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Theory and Modern Applications

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

Abstract

In this paper,we are interested in the existence and uniqueness of positive solutions for integral boundary value problem with fractional q-derivative:

$$\begin{aligned} &D_{q}^{\alpha}u(t)+f\bigl(t,u(t),u(t)\bigr)+g\bigl(t,u(t) \bigr)=0, \quad 0< t< 1, \\ & u(0)=D_{q}u(0)=0, \qquad u(1)=\mu \int_{0}^{1}u(s)\,d_{q}s, \end{aligned}$$

where \(D_{q}^{\alpha}\) is the fractional q-derivative of Riemann–Liouville type, \(0< q<1\), \(2<\alpha\leq3 \), and μ is a parameter with \(0<\mu<[\alpha]_{q}\). By virtue of fixed point theorems for mixed monotone operators, we obtain some results on the existence and uniqueness of positive solutions.

1 Introduction

The theory that fractional differential equations arise in the fields of science and engineering such as physics, chemistry, mechanics, economics, and biological sciences, etc.; see, for example, [16]. The q-difference calculus or quantum calculus is an old subject that was put forward by Jackson [7, 8]. The essential definitions and properties of q-difference calculus can be found in [9, 10]. Early development for q-fractional calculus can be seen in the papers by Al-Salam [11] and Agarwal [12] on the existence theory of fractional q-difference. These days the fractional q-difference equation have given fire to increasing scholars’ imaginations. Some works considered the existence of positive solutions for nonlinear q-fractional boundary value problem [1332]. For example, Ferreira [13] studied the existence of positive solutions to the fractional q-difference equation

$$ \textstyle\begin{cases} D_{q}^{\alpha}u(t)+ f(t,u(t))=0, \quad 0< t< 1, 1< \alpha\leq2,\\ u(0)=u(1)=0. \end{cases} $$
(1.1)

Ferreira [14] also considered the existence of positive solutions to the nonlinear q-difference boundary value problem

$$ \textstyle\begin{cases} D_{q}^{\alpha}u(t)+ f(t,u(t))=0, \quad 0< t< 1, 1< \alpha\leq3,\\ u(0)=D_{q}u(0)=0, \qquad D_{q}u(1)=\beta\geq0. \end{cases} $$
(1.2)

EI-Shahed and AI-Askar [15] studied the existence of a positive solution to the fractional q-difference equation

$$ \textstyle\begin{cases} {}_{c} D_{q}^{\alpha}u(t)+a(t)f(t)=0, \quad 0\leq t \leq1, 2< \alpha\leq 3,\\ u(0)=D_{q}^{2}(0)=0, \qquad\gamma D_{q} u(1)+ \beta D_{q}^{2} u(1)=0, \end{cases} $$
(1.3)

where \(\gamma, \beta \leqslant0\), and \({}_{c} D_{q}^{\alpha}\) is the fractional q-derivative of Caputo type.

Darzi and Agheli [16] studied the existence of a positive solution to the fractional q-difference equation

$$ \textstyle\begin{cases} D_{q}^{\alpha}u(t)+a(t)f(t)=0, \quad 0\leq t \leq1, 3< \alpha\leq4,\\ u(0)=D_{q}u(0)=D_{q}^{2}u(0)=0, \qquad D_{q}^{2} u(1)= \beta D_{q}^{2} u(\eta), \end{cases} $$
(1.4)

where \(0 <\eta<1\) and \(1-\beta\eta^{\alpha-3}>0 \).

The methods used in the papers mentioned are mainly the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Leggett–Williams fixed point theorem, and so on. Differently from methods used in the literature mentioned, on the basis of the enlightenment of the works [17, 18, 26], we will use fixed point theorems for mixed monotone operators to demonstrate the existence and uniqueness of positive solutions for integral boundary value problems of the form

$$ \textstyle\begin{cases} D_{q}^{\alpha}u(t)+f(t,u(t),u(t))+g(t,u(t))=0, \quad 0< t< 1,\\ u(0)=D_{q}u(0)=0, \qquad u(1)=\mu\int_{0}^{1}u(s)\,d_{q}s, \end{cases} $$
(1.5)

where \(D_{q}^{\alpha}\) is the fractional q-derivative of Riemann–Liouville type, \(0< q<1\), \(2<\alpha\leq3\), \(0<\mu<[\alpha]_{q}\). 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 q-fractional 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.

2 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 \(P\subset E\) is a cone if (1) \(x\in P, r\geq0\Rightarrow r x\in P\) and (2) \(x\in P,-x\in P\Rightarrow x=\theta\) (θ is the zero element of E), where \((E, \|\cdot\| )\) is a real Banach space. For all \(x,y\in E\), if there exist \(\mu,\nu >0 \) such that \(\mu x\leq y\leq \nu x\), then we write \(x\sim y\). Obviously, is an equivalence relation. Let \(P_{h}=\{x\in E| x\sim h, h> \theta\}\).

Let \({q}\in(0,1)\). Then the q-number is given by

$$[a]_{q}= \frac{1-q^{a}}{1-q},\quad a\in R. $$

The q-analogue of the power function \((a-b)^{(n)}\) with \(n\in N_{0}\) is

$$(a-b)^{(0)}=1, \qquad(a-b)^{(n)}=\prod _{k=0} ^{n-1} \bigl(a-bq^{k}\bigr),\quad n\in N, a,b \in R. $$

More generally, if \(\alpha\in R\), then

$$(a-b)^{(\alpha)}=a^{\alpha} \prod_{k=0}^{\infty}\frac {a-bq^{k}}{a-bq^{\alpha+k}}, \quad\alpha\neq0. $$

Note that if \(b=0\), then \(a^{(\alpha)}=a^{\alpha}\). The q-gamma function is defined by

$$\Gamma_{q} (x)= \frac{(1-q)^{(x-1)}}{{1-q}^{x-1}}, \quad x \in R^{+}, $$

and satisfies \(\Gamma_{q} (x+1)=[x]_{q}\Gamma_{q} (x)\).

