdifference calculus (or the socalled quantum calculus), which is an old subject. New developments in this theory were made. These include (for example) the qanalogs of the fractional integral and the fractional derivative operators, the qanalogs of the Laplace, Fourier, and other integral transforms, and so on (see, for details, [2–13], and [14]; see also a very recent work [15] dealing with qcalculus).
Throughout our present investigation, we make use of the following notations:
$$ \mathbb{N}:= \{ 1,2,3,\ldots \} \quad\mbox{and}\quad \mathbb{N}_{0}:= \mathbb{N}\cup \{ 0 \} . $$
Moreover, as usual, ℝ denotes the set of real numbers, \(\mathbb{R}_{+}\) denotes the set of positive real numbers, \(\mathbb{Z}_{}\) denotes the set of negative integers, and ℂ denotes the set of complex numbers.
AlSalam [16] and Agarwal [2] investigated several properties and results for some fractional qintegrals and fractional qderivatives which are based on the qanalog of the ordinary integral:
$$ \int_{a}^{x}f ( t )\,dt. $$
Atici and Eloe [3] constructed interesting links between the fractional qcalculus in the existing literature and the fractional qcalculus on a time scale given by
$$ T_{t_{0}}= \bigl\{ t:t=t_{0}q^{n}\ (n\in \mathbb{N}_{0}; t_{0}\in \mathbb{R}; 0< q<1) \bigr\} . $$
They also derived some properties of a qLaplace transform, which are used to solve fractional qdifference equations. Benchohra et al. [17] investigated the existence of solutions for fractionalorder functional equations by means of the Banach fixed point theorem and its nonlinear alternative of LeraySchauder type. ElSayed et al. [18] studied the stability, existence, uniqueness, and numerical solution of the fractionalorder logistic equation. The work of ElShahed [19] was concerned with the existence and nonexistence of positive solutions for some nonlinear fractional boundary value problems. Ferreira (see [20] and [21]) investigated the existence of nontrivial solutions to some nonlinear qfractional boundary value problems by applying a fixed point theorem in cones. For more information on the positive solutions (or nontrivial solutions) for a class of boundary value problems with the fractional differential equations (or qfractional differential equations), we refer the reader to such earlier works as (for example) [5, 10, 22–31], and [32].
We now review briefly some concepts of the quantum calculus.
For \(q\in(0,1)\), the qinteger \([\lambda]_{q}\) is defined by
$$ [\lambda]_{q}=\frac{1q^{\lambda}}{1q} \quad( \lambda \in \mathbb{R} ) . $$
Clearly, we have
$$ \lim_{q\rightarrow1^{}} [ \lambda ] _{q}=\lambda, $$
so we say that \([ \lambda ] _{q}\) is a qanalog of the number λ. The qanalog of the binomial formula \((ab)^{n}\) is given by
$$ (ab)^{0}=1 \quad\mbox{and}\quad (ab)^{n}=\prod _{k=0}^{n1}\bigl(abq^{k}\bigr) \quad( a,b\in \mathbb {R}; n\in\mathbb{N}_{0} ) . $$
More generally, we have
$$ (ab)^{(\delta)}=a^{\delta}\prod_{n=0}^{\infty} \biggl( \frac {abq^{n}}{abq^{\delta+n}} \biggr) \quad(\delta\in\mathbb{R}). $$
