Theory and Modern Applications

Existence of positive solutions for the fractional q-difference boundary value problem

Abstract

In this paper, we investigate the existence of positive solutions for a class of fractional boundary value problems involving q-difference. By using the fixed point theorem of cone mappings, two existence results are obtained. Examples are given to illustrate the abstract results.

Introduction

The theory of q-calculus or quantum calculus was initially developed by [6, 7] and it has many applications in the fields of hypergeometric series, particle physics, quantum mechanics and complex analysis. For a general introduction of q-calculus or quantum calculus, we refer to [1, 2, 8]. Recently, fractional boundary value problems with q-difference have been investigated by many authors; see [3, 4, 911] and the references therein. In , Ferreira considered the existence of positive solutions for the nonlinear q-fractional boundary value problem (BVP)

$$\textstyle\begin{cases} D^{\alpha }_{q}u(t)=-f(t,u(t)), \quad t\in I:=(0,1), \\ u(0)=D_{q}u(0)=0, \qquad D_{q}u(1)=\beta \geq 0, \end{cases}$$
(1.1)

where $$0< q<1$$, $$2<\alpha \leq 3$$, $$f: I^{*}\times {\mathbb{R}}^{+}\rightarrow {\mathbb{R}}^{+}$$ is a continuous function, $$I^{*}=[0,1]$$, $${\mathbb{R}}^{+}=[0, +\infty )$$. By utilizing a fixed point theorem in cones, he obtained the following existence theorem.

Theorem A

Let $$\tau =q^{n}$$with $$n\in {\mathbb{N}}$$. Suppose that $$f(t,u)$$is a nonnegative continuous function on $$[0,1]\times {\mathbb{R}}^{+}$$. If there exist two positive constants $$r_{2}>r_{1}>0$$such that the function f satisfies

$$(P1)$$:

$$\frac{\beta }{[\alpha -1]_{q}}+M\max_{(t,u)\in [0,1]\times [0, r_{1}]}f(t,u)\leq r_{1}$$;

$$(P2)$$:

$$\frac{\beta }{[\alpha -1]_{q}}+N\max_{(t,u)\in [\tau ,1] \times [\tau ^{\alpha -1}r_{2}, r_{2}]}f(t,u)\geq r_{2}$$,

where

\begin{aligned}& [\alpha -1]_{q}=\frac{1-q^{\alpha -1}}{1-q}, \\& M= \int _{0}^{1}G(1, qs)\,d_{q}s, \\& N=\max_{t\in [0,1]} \int _{\tau }^{1}G(t, qs)\,d_{q}s, \end{aligned}

$$G(t, qs)$$is the Green’s function which will be specified later, then the BVP (1.1) has a solution satisfying $$u(t)>0$$for $$t\in (0, 1]$$.

Clearly, the conditions $$(P1)$$ and $$(P2)$$ are strong in application. In 2015, Li et al.  studied a class of fractional Schrödinger equations with q-difference of the form

$$D^{\alpha }_{q}u(t)+\frac{n}{\hbar }\bigl(\aleph -\rho (t)\bigr)u(t)=0, \quad t\in I,$$
(1.2)

where $$\rho (t)$$ is the trapping potential, n is the mass of a particle, ħ is the Planck constant, is the energy of a particle. Let $$\lambda =\frac{n}{\hbar }$$ and $$h(t)=\aleph -\rho (t)$$. They transformed Eq. (1.2) to

$$D^{\alpha }_{q}u(t)+\lambda h(t)f\bigl(u(t)\bigr)=0, \quad t \in I,$$
(1.3)

subject to the boundary conditions

$$u(0)=D_{q}u(0)=D_{q}u(1)=0,$$
(1.4)

where $$0< q<1$$, $$2<\alpha \leq 3$$, $$f: I^{*}\times {\mathbb{R}}\rightarrow (0,\infty )$$ is continuous, $$h: I\rightarrow (0,\infty )$$ is continuous. By applying a fixed point theorem in cones, they proved several theorems for the existence of positive solutions of the problem (1.3)–(1.4). Here, we just list two important results of .