The q-derivative of a function f is defined by

$$(D_{q} f) (x)=\frac{f(qx)-f(x)}{(q-1)x}, \qquad(D_{q} f) (0)= \lim _{x \rightarrow0} (D_{q} f) (x), $$

and q-derivatives of higher order by

$$\bigl(D_{q}^{0} f\bigr) (x)=f(x), \qquad \bigl(D_{q}^{n} f\bigr) (x)=D_{q} \bigl(D_{q}^{n-1} f\bigr) (x), \quad n\in N. $$

The q-integral of a function f defined in the interval \([0,b]\) is given by

$$(I_{q} f) (x)= \int_{0} ^{x} f(s) \,d_{q}s = x(1-q)\sum _{k=0} ^{\infty}f\bigl(xq^{k} \bigr)q^{k},\quad x \in[0,b]. $$

If \(a \in[0,b]\) and f is defined in the interval \([0,b]\), then its integral from a to b is defined by

$$\int_{a} ^{b} f(s) \,d_{q}s= \int_{0} ^{b} f(s) \,d_{q}s - \int_{0} ^{a} f(s) \,d_{q}s. $$

Similarly to the derivatives, the operator \(I_{q}^{n}\) is given by

$$\bigl(I_{q}^{0} f\bigr) (x)=f(x),\qquad\bigl(I_{q}^{n} f\bigr) (x)=I_{q}\bigl(I_{q}^{n-1} f\bigr) (x), \quad n \in N. $$

The fundamental theorem of calculus applies to the operators \(I_{q}\) and \(D_{q}\), that is,

$$(D_{q} I_{q} f) (x)=f(x), $$

and if f is continuous at \(x=0\), then

$$(I_{q} D_{q} f) (x)=f(x)-f(0). $$

The following formulas will be used later (\({}_{t}D_{q}\) denotes the derivative with respect to variable t):

$$\begin{gathered} {}_{t}D_{q}(t-s)^{(\alpha)}=[\alpha]_{q} (t-s)^{(\alpha-1)}, \\ \biggl({}_{x} D_{q} \int_{0} ^{x} f(x,t)\,d_{q}t \biggr) (x) = \int_{0}^{x} {}_{x}D_{q} f(x,t)\,d_{q}t +f(qx,x). \end{gathered} $$

Definition 2.1

(see [4])

Let \(\alpha\geq0\), and let f be a function defined on \([0,1]\). The fractional q-integral of the Riemann–Liouville type is defined by \((I_{q}^{0} f)(x)=f(x)\) and

$$\bigl(I_{q}^{\alpha} f\bigr) (x)= \frac{1}{\Gamma_{q} (\alpha)} \int_{0} ^{x} (x-qt)^{(\alpha-1)} f(t) \,d_{q}t, \quad\alpha>0, x \in[0,1]. $$

Definition 2.2

(see [10])

The fractional q-derivative of the Riemann–Liouville type is defined by

$$\bigl(D_{q}^{0} f\bigr) (x)=f(x), \qquad \bigl(D_{q}^{\alpha} f\bigr) (x)= \bigl(D_{q}^{p} I_{q}^{p-\alpha }f\bigr) (x), \quad\alpha>0, $$

where p is the smallest integer greater than or equal to α.

Lemma 2.1

(see [10])

Let \(\alpha, \beta\geq0\), and let f be a function defined on \([0,1]\). Then the following formulas hold:

  1. (i)

    \((I_{q}^{\beta}I_{q}^{\alpha}f)(x)=(I_{q}^{\beta+\alpha} f)(x)\),

  2. (ii)

    \((D_{q}^{\alpha}I_{q}^{\alpha}f)(x)=f(x)\).

Lemma 2.2

(see [10])

Let \(\alpha>0\), and let be p be a positive integer. Then the following equality holds:

$$\bigl(I_{q}^{\alpha}D_{q}^{p} f\bigr) (x)= \bigl(D_{q}^{p} I_{q}^{\alpha}f \bigr) (x) - \sum_{k=0} ^{p-1} \frac{x^{\alpha-p+k}}{\Gamma_{q} (\alpha+k-p+1)} \bigl(D_{q}^{k} f\bigr) (0). $$

Lemma 2.3

(see [27])

Let \(2<\alpha\leq3\) and \(0<\mu<[\alpha]_{q}\). Let \(x\in C[0,1]\). Then the boundary value problem

$$\begin{aligned} &D_{q}^{\alpha}u(t)+ x(t)=0, \quad 0< t< 1, \\ \end{aligned}$$
(2.1)
$$\begin{aligned} &u(0)=D_{q}u(0)=0, \qquad u(1)=\mu \int_{0}^{1}u(s)\,d_{q}s, \end{aligned}$$
(2.2)

has a unique solution

$$u(t)= \int_{0} ^{1} G(t,qs)x(s)\,d_{q}s, $$

where

$$ G(t,s)= \textstyle\begin{cases} \frac{t^{\alpha-1}(1-s)^{(\alpha-1)} ([\alpha]_{q}-\mu+\mu q^{\alpha-1}s )- ([\alpha]_{q}-\mu )(t-s)^{\alpha -1}}{ ([\alpha]_{q}-\mu )\Gamma_{q} (\alpha)},& 0 \leq s\leq t \leq1,\\ \frac{t^{\alpha-1}(1-s)^{(\alpha-1)} ([\alpha]_{q}-\mu+\mu q^{\alpha-1}s )}{ ([\alpha]_{q}-\mu )\Gamma_{q} (\alpha)},& 0 \leq t\leq s\leq1. \end{cases} $$
(2.3)

Lemma 2.4

(see [27])

The function \(G(t,qs)\) defined by (2.3) has the following properties:

  1. (i)

    \(G(t,qs)\) is a continuous function and \(G(t,qs) \geq0\);

  2. (ii)

    \(\frac{\mu q^{\alpha}t^{\alpha-1}(1-qs)^{(\alpha-1)}s}{ ([\alpha]_{q}-\mu )\Gamma_{q} (\alpha)} \leq G(t,qs) \leq\frac {M_{0} t^{\alpha-1}}{ ([\alpha]_{q}-\mu )\Gamma_{q} (\alpha )}\), \(t, s \in[0,1]\),

where \(M_{0}=\max \{ [\alpha-1]_{q} ([\alpha]_{q}-\mu)+\mu q^{\alpha}, q^{\alpha-1} [\alpha]_{q} \}\).

Definition 2.3

(see [17])

An operator \(A: P \times P \rightarrow P\) is said to be a mixed monotone operator if \(A(x, y)\) is increasing in x and decreasing in y, that is, \(x_{i}, y_{i}\in P\) (\(i = 1, 2\)), \(x_{1} \leq x_{2}\), \(y_{1}\geq y_{2}\) imply \(A(x_{1}, y_{1})\leq A(x_{2}, y_{2})\). An element \(x\in P\) is called a fixed point of A if \(A(x, x) = x\).

Definition 2.4

(see [18])

An operator \(A: P\rightarrow P \) is said to be subhomogeneous if

$$\begin{aligned} A (tx)\geq t A(x)\quad \mbox{for any } t\in(0,1), x\in P. \end{aligned}$$
(2.4)

Definition 2.5

(see [18])

Let \(D = P \), and let γ be a real number with \(0\leq\gamma< 1 \). An operator \(A:D\rightarrow D\) is said to be γ-concave if it satisfies

$$\begin{aligned} A(tx)\geq t^{\gamma}A(x)\quad \mbox{for any } t\in(0,1),x\in D. \end{aligned}$$
(2.5)

Lemma 2.5

(see [17])

Let \(h > \theta\) and \(\gamma\in(0,1)\).

