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.

1 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, 9–11] and the references therein. In [3], 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. [9] 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, ℵ 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 [9].

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) 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) 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) 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.

2 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 ℓ be a function defined on $[0,1]$. The q-derivative of ℓ 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 ℓ 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 [3].

Definition 2.1

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

([3])

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

([3])

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

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

• (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

([5])

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

• (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

• (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 $P\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1})$.

3 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.

4 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 $u\in C[0,1]$.