Theorem B

Suppose that $$(H1)$$and one of $$(H2)$$and $$(H3)$$hold, where

$$(H1)$$:

$$h(t)$$is continuous for $$t\in (0, 1)$$such that $$\int _{0}^{1}h(t)\,d_{q}t<+\infty$$;

$$(H2)$$:

$$\lim_{u\rightarrow 0}\frac{f(u)}{u}=\infty$$;

$$(H3)$$:

$$\lim_{u\rightarrow \infty }\frac{f(u)}{u}=\infty$$.

Then the problem (1.3)(1.4) has at least one positive solution provided that

$$0< \lambda < \frac{\sup_{r>0}\frac{r}{\max_{0\leq u\leq r}f(u)}}{\max_{t\in [0,1]}\int _{0}^{1}G(t, qs)h(s)\,d_{q}s},$$
(1.5)

where $$r>0$$is constant.

Theorem C

Suppose that $$(H1)$$and one of $$(H4)$$and $$(H5)$$hold, where

$$(H4)$$:

$$\lim_{u\rightarrow 0}\frac{f(u)}{u}=0$$;

$$(H5)$$:

$$\lim_{u\rightarrow \infty }\frac{f(u)}{u}=0$$.

Then the problem (1.3)(1.4) has at least one positive solution provided that

$$\frac{\inf_{r>0}\frac{r}{\min_{\tau r\leq u\leq r}f(u)}}{\min_{t\in [\tau ,1]}\int _{\tau }^{1}G(t, qs)h(s)\,d_{q}s}< \lambda < \infty ,$$
(1.6)

where $$r>0$$is constant.

It is obvious that the conditions $$(H2)$$$$(H5)$$ are weaker than $$(P1)$$$$(P2)$$, but (1.5) and (1.6) are not easy to verify in application.

In the present work, we consider the fractional boundary value problem (Fr-BVP) with q-difference of the form

$$\textstyle\begin{cases} D^{\alpha }_{q}u(t)+\omega (t)f(t,\delta (t) u(t))=0, \quad t\in I, \\ u(0)=D_{q}u(0)=D_{q}u(1)=0, \end{cases}$$
(1.7)

where $$0< q<1$$, $$2<\alpha \leq 3$$, $$\omega \in C[0,1]$$, $$\delta \in C(I^{*}, (0,+\infty ))$$, $$f\in C(I\times {\mathbb{R}}^{+}, {\mathbb{R}}^{+})$$, f may be singular at $$t=0$$ and/or 1. Here, $$\delta (t)$$ is a scaling function of u in the nonlinearity f.

For the sake of simplicity, denote

$$\delta _{m}=\min_{t\in I^{*}}\delta (t), \qquad \delta _{M}= \max_{t\in I^{*}}\delta (t).$$

Throughout this paper, we always assume that the functions f and ω satisfy the following conditions.

$$(A1)$$:

$$\omega \in C[0,1]$$ and there exists $$\xi >0$$ such that $$\omega (t)\geq \xi$$ for $$t\in I$$;

$$(A2)$$:

$$\int _{0}^{1}G(1,qs)f(s,\delta _{M})\,d_{q}s<+\infty$$;

$$(A3)$$:

$$f(t,\delta _{m})>0$$ for any $$t\in I$$ and there exist constants $$\sigma _{1}\geq \sigma _{2}>1$$ such that, for every $$\tau \in (0,1]$$,

$$\tau ^{\sigma _{1}}f(t,x)\leq f(t,\tau x)\leq \tau ^{\sigma _{2}}f(t,x)$$
(1.8)

for any $$t\in I$$ and $$x\in {\mathbb{R}}^{+}$$;

$$(A4)$$:

$$f(t,\delta _{m})>0$$ for any $$t\in I$$ and there exist constants $$0<\sigma _{3}\leq \sigma _{4}<1$$ such that, for every $$\tau \in (0,1]$$,

$$\tau ^{\sigma _{4}}f(t,x)\leq f(t,\tau x)\leq \tau ^{\sigma _{3}}f(t,x)$$
(1.9)

for any $$t\in I$$ and $$x\in {\mathbb{R}}^{+}$$.