Let \(A: P \times P \rightarrow P\) be a mixed monotone operator satisfying

$$\begin{aligned} A\bigl(tx,t^{-1}y\bigr)\geq t^{\gamma}A(x,y) \quad \textit{for any } t \in(0,1), x,y \in P, \end{aligned}$$
(2.6)

and let \(B:P \rightarrow P \) be an increasing subhomogeneous operator. Assume that

  1. (i)

    there is \(h_{0} \in P_{h} \) such that \(A(h_{0},h_{0})\in P_{h}\) and \(B h_{0} \in P_{h} \);

  2. (ii)

    there exists a constant \(\delta_{0}\) such that \(A(x,y )\geq\delta_{0} B x\) for any \(x,y \in P \).

Then:

  1. (1)

    \(A: P_{h} \times P_{h} \rightarrow P_{h} \) and \(B:P_{h} \rightarrow P_{h}\);

  2. (2)

    there exist \(u_{0},v_{0} \in P_{h}\) and \(r \in(0,1)\) such that

    $$r v_{0} \leq u_{0} < v_{0}, \qquad u_{0} \leq A(u_{0},v_{0})+ B u_{0} \leq A ( v_{0},u_{0})+ B v_{0}\leq v_{0}; $$
  3. (3)

    the operator equation \(A(x,x )+ B x =x \) has a unique solution \(x^{*} \in P_{h}\);

  4. (4)

    for any initial values \(x_{0},y_{0} \in P_{h}\), constructing successively the sequences

    $$x_{n} = A(x_{n-1},y_{n-1})+ Bx_{n-1}, \qquad y_{n} = A(y_{n-1},x_{n-1})+ B y_{n-1}, \quad n=1,2,\ldots, $$

    we have \(x_{n}\rightarrow x^{*}\) and \(y_{n}\rightarrow x^{*} \) as \(n \rightarrow\infty\).

Remark 2.1

When \(B= \theta\) in Lemma 2.5, then the corresponding conclusion still holds.

Lemma 2.6

(see [18])

Let \(h > \theta\) and \(\gamma\in(0,1)\).

Let \(A:P \times P \rightarrow P\) be a mixed monotone operator satisfying

$$\begin{aligned} A\bigl(tx,t^{-1}y\bigr)\geq t A(x,y), \quad \textit{for any } t \in(0,1), x, y \in P, \end{aligned}$$
(2.7)

and let \(B:P \rightarrow P \) be an increasing γ-concave operator. Assume that

  1. (i)

    there is \(h_{0} \in P_{h} \) such that \(A(h_{0},h_{0})\in P_{h}\) and \(B h_{0} \in P_{h} \);

  2. (ii)

    there exists a constant \(\delta_{0}\) such that \(A(x,y )\leq\delta_{0} B x\) for any \(x,y \in P \).

Then:

  1. (1)

    \(A: P_{h} \times P_{h} \rightarrow P_{h} \) and \(B:P_{h} \rightarrow P_{h}\);

  2. (2)

    there exist \(u_{0},v_{0} \in P_{h}\) and \(r \in(0,1)\) such that

    $$r v_{0} \leq u_{0} < v_{0}, \qquad u_{0} \leq A(u_{0},v_{0})+ B u_{0} \leq A( v_{0},u_{0})+ B v_{0}\leq v_{0}; $$
  3. (3)

    the operator equation \(A (x,x )+ B x =x \) has a unique solution \(x^{*} \in P_{h}\);

  4. (4)

    for any initial values \(x_{0},y_{0} \in P_{h}\), constructing successively the sequences

    $$x_{n} = A (x_{n-1},y_{n-1})+ Bx_{n-1}, \qquad y_{n} =A(y_{n-1},x_{n-1})+B y_{n-1}, \quad n=1,2,\ldots, $$

    we have \(x_{n}\rightarrow x^{*}\) and \(y_{n}\rightarrow x^{*} \) as \(n \rightarrow\infty\).

Remark 2.2

When \(A= \theta\) in Lemma 2.6, then the corresponding conclusion still holds.

3 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 \(\|x\|=\sup\{|x(t)|:{t\in[0,1]}\}\). Clearly, this space can be equipped with a partial order given by

$$x,y \in C[0,1], \quad x\leq y \quad \Leftrightarrow \quad x(t)\leq y(t) \quad \mbox{for }t \in[0,1]. $$

We define the cone \(P=\{x\in X:x(t) \geq 0,t\in[0,1]\}\). Notice that P is a normal cone in \(C[0,1]\) and the normality constant is 1.

Theorem 3.1

Suppose that

\((F_{1})\) :

a function \(f(t,x,y):[0,1]\times[0, +\infty)\times[0,+\infty)\rightarrow[0,+\infty)\) is continuous, increasing with respect to the second variable, and decreasing with respect to the third variable;

\((F_{2})\) :

a function \(g(t,x): [0,1]\times[0, +\infty)\rightarrow [0,+\infty) \) is continuous and increasing with respect to the second variable;

\((F_{3})\) :

there exists a constant \(\gamma\in(0,1)\) such that \(f(t,\lambda x,\lambda^{-1} y)\geq\lambda^{\gamma}f(t,x,y)\) for any \(t \in[0,1]\), \(\lambda\in(0,1)\), \(x, y \in [0,+\infty)\), and \(g(t,\lambda x)\geq\lambda g(t,x)\) for \(\lambda\in(0,1)\), \(t\in[0,1]\), \(u\in[0, +\infty)\), and \(g(t,0)\not\equiv0 \);

\((F_{4})\) :

there exists a constant \(\delta_{0} > 0\) such that \(f(t,x,y)\geq\delta_{0} g(t,x)\), \(t\in[0,1]\), \(x,y\geq0\).

Then:

  1. (1)

    there exist \(x_{0},y_{0}\in P_{h}\) and \(r\in(0,1)\) such that \(r y_{0}\leq x_{0} < y_{0}\) and

    $$\begin{gathered} x_{0}\leq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,x_{0}(s),y_{0}(s) \bigr)+g\bigl(s,x_{0}(s)\bigr)\bigr]\,d_{q}s, \quad t\in[0,1], \\ y_{0}\geq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,y_{0}(s),y_{0}(s) \bigr)+g\bigl(s,y_{0}(s)\bigr)\bigr]\,d_{q}s, \quad t\in[0,1], \end{gathered} $$

    where \(G(t,qs)\) is defined by (2.3), and \(h(t)= t^{\alpha-1}\), \(t \in[0,1]\);

  2. (2)

    the boundary value problem (1.5) has a unique positive solution \(u^{*} \) in \(P_{h}\), and for any \(x_{0},y_{0} \in P_{h}\), constructing successively the sequences

    $$\begin{gathered} x_{n+1}= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,x_{n}(s),y_{n}(s) \bigr)+g\bigl(s,x_{n}(s)\bigr)\bigr]\,d_{q}s,\quad n=0,1,2,\ldots, \\ y_{n+1}= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,y_{n}(s),x_{n}(s) \bigr)+g\bigl(s,y_{n}(s)\bigr)\bigr]\,d_{q}s,\quad n=0,1,2,\ldots, \end{gathered} $$

    we have \(\|x_{n}-u^{*}\|\rightarrow0\) and \(\|y_{n}-u^{*}\| \rightarrow0\) as \(n \rightarrow\infty\).