(1.1)
Clearly, if we set \(b=0\) in Eq. (1.1), it reduces immediately to
$$ a^{(\delta)}=a^{\delta} \quad(\delta\in\mathbb{R}). $$
The qgamma function is defined as follows:
$$ \Gamma_{q}(x)=\frac{(1q)^{(x1)}}{(1q)^{x1}} \quad \bigl( x\in \mathbb{R} \setminus \bigl\{ \{ 0 \} \cup\mathbb {Z}_{} \bigr\} \bigr) $$
and satisfies the formula:
$$ \Gamma_{q}(x+1)=[x]\Gamma_{q}(x). $$
The qderivative of a function \(f(x)\) is given by
$$ (D_{q}f) (x)=\frac{f(x)f(qx)}{(1q)x}\quad \mbox{and}\quad \lim_{q\rightarrow1^{}}(D_{q}f) (x)=f^{\prime}(x)=\frac{d}{dx}\bigl\{ f(x)\bigr\} . $$
For the qderivatives of higher order, we have
$$ \bigl(D_{q}^{0}f\bigr) (x)=f(x) \quad\mbox{and}\quad \bigl(D_{q}^{n}f\bigr) (x)=D_{q} \bigl(D_{q}^{n1}f\bigr) (x) \quad( n\in\mathbb {N} ). $$
Suppose now that \(0< a<b\). Then the definite qintegral is defined as follows:
$$ (I_{q}f) (x)=\int_{0}^{x}f(t) \,d_{q}t=x(1q)\sum_{n=0}^{\infty }f \bigl(xq^{n}\bigr)q^{n} \quad\bigl( x\in [0,b] \bigr) $$
and
$$ \int_{a}^{b}f(t)\,d_{q}t=\int _{0}^{b}f(t)\,d_{q}t\int _{0}^{a}f(t)\,d_{q}t. $$
The operator \(I_{q}^{n}\) can be defined by
$$ \bigl(I_{q}^{0}f\bigr) (x)=f(x) \quad\mbox{and}\quad \bigl(I_{q}^{n}f\bigr) (x)=I_{q} \bigl(I_{q}^{n1}f\bigr) (x) \quad( n\in\mathbb {N} ). $$
The Fundamental Theorem of Calculus does indeed apply mutatis mutandis to the operators \(I_{q}\) and \(D_{q}\). We thus have
$$ (D_{q}I_{q}f) (x)=f(x), $$
and if f is continuous at \(x=0\), then
$$ (I_{q}D_{q}f) (x)=f(x)f(0). $$
Denoting by \(_{x}D_{q}\) the qderivative with respect to the variable x, we now recall the following three formulas which will be used in the remainder of this paper:
$$\begin{aligned}& \bigl[ a(ts)\bigr]^{(\delta)}=a^{\delta}(ts)^{(\delta)} , \end{aligned}$$
(1.2)
$$\begin{aligned}& {_{x}}D_{q}(ts)^{(\delta)}=[\delta]_{q}(xs)^{(\delta1)} , \end{aligned}$$
(1.3)
$$\begin{aligned}& \biggl({_{x}}D_{q}\int_{0}^{x}f(x,t) \,d_{q}t \biggr) (x)=\int_{0}^{x}{_{x}}D_{q}f(x,t) \,d_{q}t+f(qx,x). \end{aligned}$$
(1.4)
Definition 1
(see [21])
Let \(\delta\geqq0\) and f be a function defined on \([0,1]\). The fractional qintegral of the RiemannLiouville type is given by
$$ \bigl(I_{q}^{0}f\bigr) (x)=f(x) $$
and
$$ \bigl(I_{q}^{\delta}f\bigr) (x)=\frac{1}{\Gamma_{q}(\delta)}\int _{0}^{x}(xqt)^{( \delta1)}f(t)\,d_{q}t\quad \bigl( \delta>0; x\in [ 0,1] \bigr). $$
Definition 2
(see [21] and [13])
The fractional qderivative of the RiemannLiouville type of order δ (\(\delta\geqq0 \)) is defined by
$$ \bigl(D_{q}^{0}f\bigr) (x)=f(x) $$
and
$$ \bigl(D_{q}^{\delta}f\bigr) (x)=\bigl(D_{q}^{m}I_{q}^{m\delta}f \bigr) (x) \quad( \delta >0 ), $$
where m is the smallest integer greater than or equal to δ.
Lemma 1
(see [21])
Let
\(\delta \geqq0\), \(\beta\geqq0\), and
f
be a function defined on
\([0,1]\). Then the following two formulas hold true:

(1)
\((I_{q}^{\beta}I_{q}^{\delta }f)(x)=(I_{q}^{\delta +\beta}f)(x)\);

(2)
\((D_{q}^{\delta}I_{q}^{\delta}f)(x)=f(x)\).