Remark 1.1

1. (1)

If f satisfies the assumption $$(A3)$$ or $$(A4)$$, then $$f(t,x)$$ is non-decreasing with respect to $$x\in {\mathbb{R}}^{+}$$ for every $$t\in I$$.

2. (2)

The condition (1.8) is equivalent to

$$\tau ^{\sigma _{2}}f(t,x)\leq f(t,\tau x)\leq \tau ^{\sigma _{1}}f(t,x), \quad \forall \tau \geq 1.$$
(1.10)
3. (3)

The condition (1.9) is equivalent to

$$\tau ^{\sigma _{3}}f(t,x)\leq f(t,\tau x)\leq \tau ^{\sigma _{4}}f(t,x), \quad \forall \tau \geq 1.$$

Remark 1.2

The assumptions $$(A3)$$ and $$(A4)$$ are order conditions. They are much easier to verify in application than the conditions $$(H2)$$$$(H5)$$ and (1.5), (1.6).

Remark 1.3

If $$\delta (t)\equiv 1$$ for $$t\in [0,1]$$ and $$f(t, u)=f(u)$$, then the Fr-BVP (1.7) becomes to the problem (1.3)–(1.4) with $$\omega (t)=\lambda h(t)$$. Therefore, the Fr-BVP (1.7) is more general than the problem (1.3)–(1.4).

By using the fixed point theorem of cone mappings, we obtain the following theorems.

Theorem 1.1

Let the assumptions $$(A1)$$$$(A3)$$hold. Then the Fr-BVP (1.7) has at least one positive solution $$u\in C[0,1]$$.

Theorem 1.2

Let the assumptions $$(A1)$$, $$(A2)$$and $$(A4)$$hold. Then the Fr-BVP (1.7) has at least one positive solution $$u\in C[0,1]$$.

The rest of this paper is organized as follows. In Sect. 2 we introduce some preliminaries and notations which are useful in our proof. In Sect. 3, we will prove Theorems 1.1 and 1.2. Examples are given in Sect. 4 to illustrate the abstract results.

Preliminaries

In this section, we introduce some definitions and notations on fractional q-difference equations. Some related lemmas are also given in this section. For $$q\in (0,1)$$ and $$a, b, \alpha \in {\mathbb{R}}$$, we denote

$$[\alpha ]_{q}:=\frac{1-q^{\alpha }}{1-q}$$

and

$$(a-b)^{(\alpha )}:=a^{\alpha }\prod_{n=0}^{\infty } \frac{a-bq^{n}}{a-bq^{n+\alpha }}.$$

The q-analogue of the power function $$(a-b)^{n}$$ is defined by

$$(a-b)^{0}=1$$

and

$$(a-b)^{n}=\prod_{k=1}^{\infty } \bigl(a-bq^{k}\bigr), \quad n\in {\mathbb{N}}.$$

The q-gamma function is given by

$$\varGamma _{q}(\alpha )=\frac{(1-q)^{(\alpha -1)}}{(1-q)^{\alpha -1}}, \quad \alpha \in {\mathbb{R}}\setminus \{0, -1, -2, \ldots \},$$

and it satisfies $$\varGamma _{q}(\alpha +1)=[\alpha ]_{q}\varGamma _{q}(\alpha )$$.

Let be a function defined on $$[0,1]$$. The q-derivative of is

$$(D_{q}\ell ) (t)=\frac{\ell (t)-\ell (qt)}{(1-q)t}, \quad t>0,$$

and

$$(D_{q}\ell ) (0)=\lim_{t\rightarrow 0}(D_{q} \ell ) (t).$$

The q-derivative of of high order is given by

$$\bigl(D_{q}^{0}\ell \bigr) (t)=\ell (t),\quad t\in [0, 1],$$

and

$$\bigl(D_{q}^{n}\ell \bigr) (t)=D_{q} \bigl(D_{q}^{n-1}\ell \bigr) (t), \quad t\in [0, 1], n\in {\mathbb{N}}.$$

The following definitions of fractional q-calculus are cited from .