Proof

We note that if u is a solution of boundary value problem (1.5), then

$$\begin{aligned} u(t)= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,u(s),u(s)\bigr)+g \bigl(s,u(s)\bigr)\bigr]\,d_{q}s, \quad0 \leq t \leq1. \end{aligned}$$
(3.1)

Define two operators \(T_{1}:P\times P \rightarrow E\) and \(T_{2}: P \rightarrow E \) by

$$\begin{aligned} \begin{aligned} &T_{1}(u,v) (t)= \int_{0}^{1} G(t,qs)f\bigl(s,u(s),v(s) \bigr)\,d_{q}s, \\ &(T_{2}u) (t)= \int _{0}^{1} G(t,qs)g\bigl(s,u(s) \bigr)\,d_{q}s. \end{aligned} \end{aligned}$$
(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 \(T_{1}: P\times P \rightarrow P\) and \(T_{2}: P \rightarrow P\). Next, we want to prove that \(T_{1}\) and \(T_{2}\) satisfy the conditions of Lemma 2.5.

To begin with, we prove that \(T_{1}\) is a mixed monotone operator. In fact, for \(u_{1},u_{2},v_{1},v_{2} \in P\) with \(u_{1}\geq u_{2}\) and \(v_{1}\leq v_{2}\), it is easy to see that \(u_{1}(t) \geq u_{2}(t)\), \(v_{1}(t)\leq v_{2}(t)\), \(t \in[0,1]\), and by Lemma 2.4 and \((F_{1})\),

$$\begin{aligned} T_{1}(u_{1},v_{1}) (t)&= \int_{0}^{1} G(t,qs)f\bigl(s,u_{1}(s),v_{1}(s) \bigr)\,d_{q}s \\ &\geq \int_{0}^{1} G(t,qs)f\bigl(s,u_{2}(s),v_{2}(s) \bigr)\,d_{q}s =T_{1}(u_{2},v_{2}) (t). \end{aligned}$$
(3.3)

For any \(\lambda\in(0,1)\) and \(u,v \in P \), by \((F_{3}) \) we have

$$ \begin{aligned}[b] T_{1}\bigl(\lambda u, \lambda^{-1}v\bigr) (t)&= \int_{0}^{1}G(t,qs)f\bigl(s,\lambda u(s), \lambda^{-1}v(s)\bigr)\,d_{q}s \\ &\geq\lambda^{\gamma}\int_{0}^{1}G(t,qs)f\bigl(s, u(s),v(s) \bigr)\,d_{q}s \geq\lambda^{\gamma}T_{1}(u,v) (t). \end{aligned} $$
(3.4)

So, the operator \(T_{1}\) satisfies (2.6).

For any \(u_{1}(t)\geq u_{2}(t)\), \(t \in[0,1]\), from \(G(t,qs) \geq0\) and \((F_{2})\) we know that

$$T_{2}u_{1}(t)= \int_{0}^{1} G(t,qs)g\bigl(s,u_{1}(s) \bigr)\,d_{q}s \geq \int_{0}^{1} G(t,qs)g\bigl(s,u_{2}(s) \bigr)\,d_{q}s=T_{2}u_{2}(t). $$

So \(T_{2}\) is increasing. Further, for any \(\lambda\in(0,1)\) and \(u \in P\), from hypothesis \((F_{3})\) we get

$$\begin{aligned} T_{2}(\lambda u) (t)= \int_{0}^{1}G(t,qs)g\bigl(s,\lambda u(s) \bigr)\,d_{q}s\geq\lambda \int_{0}^{1}G(t,qs)g\bigl(s,u(s)\bigr)\,d_{q}s =\lambda T_{2}u(t), \end{aligned}$$
(3.5)

that is, the operator \(T_{2}\) is subhomogeneous. By \((F_{1})\) and Lemma 2.4, for any \(t \in[0,1]\), we have

$$ \begin{aligned}[b] T_{1}(h,h) (t)&= \int_{0}^{1}G(t,qs)f\bigl(s,h(s),h(s) \bigr)\,d_{q}s \\ &= \int_{0}^{1}G(t,qs)f\bigl(s, s^{\alpha-1},s^{\alpha-1} \bigr)\,d_{q}s \\ &\leq\frac{M_{0}}{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )}h(t) \int_{0}^{1} f(s,1,0)\,d_{q}s \end{aligned} $$
(3.6)

and

$$ \begin{aligned}[b] T_{1}(h,h) (t)&= \int_{0}^{1}G(t,qs)f\bigl(s,h(s),h(s) \bigr)\,d_{q}s \\ &= \int_{0}^{1}G(t,qs)f\bigl(s, s^{\alpha-1},s^{\alpha-1} \bigr)\,d_{q}s \\ &\geq\frac{\mu q^{\alpha}}{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )}h(t) \int_{0}^{1}s (1-qs)^{(\alpha-1)}f(s,0,1)\,d_{q}s. \end{aligned} $$
(3.7)

From \((F_{2})\) and \((F_{4})\) we have the inequality

$$f(s,1,0)\geq f(s,0,1)\geq\delta_{0} g(s,0)\geq0. $$

Since \(g(t,0)\not\equiv0 \), we also obtain

$$ \int_{0}^{1} f(s,1,0)\,d_{q}s\geq \int_{0}^{1} f(s,0,1)\,d_{q}s \geq \delta_{0} \int _{0}^{1} g(s,0)\,d_{q}s > 0. $$
(3.8)

Let

$$\begin{aligned} &M_{1}=\frac{M_{0}}{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )} \int_{0}^{1} f(s,1,0)\,d_{q}s, \\ &M_{2}=\frac{\mu q^{\alpha}}{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )} \int_{0}^{1}s (1-qs)^{(\alpha-1)}f(s,0,1)\,d_{q}s, \\ &M_{3}=\frac{\mu q^{\alpha}}{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )} \int_{0}^{1}s (1-qs)^{(\alpha-1)}g(s,0)\,d_{q}s, \\ &M_{4}=\frac{M_{0} }{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )} \int_{0}^{1} g(s,1)\,d_{q}s. \end{aligned}$$

Thus we have \(M_{2}h(t)\leq T_{1}(h,h) \leq M_{1}h(t)\), \(M_{3}h(t)\leq T_{2}h\leq M_{4}h(t)\), \(t\in[0,1]\). So, \(T_{1}(h,h)\in P_{h}\). From \(g(t,0)\not\equiv0 \) it is easy to see that \(T_{2}h\in P_{h}\). So, there is \(h(t)=t^{\alpha-1}\in P_{h} \) such that \(T_{1}(h,h)\in P_{h}\) and \(T_{2}h \in P_{h} \).