Lemma 2
(see [21] and [13])
Let
\(\delta>0\)
and
p
be a positive integer. Then the following equality holds:
$$ \bigl(I_{q}^{\delta}D_{q}^{p}f\bigr) (x)= \bigl(D_{q}^{p}I_{q}^{\delta }f\bigr) (x) \sum_{k=0}^{p1}\frac{x^{\deltap+k}}{\Gamma_{q}(\delta+kp+1)} \bigl(D_{q}^{k}f\bigr) (0). $$
Theorem 1
(see [33, 34], and [35])
(a) Let
\(( E,\leqq )\)
be a partially ordered set and suppose that there exists a metric
d
in
E
such that
\(( E,d )\)
is a complete metric space. Assume that
E
satisfies the condition that, if
\(\{ x_{n} \}\)
is a nondecreasing sequence in
E
such that
\(x_{n}\rightarrow x\), then
$$ x_{n}\leqq x \quad( n\in\mathbb{N} ). $$
Let
\(T:E\rightarrow E\)
be a nondecreasing mapping such that
$$ d ( Tx,Ty ) \leqq d ( x,y ) \psi \bigl( d ( x,y ) \bigr) \quad(x\geqq y), $$
where
$$ \psi:[ 0,+\infty ) \rightarrow [ 0,+\infty ) $$
is a continuous and nondecreasing function such that
ψ
is positive in
\(( 0,\infty )\), \(\psi ( 0 ) =0\), and
$$ \lim_{t\rightarrow\infty}\psi ( t ) =\infty. $$
If there exists
\(x_{0}\in E\)
with
\(x_{0}\leqq T ( x_{0} )\), then
T
has a fixed point.
(b) If we assume that
\(( E,\leqq )\)
satisfies the condition that, for
\(x\in E\)
and
\(y\in E\), there exists
\(z\in E\)
which is comparable to
x
and
y
and the hypothesis of (a), then it leads to the uniqueness of the fixed point.
Mena et al. [27] investigated the existence and uniqueness of positive and nondecreasing solutions for the following singular fractional boundary value problem:
$$\begin{aligned}& D_{0^{+}}^{\alpha}u(t)+f\bigl(t,u(t)\bigr)=0 \quad(0< t<1; 2<\alpha <3 ),\\& u(0)=u^{\prime} ( 1 ) =u^{\prime\prime} ( 0 ) =0. \end{aligned}$$
Miao and Liang [10], on the other hand, studied the existence and uniqueness of a positive and nondecreasing solution for the following fractional qdifference boundary value problem:
$$\begin{aligned}& D_{q}^{\gamma}\bigl(\phi_{p}\bigl(D_{q}^{\alpha}u(t) \bigr)\bigr)+f\bigl(t,u(t)\bigr)=0\quad (0< t<1; 2<\alpha<3 ),\\& u(0)=(D_{q}u) (0)=0,\qquad (D_{q}u) (1)=0, \quad\mbox{and}\quad D_{0+}^{\gamma}u(t)_{t=0}=0. \end{aligned}$$
Motivated essentially by the aforementioned work by Miao and Liang [10], we introduce and investigate here the following qdifference boundary value problem by using the pLaplacian operator:
$$\begin{aligned}& D_{q}^{\gamma}\bigl(\phi_{p}\bigl(D_{q}^{\delta}y(t) \bigr)\bigr)+f\bigl(t,y(t)\bigr)=0 \quad( 0< t<1; 3<\delta<4 ) , \end{aligned}$$
(1.5)
$$\begin{aligned}& \left \{ \begin{array}{@{}l} y(0)=(D_{q}y)(0)=(D_{q}^{2}y)(0)=0, \\ a_{1}(D_{q}y)(1)+a_{2}(D_{q}^{2}y)(1)=0, \quad\mbox{and}\quad D_{0+}^{\gamma}y(t)_{t=0}=0.\end{array} \right . \end{aligned}$$
(1.6)
We prove the existence and uniqueness of a positive and nondecreasing solution for the boundary value problem given by Eqs. (1.5) and (1.6) by means of a fixed point theorem involving partially ordered sets.