Definition 2.1

The fractional q-integral of the Riemann–Liouville type of order $$\alpha \geq 0$$ for the function is defined by

$$\bigl(I_{q}^{0}\ell \bigr) (t)=\ell (t),\quad t\in [0, 1],$$

and

$$\bigl(I_{q}^{\alpha }\ell \bigr) (t)=\frac{1}{\varGamma _{q}(\alpha )} \int _{0}^{t}(t-qs)^{( \alpha -1)}\ell (s) \,d_{q}s,\quad \alpha >0, t\in [0,1].$$

Definition 2.2

The fractional q-derivative of the Riemann–Liouville type of order $$\alpha \geq 0$$ for the function is defined by

$$\bigl(D_{q}^{0}\ell \bigr) (t)=\ell (t), \quad t\in [0, 1],$$

and

$$\bigl(D_{q}^{\alpha }\ell \bigr) (t)=\bigl(D_{q}^{m}I_{q}^{m-\alpha } \ell \bigr) (t) \alpha >0,\quad t\in [0,1],$$

where $$m:=\lceil \alpha \rceil$$ is the smallest integer greater than or equal to α.

We refer the reader to the papers [3, 10] and the monographs [1, 2] for more details on the definitions of fractional q-calculus.

In order to prove the existence of positive solutions of the Fr-BVP (1.7), for any $$h\in C[0,1]$$, we first consider the linear Fr-BVP

$$\textstyle\begin{cases} D^{\alpha }_{q}u(t)+h(t)=0, \quad t\in I, \\ u(0)=D_{q}u(0)=D_{q}u(1)=0. \end{cases}$$
(2.1)

Lemma 2.1

()

Let $$0< q<1$$and $$2<\alpha \leq 3$$. For any $$h\in C[0,1]$$, the linear Fr-BVP (2.1) has a unique solution expressed by

$$u(t)= \int _{0}^{1}G(t,qs)h(s)\,d_{q}s,$$

where

$$G(t,qs)=\frac{1}{\varGamma _{q}(\alpha )} \textstyle\begin{cases} (1-qs)^{(\alpha -2)}t^{\alpha -1}-(t-qs)^{(\alpha -1)}, & 0\leq qs \leq t< 1, \\ (1-qs)^{(\alpha -2)}t^{\alpha -1}, & 0\leq t \leq qs< 1, \end{cases}$$
(2.2)

is the Green’s function of the linear Fr-BVP (2.1).

Lemma 2.2

()

The Green’s function $$G(t,qs)$$has the following properties:

1. (i)

$$G(t,qs)\geq 0$$for all $$t,s\in I^{*}$$;

2. (ii)

$$t^{\alpha -1}G(1,qs)\leq G(t,qs)\leq G(1, qs)$$for all $$t,s\in I^{*}$$.

By Lemma 2.1, we can define the solution of the Fr-BVP (1.7) as follows.

Definition 2.3

A function $$u\in C[0, 1]$$ is called a solution of the Fr-BVP (1.7) if it satisfies the integral equation

$$u(t)= \int _{0}^{1}G(t,qs)\omega (s)f\bigl(s,\delta (s)u(s)\bigr)\,d_{q}s, \quad t \in I^{*}.$$

If $$u(t)>0$$ for $$t\in I$$, then it is called a positive solution of the Fr-BVP (1.7).

Let $$E:=C[0,1]$$. Then E is a Banach space endowed with the norm

$$\Vert u \Vert =\max_{t\in I^{*}} \bigl\vert u(t) \bigr\vert , \quad \forall u\in E.$$

Let $$\eta \in (0,1)$$. Define a cone K in E by

$$K=\Bigl\{ u\in E: u(t)\geq 0, t\in I^{*}, \min_{t\in [\eta ,1]}u(t) \geq \eta ^{\alpha -1} \Vert u \Vert \Bigr\} .$$

Then K is a nonempty closed convex cone of E.