Next, we prove that the operators \(T_{1}\) and \(T_{2}\) satisfy condition (ii) of Lemma 2.5. In fact, for \(u,v \in P \) and any \(t \in[0,1]\), by \((F_{4}) \) we have

$$ \begin{aligned}[b] T_{1}(u,v) (t)&= \int_{0}^{1}G(t,qs)f\bigl(s,u(s),v(s) \bigr)\,d_{q}s\\ &\geq\delta_{0} \int _{0}^{1}G(t,qs)g\bigl(s,u(s) \bigr)\,d_{q}s=\delta_{0} (T_{2}u) (t). \end{aligned} $$
(3.9)

Then we have \(T_{1}(u,v)\geq\delta_{0} T_{2} u\) for \(u,v\in P\). By Lemma 2.5 we can deduce: there exist \(u_{0},v_{0}\in P_{h}\) and \(r \in (0,1)\) such that \(rv_{0}\leq u_{0}\leq v_{0}\), \(u_{0}\leq T_{1}(u_{0},v_{0})+ T_{2} u_{0}\leq T_{1}(v_{0},u_{0})+T_{2}v_{0}\leq v_{0}\); the operator equation \(T_{1}(u,u)+T_{2}u=u\) has a unique solution \(u^{*} \in P_{h}\); and for any initial values \(x_{0}, y_{0} \in P_{h}\), constructing successively the sequences

$$x_{n} = T_{1} (x_{n-1},y_{n-1})+T_{2}x_{n-1}, \qquad y_{n} =T_{1}(y_{n-1},x_{n-1})+T_{2} y_{n-1},\quad n=1,2,\ldots, $$

we get \(x_{n}\rightarrow u^{*}\) and \(y_{n}\rightarrow u^{*} \) as \(n \rightarrow\infty\). We have the following two inequalities:

$$\begin{gathered} u_{0}(t)\leq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,u_{0}(s),v_{0}(s) \bigr)+g\bigl(s,u_{0}(s)\bigr)\bigr]\,d_{q}s, \quad t\in[0,1], \\ v_{0}(t)\geq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,v_{0}(s),u_{0}(s) \bigr)+g\bigl(s,v_{0}(s)\bigr)\bigr]\,d_{q}s, \quad t\in[0,1]. \end{gathered} $$

Thus problem (1.5) has a unique positive solution \(u^{*} \in P_{h}\); for any \(u_{0},v_{0} \in P_{h}\), constructing successively the sequences

$$\begin{gathered} x_{n+1}(t)= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,x_{n}(s),y_{n}(s) \bigr)+g\bigl(s,x_{n}(s)\bigr)\bigr]\,d_{q}s,\quad n=0,1,2, \ldots, \\ y_{n+1}(t)= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,y_{n}(s),x_{n}(s) \bigr)+g\bigl(s,y_{n}(s)\bigr)\bigr]\,d_{q}s,\quad n=0,1,2, \ldots, \end{gathered} $$

we have \(\|x_{n}-u^{*}\|\rightarrow0\) and \(\|y_{n}-u^{*}\|\rightarrow0 \) as \(n\rightarrow\infty\). □

Corollary 3.1

Suppose that f satisfies the conditions of Theorem 3.1 and \(g\equiv 0\), \(f(t,0,1)\not\equiv0\). Then:

  1. (i)

    there exist \(u_{0},v_{0} \in P_{h} \) and \(r\in(0,1)\) such that \(r v_{0} \leq u_{0} < v_{0}\), and

    $$\begin{gathered} u_{0}(t)\leq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,u_{0}(s),v_{0}(s) \bigr)\bigr]\,d_{q}s, \quad t\in [0,1], \\ v_{0}(t)\geq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,v_{0}(s),u_{0}(s) \bigr)\bigr]\,d_{q}s, \quad t\in[0,1], \end{gathered} $$

    where \(G(t,qs)\) is defined by (2.3), and \(h(t)= t^{\alpha-1}\), \(t \in[0,1]\);

  2. (ii)

    the BVP

    $$ \textstyle\begin{cases} D_{q}^{\alpha}u(t)+ f(t,u(t),u(t))=0, \quad 0< t< 1, 2< \alpha\leq3,\\ u(0)=D_{q}u(0)=0, \qquad u(1)=\mu\int_{0}^{1}u(s)\,d_{q}s, \end{cases} $$
    (3.10)

    has a unique positive solution \(u^{*} \) in \(P_{h}\);

  3. (iii)

    for any \(x_{0},y_{0} \in P_{h}\), the sequences

    $$\begin{gathered} x_{n+1}= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,x_{n}(s),y_{n}(s) \bigr)\bigr]\,d_{q}s, \quad n=0,1,2,\ldots, \\ y_{n+1}= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,y_{n}(s),x_{n}(s) \bigr)\bigr]\,d_{q}s, \quad n=0,1,2,\ldots, \end{gathered} $$

    satisfy \(\|x_{n}-u^{*}\|\rightarrow0\) and \(\|y_{n}-u^{*}\|\rightarrow0\) as \(n \rightarrow\infty\).

Theorem 3.2

Suppose that \((F_{1})\)\((F_{2}) \) hold. In addition, suppose that f, g satisfy the following conditions:

\((F_{5})\) :

there exists a constant \(\gamma\in(0,1)\) such that \(g(t,\lambda u)\geq\lambda^{\gamma}g(t,u)\) for any \(t \in[0,1] \), \(\lambda\in(0,1)\), \(u \in[0, +\infty)\), and \(f(t,\lambda u,\lambda^{-1} v)\geq\lambda f(t,u,v)\) for \(\lambda\in (0,1)\), \(t\in[0,1]\), \(u,v\in[0,+\infty)\);

\((F_{6})\) :

\(f(t,0,1)\not\equiv0\) for \(t \in[0,1]\), and there exists a constant \(\delta_{0} > 0\) such that \(f(t,u,v)\leq\delta_{0} g(t,u)\), \(t\in[0,1]\), \(u,v \geq0\).

Then:

  1. (1)

    there exist \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that \(r v_{0}\leq u_{0} < v_{0}\) and

    $$\begin{gathered} u_{0}\leq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,u_{0}(s),v_{0}(s) \bigr)+g\bigl(s,u_{0}(s)\bigr)\bigr]\,d_{q}s, \quad t\in[0,1], \\ v_{0}\geq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,v_{0}(s),u_{0}(s) \bigr)+g\bigl(s,v_{0}(s)\bigr)\bigr]\,d_{q}s, \quad t\in[0,1], \end{gathered} $$

    where \(G(t,qs)\) is defined by (2.3), and \(h(t)= t^{\alpha-1}\), \(t \in[0,1]\);

  2. (2)

    the boundary value problem (1.5) has a unique positive solution \(u^{*} \) in \(P_{h}\); and for any \(x_{0},y_{0} \in P_{h}\), the sequences

    $$\begin{gathered} x_{n+1}= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,x_{n}(s),y_{n}(s) \bigr)+g\bigl(s,x_{n}(s)\bigr)\bigr]\,d_{q}s,\quad n=0,1,2, \ldots, \\ y_{n+1}= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,y_{n}(s),x_{n}(s) \bigr)+g\bigl(s,y_{n}(s)\bigr)\bigr]\,d_{q}s,\quad n=0,1,2, \ldots, \end{gathered} $$

    satisfy \(\|x_{n}-u^{*}\|\rightarrow0\) and \(\|y_{n}-u^{*}\|\rightarrow0\) as \(n \rightarrow\infty\).

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 \(T_{1}: P\times P \rightarrow P \) is a mixed monotone operator and \(T_{2}: P\rightarrow P\) is increasing. By \((F_{5})\) we obtain

$$T_{1}\bigl(\lambda u,\lambda^{-1}v\bigr)\geq\lambda T_{1}(u,v), \qquad T_{2}(\lambda u)\geq\lambda^{\gamma}T_{2}u, \quad \mbox{for }\lambda\in(0,1), u,v \in P. $$

According to \((F_{2})\) and \((F_{6})\), we have

$$f(s,0,1)\leq\delta_{0} g(s,0), \qquad f(s,0,1)\leq f(s,1,0),\quad s \in[0,1]. $$

From \(f(t,0,1) \not\equiv0\) we get

$$\begin{aligned} &0< \int_{0}^{1}f(s,0,1)\,d_{q}s\leq \int_{0}^{1}f(s,1,0)\,d_{q}s, \\ &0< \frac{1}{\delta_{0}} \int_{0}^{1}f(s,0,1)\,d_{q}s\leq \int_{0}^{1}g(s,0)\,d_{q}s\leq \int_{0}^{1} g(s,1)\,d_{q}s, \end{aligned} $$

and the following inequalities hold:

$$\begin{aligned} & \begin{aligned}[b] 0&< \frac{\mu q^{\alpha}}{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )} \int_{0}^{1}s (1-qs)^{(\alpha-1)}f(s,0,1)\,d_{q}s \\ &\leq\frac{M_{0} }{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )} \int_{0}^{1} f(s,1,0)\,d_{q}s, \end{aligned} \end{aligned}$$
(3.11)
$$\begin{aligned} &\begin{aligned}[b] 0&< \frac{\mu q^{\alpha}}{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )} \int_{0}^{1}s (1-qs)^{(\alpha-1)}g(s,0)\,d_{q}s \\ &\leq\frac{M_{0} }{\Gamma_{q} (\alpha) ([\alpha]_{q}-\mu )} \int_{0}^{1} g(s,1)\,d_{q}s. \end{aligned} \end{aligned}$$
(3.12)

Hence we can easily check that \(T_{1}(h,h) \in P\), \(T_{2}h \in P\), \(t\in [0,1]\), and, by using \((F_{6})\), we have

$$\begin{aligned} T_{1}(u,v) (t)&= \int_{0}^{1} G(t,s)f\bigl(s,u(s),v(s) \bigr)\,d_{q}s \\ &\leq\delta_{0} \int_{0}^{1} G(t,s)g\bigl(s,u(s)\bigr)\,d_{q}s= \delta_{0} T_{2}u(t). \end{aligned}$$
(3.13)

Then we have \(T_{1}(u,v)\leq\delta_{0} T_{2}u\) for \(u,v\in P\). Thus, from Lemma 2.6 we get that there exist \(u_{0},v_{0}\in P_{h}\) and \(r \in(0,1)\) such that \(rv_{0}\leq u_{0}\leq v_{0},u_{0}\leq T_{1}(u_{0},v_{0})+ T_{2} u_{0}\leq T_{1}(v_{0},u_{0})+T_{2}v_{0}\leq v_{0}\); the operator equation \(T_{1}(u,u)+T_{2}u=u\) has a unique solution \(u^{*} \in P_{h}\); and for any initial values \(x_{0}\), \(y_{0} \in P_{h}\), the sequences

$$x_{n} = T_{1} (x_{n-1},y_{n-1})+T_{2}x_{n-1}, \qquad y_{n} =T_{1}(y_{n-1},x_{n-1})+T_{2} y_{n-1},\quad n=1,2,\ldots, $$

satisfy \(x_{n}\rightarrow u^{*}\) and \(y_{n}\rightarrow u^{*} \) as \(n \rightarrow\infty\). That is,

$$\begin{gathered} u_{0}(t)\leq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,u_{0}(s),v_{0}(s) \bigr)+g\bigl(s,u_{0}(s)\bigr)\bigr]\,d_{q}s, \quad t\in[0,1], \\ v_{0}(t)\geq \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,v_{0}(s),u_{0}(s) \bigr)+g\bigl(s,v_{0}(s)\bigr)\bigr]\,d_{q}s, \quad t\in[0,1], \end{gathered} $$

The boundary value problem (1.5) has a unique positive solution \(u^{*} \in P_{h}\); for \(u_{0},v_{0} \in P_{h}\), the sequences

$$\begin{gathered} x_{n+1}(t)= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,x_{n}(s),y_{n}(s) \bigr)+g\bigl(s,x_{n}(s)\bigr)\bigr]\,d_{q}s,\quad n=0,1,2, \ldots, \\ y_{n+1}(t)= \int_{0}^{1} G(t,qs)\bigl[f\bigl(s,y_{n}(s),x_{n}(s) \bigr)+g\bigl(s,y_{n}(s)\bigr)\bigr]\,d_{q}s,\quad n=0,1,2, \ldots, \end{gathered} $$

satisfy \(\|x_{n}-u^{*}\|\rightarrow0\) and \(\|y_{n}-u^{*}\|\rightarrow0 \) as \(n\rightarrow\infty\). □

Corollary 3.2

Suppose that g satisfies the conditions of Theorem 3.2, \(f\equiv0\), and \(g(t,0)\not\equiv0\) for \(t \in[0,1]\). Then:

  1. (i)

    there exist \(u_{0},v_{0} \in P_{h} \) and \(r\in(0,1)\) such that \(r v_{0} \leq u_{0} < v_{0}\), and

    $$\begin{gathered} u_{0}(t)\leq \int_{0}^{1} G(t,qs)\bigl[g\bigl(s,u_{0}(s) \bigr)\bigr]\,d_{q}s, \\ v_{0}(t)\geq \int_{0}^{1} G(t,qs)\bigl[g\bigl(s,v_{0}(s) \bigr)\bigr]\,d_{q}s, \quad t\in[0,1], \end{gathered} $$

    where \(G(t,qs)\) is defined by (2.3), and \(h(t)= t^{\alpha-1}\), \(t \in[0,1]\);

  2. (ii)

    the BVP

    $$ \textstyle\begin{cases} D_{q}^{\alpha}u(t)+ g(t,u(t))=0, \quad 0< t< 1, 2< \alpha\leq3,\\ u(0)=D_{q}u(0)=0, \qquad u(1)=\mu\int_{0}^{1}u(s)\,d_{q}s, \end{cases} $$
    (3.14)

    has a unique positive solution \(u^{*} \) in \(P_{h}\); and for any \(x_{0},y_{0} \in P_{h}\), the sequences

    $$\begin{gathered} x_{n+1}= \int_{0}^{1} G(t,qs)g\bigl(s,x_{n}(s) \bigr)\,d_{q}s,\quad n=0,1,2,\ldots, \\ y_{n+1}= \int_{0}^{1} G(t,qs)g\bigl(s,y_{n}(s) \bigr)\,d_{q}s,\quad n=0,1,2,\ldots, \end{gathered} $$

    satisfy \(\|x_{n}-u^{*}\|\rightarrow0\) and \(\| y_{n}-u^{*}\|\rightarrow0\) as \(n \rightarrow\infty\).