Define an operator $$Q: K\rightarrow E$$ by

$$(Qu) (t)= \int _{0}^{1}G(t,qs)\omega (s)f\bigl(s,\delta (s)u(s)\bigr)\,d_{q}s, \quad t\in I^{*}.$$
(2.3)

Lemma 2.3

Let the assumptions $$(A1)$$$$(A3)$$hold. Then $$Q: K\rightarrow E$$is well defined, and $$u\in E$$is a positive solution of the Fr-BVP (1.7) if and only if u is a positive fixed point of Q.

Proof

For fixed $$u\in E$$ with $$u(t)\geq 0$$ for all $$t\in I^{*}$$, choosing a constant $$a\in (0,1)$$ such that $$a\|u\|<1$$. Then, for any $$t\in I^{*}$$, by (1.8) and (1.10), we have

\begin{aligned} f\bigl(t,\delta (t)u(t)\bigr) \leq &\biggl(\frac{1}{a} \biggr)^{\sigma _{1}}f\bigl(t,a\delta (t)u(t)\bigr) \\ \leq &\biggl(\frac{1}{a}\biggr)^{\sigma _{1}}\bigl[au(t) \bigr]^{\sigma _{2}}f\bigl(t,\delta (t)\bigr) \\ \leq &a^{\sigma _{2}-\sigma _{1}} \Vert u \Vert ^{\sigma _{2}}f(t,\delta _{M}). \end{aligned}

So, for any $$t\in I^{*}$$, by (2.2), we have

\begin{aligned} 0 < & \int _{0}^{1}G(t,qs)\omega (s)f\bigl(s,\delta (s)u(s)\bigr)\,d_{q}s \\ \leq & \int _{0}^{1}G(1,qs)\omega (s)f\bigl(s,\delta (s)u(s)\bigr)\,d_{q}s \\ \leq &a^{\sigma _{2}-\sigma _{1}} \Vert u \Vert ^{\sigma _{2}} \Vert \omega \Vert \int _{0}^{1}G(1,qs)f(s, \delta _{M})\,d_{q}s \\ < &+\infty . \end{aligned}

This implies that the operator $$Q: K\rightarrow E$$ is well defined. By Definition 2.3, $$u\in E$$ is a positive solution of the Fr-BVP (1.7) if and only if u is a positive fixed point of Q. □

Lemma 2.4

If the assumptions $$(A1)$$, $$(A2)$$and $$(A4)$$hold, then $$Q: K\rightarrow E$$is well defined, and $$u\in E$$is a positive solution of the Fr-BVP (1.7) if and only if u is a positive fixed point of Q.

Lemma 2.5

$$Q: K\rightarrow K$$is a completely continuous operator.

Proof

For any $$u\in K$$ and $$t\in I^{*}$$, by Lemma 2.2 and (1.10), we have $$(Qu)(t)\geq 0$$ on $$I^{*}$$ and

\begin{aligned} \min_{t\in [\eta ,1]}(Qu) (t) \geq &\min_{t\in [\eta ,1]} \int _{0}^{1}t^{ \alpha -1}G(1,qs)\omega (s)f\bigl(s,\delta (s)u(s)\bigr)\,d_{q}s \\ =&\eta ^{\alpha -1} \int _{0}^{1}G(1,qs)\omega (s)f\bigl(s,\delta (s)u(s)\bigr)\,d_{q}s \\ \geq &\eta ^{\alpha -1} \Vert Qu \Vert . \end{aligned}

Hence, $$Q: K\rightarrow K$$. By the Ascoli–Arzela theorem, one can prove that $$Q: K\rightarrow K$$ is completely continuous. □

At last, we state a fixed point theorem of cone mapping to end this section, which is useful in the proof of our main results.

Lemma 2.6

()

Let E be a Banach space, $$P\subset E$$a cone in E. Assume that $$\varOmega _{1}$$and $$\varOmega _{2}$$are two bounded and open subset of E with $$\theta \in \varOmega _{1}$$, $$\overline{\varOmega }_{1}\subset \varOmega _{2}$$. If

$$Q: P\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1})\rightarrow P$$

is a completely continuous operator such that either

1. (i)

$$\|Qu\|\leq \|u\|$$, $$\forall u\in P\cap \partial \varOmega _{1}$$and $$\|Qu\|\geq \|u\|$$, $$\forall u\in P\cap \partial \varOmega _{2}$$, or

2. (ii)

$$\|Qu\|\geq \|u\|$$, $$\forall u\in P\cap \partial \varOmega _{1}$$and $$\|Qu\|\leq \|u\|$$, $$\forall u\in P\cap \partial \varOmega _{2}$$,

Then Q has at least one fixed point in $$P\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1})$$.

Proof of the main results

In this section, we will apply Lemma 2.6 to prove the existence of positive solutions of the Fr-BVP (1.7). For any $$0< r< R$$, let

$$\varOmega _{r}=\bigl\{ u\in E: \Vert u \Vert < r\bigr\} , \qquad \varOmega _{R}=\bigl\{ u\in E: \Vert u \Vert < R \bigr\} .$$

Then $$\partial \varOmega _{r}=\{u\in E: \|u\|=r\}$$, $$\partial \varOmega _{R}=\{u \in E: \|u\|=R\}$$.

Proof of Theorem 1.1

On the one hand, defining an operator $$Q: K\rightarrow E$$ as in (2.3), we prove that there exists a constant $$r\in (0,1]$$ such that

$$\Vert Qu \Vert \leq \Vert u \Vert , \quad \forall u\in K\cap \partial \varOmega _{r}.$$

In fact, for $$u\in K$$ with $$\|u\|\leq 1$$, we have

$$f\bigl(t, \delta (t)u(t)\bigr)\leq u^{\sigma _{2}}(t)f\bigl(t, \delta (t) \bigr)\leq \Vert u \Vert ^{ \sigma _{2}}f(t, \delta _{M}),\quad \forall t\in I^{*}.$$

So, by Lemma 2.2, we have

\begin{aligned} \Vert Qu \Vert =&\max_{t\in I^{*}} \biggl\vert \int _{0}^{1}G(t,qs)\omega (s)f\bigl(s, \delta (s)u(s)\bigr)\,d_{q}s \biggr\vert \\ \leq & \Vert u \Vert ^{\sigma _{2}} \Vert \omega \Vert \int _{0}^{1}G(1,qs)f(s, \delta _{M})\,d_{q}s \\ =&\beta _{1} \Vert u \Vert ^{\sigma _{2}}, \end{aligned}

where $$\beta _{1}=\|\omega \|\int _{0}^{1}G(1,qs)f(s, \delta _{M})\,d_{q}s$$.

If $$\beta _{1}>1$$, choosing $$r=(\frac{1}{\beta _{1}})^{\frac{1}{\sigma _{2}-1}}$$, then $$r\in (0,1)$$. For any $$u\in K\cap \partial \varOmega _{r}$$, we have

$$\Vert Qu \Vert \leq \beta _{1} \Vert u \Vert ^{\sigma _{2}}=\beta _{1}^{1- \frac{\sigma _{2}}{\sigma _{2}-1}}=r= \Vert u \Vert .$$

If $$\beta _{1}\leq 1$$, choosing $$r=1$$, then, for any $$u\in K\cap \partial \varOmega _{r}$$, we have

$$\Vert Qu \Vert \leq \beta _{1} \Vert u \Vert ^{\sigma _{2}}=\beta _{1}\leq 1=r= \Vert u \Vert .$$

On the other hand, we prove that there exists a constant $$R>1$$ such that

$$\Vert Qu \Vert \geq \Vert u \Vert , \quad \forall u\in K\cap \partial \varOmega _{R}.$$

In fact, for $$u\in K$$ with $$u(t)\geq 1$$ for $$t\in I^{*}$$, we have

$$f\bigl(t, \delta (t)u(t)\bigr)\geq u^{\sigma _{2}}(t)f\bigl(t, \delta (t) \bigr)\geq u^{ \sigma _{2}}(t)f(t, \delta _{m}), \quad \forall t\in I^{*}.$$