4 Example

Now, we give two examples to illustrate our results.

Example 4.1

Consider the following boundary value problem:

$$ \textstyle\begin{cases} -D_{\frac{1}{2}}^{\frac{5}{2}} u(t)=u(t)^{\frac{1}{3}}+[u(t)+1]^{- \frac{1}{2}}+{\frac{u(t)}{1+u(t)}}t^{3}+t^{2}+4, \quad 0< t< 1,\\ u(0)=D_{\frac{1}{2}}u(0)=0, \qquad u(1)=\mu\int_{0}^{1}u(s)\,d_{\frac{1}{2}}s. \end{cases} $$
(4.1)

In this example, we let

$$\begin{aligned} &f(t,u,v )=u^{\frac{1}{3}}+[v+1]^{- \frac{1}{2}}+t^{2}+2, \qquad g(t,u)={\frac{u}{1+u}}t^{3}+2, \\ &\gamma={\frac{1}{2}}, \qquad\mu=\frac{1}{2}. \end{aligned}$$

It is not difficult to find that \(f(t,x,y):[0,1]\times[0, +\infty )\times[0,+\infty)\rightarrow[0,+\infty)\) is continuous, increasing with respect to the second variable, and decreasing with respect to the third variable and that \(g(t,x): [0,1]\times[0, +\infty)\rightarrow[0,+\infty) \) is continuous with \(g(t,0)=2>0\) and increasing with respect to the second variable. We also have

$$\begin{aligned}& g(t,\lambda u )={\frac{{\lambda}u}{1+\lambda{u}}}t^{3}+2 \geq{ \frac {{\lambda}u}{1+ {u}}}t^{3}+2 \lambda=\lambda g(t,u), \quad \lambda\in(0,1), \\& \begin{aligned} f\bigl(t,\lambda u,\lambda^{-1}v \bigr)&=\lambda^{\frac{1}{3}}u^{\frac {1}{3}}+ \lambda^{\frac{1}{2}}[v+\lambda]^{-\frac{1}{2}}+t^{2}+2 \\ &\geq\lambda^{\frac{1}{2}} \bigl\{ u^{\frac{1}{3}}+[v+1]^{-\frac {1}{2}}+t^{2}+2 \bigr\} \\ &=\lambda^{\gamma}f(t,u,v). \end{aligned} \end{aligned}$$

Further, if we take \(\delta_{0} \in(0,\frac{2}{3}]\), then we easily get

$$\begin{aligned} f(t,u,v )&=u^{\frac{1}{3}}+[v+1]^{- \frac{1}{2}}+t^{2}+2\geq2= \frac {2}{3} \cdot3 \\ &\geq\delta_{0}\biggl[{\frac{u}{1+u}}t^{3}+2\biggr]= \delta_{0} g(t,u). \end{aligned}$$

So f and g satisfy the conditions of Theorem 3.1. Thus by Theorem 3.1 the boundary value problem (4.1) has a unique positive solution in \(P_{h}\), where \(h(t)=t^{\alpha-1}=t^{\frac{3}{2}}\), \(t\in[0,1]\).

Example 4.2

Consider the following boundary value problem:

$$ \textstyle\begin{cases} -D_{\frac{1}{2}}^{\frac{5}{2}} u(t)={({\frac{u(t)}{1+u(t)}})}^{\frac {1}{4}}+[u(t)+1]^{- \frac{1}{3}}+t^{3}+u(t)^{\frac{1}{3}}+t^{2}+1, \quad 0< t< 1,\\ u(0)=D_{\frac{1}{2}}u(0)=0, \qquad u(1)=\mu\int_{0}^{1}u(s)\,d_{\frac{1}{2}}s. \end{cases} $$
(4.2)

We let

$$f(t,u,v )={\biggl({\frac{u}{1+u}}\biggr)}^{\frac{1}{4}}+[v+1]^{- \frac {1}{3}}+t^{3}, \qquad g(t,u)=u^{\frac{1}{3}}+t^{2}+1, \qquad \gamma={ \frac {1}{3}},\qquad \mu=\frac{1}{2}. $$

It is not difficult to find that \(f(t,x,y):[0,1]\times[0, +\infty )\times[0,+\infty)\rightarrow[0,+\infty)\) is continuous, increasing with respect to the second variable, and decreasing with respect to the third variable and that \(g(t,x): [0,1]\times[0, +\infty)\rightarrow[0,+\infty)\) and increasing with respect to the second variable. We also have

$$\begin{aligned} &g(t,\lambda u )=\lambda^{\frac{1}{3}}u^{\frac{1}{3}}+t^{2}+1 \geq \lambda^{\frac{1}{3}}\bigl[u^{\frac{1}{3}}+t^{2}+1\bigr] = \lambda^{\gamma}g(t,u), \quad \lambda\in(0,1), \\ & \begin{aligned} f\bigl(t,\lambda u,\lambda^{-1}v \bigr)&={\biggl({ \frac{\lambda u}{1+\lambda u}}\biggr)}^{\frac{1}{4}}+\bigl[\lambda^{-1}v+1 \bigr]^{- \frac{1}{3}}+t^{3} \\ &\geq\lambda^{\frac{1}{3}} \biggl\{ {\biggl({\frac{ u}{1+ u}} \biggr)}^{\frac {1}{4}}+[v+\lambda]^{- \frac{1}{3}}+t^{3} \biggr\} \\ &\geq\lambda \biggl\{ {\biggl({\frac{ u}{1+ u}}\biggr)}^{\frac{1}{4}}+[v+1]^{- \frac{1}{3}}+t^{3} \biggr\} \\ &=\lambda f(t,u,v). \end{aligned} \end{aligned}$$

If we take \(\delta_{0} =1>0\), then we have

$$f(t,u,v)={\biggl({\frac{u}{1+u}}\biggr)}^{\frac{1}{4}}+[v+1]^{- \frac {1}{3}}++t^{3} \leq u^{\frac{1}{4}}+t^{2}+1\leq u^{\frac{1}{3}}+t^{2}+1= \delta_{0} g(u,t). $$

So f and g satisfy the conditions of Theorem 3.2. Thus by Theorem 3.2 the boundary value problem (4.2) has a unique positive solution in \(P_{h}\), where \(h(t)=t^{\alpha-1}=t^{\frac{3}{2}}\), \(t\in[0,1]\).

References

  1. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  2. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York (1993)

    MATH  Google Scholar 

  3. Glockle, W.G., Nonnenmacher, T.F.: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46–53 (1995)

    Article  Google Scholar 

  4. Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999)

    MATH  Google Scholar 

  5. Field, C., Joshi, N., Nijhoff, F.: q-Difference equations of KdV type and Chazy-type second-degree difference equations. J. Phys. A, Math. Theor. 41, 1–13 (2008)

    Article  MathSciNet  Google Scholar 

  6. Abreu, L.: Sampling theory associated with q-difference equations of the Sturm–Liouville type. J. Phys. A 38(48), 10311–10319 (2005)

    Article  MathSciNet  Google Scholar 

  7. Jackson, F.: On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253–281 (1908)

    Article  Google Scholar 

  8. Jackson, F.: On q-definite integrals. Pure Appl. Math. Q. 41, 193–203 (1910)

    MATH  Google Scholar 

  9. Rajković, P., Marinković, S., Stanković, M.: Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1(1), 311–323 (2007)

    Article  MathSciNet  Google Scholar 

  10. Annaby, M.H., Mansour, Z.S.: q-Fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056. Springer, Berlin (2012)

    MATH  Google Scholar 

  11. Al-Salam, W.A.: Some fractional q-integrals and q-derivatives. Proc. Edinb. Math. Soc. 15, 135–140 (1966)

    Article  MathSciNet  Google Scholar 

  12. Agarwal, R.P.: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)

    Article  MathSciNet  Google Scholar 

  13. Ferreira, R.A.C.: Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ., 2010, 70 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Ferreira, R.A.C.: Positive solutions for a class of boundary value problems with fractional q-differences. Comput. Math. Appl. 61(2), 367–373 (2011)

    Article  MathSciNet  Google Scholar 

  15. EI-Shahed, M., Al-Askar, F.: Positive solution for boundary value problem of nonlinear fractional q-difference equation. ISRN Math. Anal. 2011, Article ID 385459 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Darzi, R., Agheli, B.: Existence results to positive solution of fractional BVP with q-derivatives. J. Appl. Math. Comput. 55, 353–367 (2017)

    Article  MathSciNet  Google Scholar 

  17. Zhai, C.B., Hao, M.R.: Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems. Nonlinear Anal. 75, 2542–2551 (2012)

    Article  MathSciNet  Google Scholar 

  18. Zhai, C., Yang, C., Zhang, X.: Positive solutions for nonlinear operator equations and several classes of applications. Math. Z. 266, 43–63 (2010)

    Article  MathSciNet  Google Scholar 

  19. Ahmad, B., Ntouyas, S.K., Purnaras, I.K.: Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations. Adv. Differ. Equ. 2012, 140 (2012)

    Article  MathSciNet  Google Scholar 

  20. Graef, J.R., Kong, L.: Positive solutions for a class of higher order boundary value problems with fractional q-derivatives. Appl. Math. Comput. 218, 9682–9689 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Almeida, R., Martins, N.: Existence results for fractional q-difference equations of order \(\alpha\in[2,3]\) with three-point boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 19, 1675–1685 (2014)

    Article  MathSciNet  Google Scholar 

  22. Yang, W.: Positive solution for fractional q-difference boundary value problems with Φ-Laplacian operator. Bull. Malays. Math. Sci. Soc. 36, 1195–1203 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Ahmad, B., Etemad, S., Ettefagh, M., Rezapour, S.: On the existence of solutions for fractional q-difference inclusions with q-antiperiodic boundary conditions. Bull. Math. Soc. Sci. Math. Roum. 59, 119–134 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Agarwal, R.P., Ahmad, B., Alsaedi, A., Al-Hutami, H.: Existence theory for q-antiperiodic boundary value problems of sequential q-fractional integro-differential equations. Abstr. Appl. Anal. 2014, Article ID 207547 (2014)

    Google Scholar 

  25. Wang, J.R., Zhang, Y.R.: On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives. Appl. Math. Lett. 39, 85–90 (2015)

    Article  MathSciNet  Google Scholar 

  26. Zhai, C.B., Yan, W.P., Yang, C.: A sum operator method for the existence and uniqueness of positive solution to Riemann–Liouville fractional differential equation boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 18, 858–866 (2013)

    Article  MathSciNet  Google Scholar 

  27. Zhao, Y., Ye, G., Chen, H.: Multiple positive solutions of a singular semipositone integral boundary value problem for fractional q-derivatives equation. Abstr. Appl. Anal. 2013, Article ID 643571 (2013). https://doi.org/10.1155/2013/643571

    Article  MathSciNet  MATH  Google Scholar 

  28. Ahmad, B., Ntouyas, S.K., Alsaedi, A., Al-Hutami, H.: Nonlinear q-fractional differential equations with nonlocal and sub-strip type boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2014, 26 (2014)

    Article  MathSciNet  Google Scholar 

  29. Almeida, R., Martins, N.: Existence results for fractional q-difference equations of order \(\alpha\in[2,3]\) with three-point boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 19, 1675–1685 (2014)

    Article  MathSciNet  Google Scholar 

  30. Sitthiwirattham, T.: On nonlocal fractional q-integral boundary value problems of fractional q-difference equations and fractional q-integrodifference equations involving different numbers of order and q. Bound. Value Probl. 2016, Article ID 12 (2016)

    Article  MathSciNet  Google Scholar 

  31. Patanarapeelert, N., Sriphanomwan, U., Sitthiwirattham, T.: On a class of sequential fractional q-integrodifference boundary value problems involving different numbers of q in derivatives and integrals. Adv. Differ. Equ. 2016, Article ID 148 (2016)

    Article  MathSciNet  Google Scholar 

  32. Sriphanomwan, U., Tariboon, J., Patanarapeelert, N., Sitthiwirattham, T.: Existence results of nonlocal boundary value problems for nonlinear fractional q-integral difference equations. J. Nonlinear Funct. Anal. 2017, Article ID 28 (2017)

    Google Scholar 

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Acknowledgements

The authors are very grateful to the reviewers for their valuable suggestions and useful comments, which led to an improvement of this paper.

Funding

This project was supported by the National Natural Science Foundation of China (Grant No. 11271235).

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FG carried out the molecular genetic studies, participated in the sequence alignment, and drafted the manuscript. SK conceived the study and participated in its design and coordination. FC helped to draft the manuscript. All authors read and approved the final manuscript.

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Correspondence to Fu Chen.

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Guo, F., Kang, S. & Chen, F. Existence and uniqueness results to positive solutions of integral boundary value problem for fractional q-derivatives. Adv Differ Equ 2018, 379 (2018). https://doi.org/10.1186/s13662-018-1796-3

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