Thus, for every $$u\in K\cap \partial \varOmega _{R}$$, by Lemma 2.5, we have $$Qu\in K$$ and, for any $$\eta \in (0,1)$$,

\begin{aligned} \Vert Qu \Vert \geq &\min_{t\in [\eta , 1]}(Qu) (t) \\ =&\min_{t\in [\eta , 1]} \int _{0}^{1}G(t,qs)\omega (s)f\bigl(s, \delta (s)u(s)\bigr)\,d_{q}s \\ \geq &\min_{t\in [\eta , 1]} \int _{0}^{1}G(t,qs)\omega (s)u^{ \sigma _{2}}(s)f(s, \delta _{m})\,d_{q}s \\ \geq &\min_{t\in [\eta , 1]}t^{\alpha -1}\xi \int _{\eta }^{1}G(1,qs)u^{ \sigma _{2}}(s)f(s, \delta _{m})\,d_{q}s \\ \geq &\eta ^{(\sigma _{2}+1)(\alpha -1)}\xi \Vert u \Vert ^{\sigma _{2}} \int _{ \eta }^{1}G(1,qs)f(s, \delta _{m})\,d_{q}s \\ =&\beta _{2} \Vert u \Vert ^{\sigma _{2}}, \end{aligned}

where $$\beta _{2}=\eta ^{(\sigma _{2}+1)(\alpha -1)}\xi \int _{\eta }^{1}G(1,qs)f(s, \delta _{m})\,d_{q}s$$.

If $$\beta _{2}<1$$, choosing $$R=(\frac{1}{\beta _{2}})^{\frac{1}{\sigma _{2}-1}}$$, then $$R>1\geq r$$. For any $$u\in K\cap \partial \varOmega _{R}$$, we have

$$\Vert Qu \Vert \geq \beta _{2} \Vert u \Vert ^{\sigma _{2}}=\beta _{2}^{1- \frac{\sigma _{2}}{\sigma _{2}-1}}=R= \Vert u \Vert .$$

If $$\beta _{2}\geq 1$$, choosing $$R=\beta _{2}+1$$, then $$R>1\geq r$$. For $$u\in K\cap \partial \varOmega _{R}$$, we have

$$\Vert Qu \Vert \geq \beta _{2} \Vert u \Vert ^{\sigma _{2}}\geq \beta _{2} \Vert u \Vert \geq \Vert u \Vert .$$

Hence, by Lemma 2.6, Q has at least one fixed point $$u^{*}\in K\cap (\overline{\varOmega }_{R}\setminus \varOmega _{r})$$ satisfying $$0< r\leq \|u^{*}\|\leq R$$. Hence for $$\eta \in (0,1)$$, $$\min_{t\in [\eta ,1]}u^{*}(t)\geq \eta ^{\alpha -1}\|u^{*} \|>0$$ and it is a positive solution of the Fr-BVP (1.7). □

Proof of Theorem 1.2

Similar to the proof of Theorem 1.1, we can prove this theorem. So we omit the details here. □

Remark 3.1

If $$\omega \in L^{\infty }[0,T]$$, the results in Theorem 1.1 and 1.2 are still true.

Examples

Example 4.1

Consider the following BVP:

$$\textstyle\begin{cases} D_{\frac{1}{2}}^{\frac{5}{2}}u(t)+\frac{5-\sin \pi t^{2}}{t(1-t)}(e^{3t}u^{3}(t)+e^{2t}u^{2}(t))=0, \quad t\in (0,1), \\ u(0)=0, \qquad D_{\frac{1}{2}}u(0)=D_{\frac{1}{2}}u(1)=0. \end{cases}$$
(4.1)

Let $$q=\frac{1}{2}$$, $$\alpha =\frac{5}{2}$$, $$f(t,\delta (t)u(t))=\frac{1}{t(1-t)}(e^{3t}u^{3}(t)+e^{2t}u^{2}(t))$$ and $$\omega (t)=5-\sin \pi t^{2}$$, where $$\delta (t)=e^{t}>0$$. Then $$\xi =4$$, $$\delta _{m}=1$$ and $$\delta _{M}=e$$. Clearly, $$f(t, \delta _{m})=\frac{2}{t(1-t)}>0$$ and

$$\int _{0}^{1}G\biggl(1,\frac{1}{2}s \biggr)f(s,\delta _{M})\,d_{\frac{1}{2}}s\leq \int _{0}^{1} \frac{(1-\frac{1}{2}s)^{(\frac{1}{2})}(e^{2}+e^{3})}{\varGamma _{\frac{1}{2}} (\frac{5}{2})s(1-s)} \,d_{ \frac{1}{2}}s< +\infty ,$$

where $$\varGamma _{\frac{1}{2}}(\frac{5}{2})= \frac{(\frac{1}{2})^{(\frac{3}{2})}}{(\frac{1}{2})^{\frac{3}{2}}}$$. Hence the conditions $$(A1)$$ and $$(A2)$$ hold.

For $$\tau \in (0,1]$$, since

$$\frac{\tau ^{3}}{t(1-t)}\bigl(x^{3}+x^{2}\bigr)\leq f(t, \tau x)= \frac{1}{t(1-t)}\bigl(\tau ^{3}x^{3}+\tau ^{2}x^{2}\bigr)\leq \frac{\tau ^{2}}{t(1-t)} \bigl(x^{3}+x^{2}\bigr),$$

then the condition $$(A3)$$ is satisfied with $$\sigma _{1}=3$$, $$\sigma _{2}=2$$. Hence, by Theorem 1.1, the BVP (4.1) has at least one positive solution $$u\in C[0,1]$$.

Example 4.2

Consider the following BVP:

$$\textstyle\begin{cases} D_{\frac{1}{2}}^{\frac{5}{2}}u(t)+\frac{\cos 3t^{2}+2}{t(1-t)}(e^{ \frac{t}{3}}u^{\frac{1}{3}}(t)+e^{\frac{t}{4}}u^{\frac{1}{4}}(t))=0, \quad t\in (0,1), \\ u(0)=0,\qquad D_{\frac{1}{2}}u(0)=D_{\frac{1}{2}}u(1)=0. \end{cases}$$
(4.2)

Let $$q=\frac{1}{2}$$, $$\alpha =\frac{5}{2}$$, $$f(t,\delta (t)u(t))=\frac{1}{t(1-t)}(e^{\frac{t}{3}}u^{\frac{1}{3}}(t)+e^{ \frac{t}{4}}u^{\frac{1}{4}}(t))$$ and $$\omega (t)=\cos 3t^{2}+2$$, where $$\delta (t)=e^{t}>0$$. Then $$\xi =1$$, $$\delta _{m}=1$$ and $$\delta _{M}=e$$. Only we verify $$(A4)$$. For $$\tau \in (0,1]$$, since

$$\frac{\tau ^{\frac{1}{3}}}{t(1-t)}\bigl(x^{\frac{1}{3}}+x^{\frac{1}{4}}\bigr) \leq f(t, \tau x)=\frac{1}{t(1-t)}\bigl(\tau ^{\frac{1}{3}}x^{\frac{1}{3}}+ \tau ^{\frac{1}{4}}x^{\frac{1}{3}}\bigr)\leq \frac{\tau ^{\frac{1}{4}}}{t(1-t)} \bigl(x^{\frac{1}{3}}+x^{\frac{1}{4}}\bigr),$$

then the condition $$(A4)$$ is satisfied with $$\sigma _{3}=\frac{1}{4}$$, $$\sigma _{4}=\frac{1}{3}$$. Hence, by Theorem 1.2, the BVP (4.2) has at least one positive solution $$u\in C[0,1]$$.

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Acknowledgements

The authors are grateful to the editor and the reviewers for their constructive comments and suggestions for the improvement of the paper.

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The research is supported by the National Natural Science Function of China (No. 11701457) and the fund of College of Science, Gansu Agricultural University (No. GAU-XKJS-2018-142).

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