 Research
 Open Access
 Published:
Existence of positive solutions for the fractional qdifference boundary value problem
Advances in Difference Equations volume 2020, Article number: 416 (2020)
Abstract
In this paper, we investigate the existence of positive solutions for a class of fractional boundary value problems involving qdifference. 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 qcalculus 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 qcalculus or quantum calculus, we refer to [1, 2, 8]. Recently, fractional boundary value problems with qdifference 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 qfractional boundary value problem (BVP)
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
\(G(t, qs)\)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 qdifference of the form
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
subject to the boundary conditions
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
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
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 (FrBVP) with qdifference of the form
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
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 nondecreasing 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 FrBVP (1.7) becomes to the problem (1.3)–(1.4) with \(\omega (t)=\lambda h(t)\). Therefore, the FrBVP (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 FrBVP (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 FrBVP (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 qdifference equations. Some related lemmas are also given in this section. For \(q\in (0,1)\) and \(a, b, \alpha \in {\mathbb{R}}\), we denote
and
The qanalogue of the power function \((ab)^{n}\) is defined by
and
The qgamma function is given by
and it satisfies \(\varGamma _{q}(\alpha +1)=[\alpha ]_{q}\varGamma _{q}(\alpha )\).
Let ℓ be a function defined on \([0,1]\). The qderivative of ℓ is
and
The qderivative of ℓ of high order is given by
and
The following definitions of fractional qcalculus are cited from [3].
Definition 2.1
The fractional qintegral of the Riemann–Liouville type of order \(\alpha \geq 0\) for the function ℓ is defined by
and
Definition 2.2
The fractional qderivative of the Riemann–Liouville type of order \(\alpha \geq 0\) for the function ℓ is defined by
and
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 qcalculus.
In order to prove the existence of positive solutions of the FrBVP (1.7), for any \(h\in C[0,1]\), we first consider the linear FrBVP
Lemma 2.1
([3])
Let \(0< q<1\)and \(2<\alpha \leq 3\). For any \(h\in C[0,1]\), the linear FrBVP (2.1) has a unique solution expressed by
where
is the Green’s function of the linear FrBVP (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 FrBVP (1.7) as follows.
Definition 2.3
A function \(u\in C[0, 1]\) is called a solution of the FrBVP (1.7) if it satisfies the integral equation
If \(u(t)>0\) for \(t\in I\), then it is called a positive solution of the FrBVP (1.7).
Let \(E:=C[0,1]\). Then E is a Banach space endowed with the norm
Let \(\eta \in (0,1)\). Define a cone K in E by
Then K is a nonempty closed convex cone of E.
Define an operator \(Q: K\rightarrow E\) by
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 FrBVP (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
So, for any \(t\in I^{*}\), by (2.2), we have
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 FrBVP (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 FrBVP (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
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
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 \(P\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1})\).
Proof of the main results
In this section, we will apply Lemma 2.6 to prove the existence of positive solutions of the FrBVP (1.7). For any \(0< r< R\), let
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
In fact, for \(u\in K\) with \(\u\\leq 1\), we have
So, by Lemma 2.2, we have
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
If \(\beta _{1}\leq 1\), choosing \(r=1\), then, for any \(u\in K\cap \partial \varOmega _{r}\), we have
On the other hand, we prove that there exists a constant \(R>1\) such that
In fact, for \(u\in K\) with \(u(t)\geq 1\) for \(t\in I^{*}\), we have
Thus, for every \(u\in K\cap \partial \varOmega _{R}\), by Lemma 2.5, we have \(Qu\in K\) and, for any \(\eta \in (0,1)\),
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
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
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 FrBVP (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:
Let \(q=\frac{1}{2}\), \(\alpha =\frac{5}{2}\), \(f(t,\delta (t)u(t))=\frac{1}{t(1t)}(e^{3t}u^{3}(t)+e^{2t}u^{2}(t))\) 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, \(f(t, \delta _{m})=\frac{2}{t(1t)}>0\) and
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
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:
Let \(q=\frac{1}{2}\), \(\alpha =\frac{5}{2}\), \(f(t,\delta (t)u(t))=\frac{1}{t(1t)}(e^{\frac{t}{3}}u^{\frac{1}{3}}(t)+e^{ \frac{t}{4}}u^{\frac{1}{4}}(t))\) 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
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 \(u\in C[0,1]\).
References
 1.
Ahmad, B., Ntouyas, S., Tariboon, J.: Quantum Calculus: New Concepts, Impulsive IVPs and BVPs, Inequalities. World Scientific, Singapore (2016)
 2.
Ernst, T.: A Comprehensive Treatment of qCalculus. Springer, Basel (2012)
 3.
Ferreira, R.: Positive solutions for a class of boundary value problems with fractional qdifferences. Comput. Math. Appl. 61, 367–373 (2011)
 4.
Gao, C., Lv, L., Wang, Y.: Spectra of a discrete Sturm–Liouville problem with eigenparameterdependent boundary conditions in Pontryagin space. Quaest. Math. (2019). https://doi.org/10.2989/16073606.2019.1680456
 5.
Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cone. Academic Press, New York (1988)
 6.
Jackson, D., Fukuda, T., Dunn, O., Majors, E.: On qdefinite integrals. J. Pure Appl. Math. 41, 193–203 (1910)
 7.
Jackson, F.: On qfunctions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253–281 (1908)
 8.
Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2001)
 9.
Li, X., Han, Z., Li, X.: Boundary value problems of fractional qdifference Schrödinger equations. Appl. Math. Lett. 46, 100–105 (2015)
 10.
Mao, J., Zhao, Z., Wang, C.: The unique iterative positive solution of fractional boundary value problem with qdifference. Appl. Math. Lett. 10, 106002 (2020)
 11.
Wang, G.: Twin iterative positive solutions of fractional qdifference Schrödinger equations. Appl. Math. Lett. 76, 103–109 (2018)
Acknowledgements
The authors are grateful to the editor and the reviewers for their constructive comments and suggestions for the improvement of the paper.
Availability of data and materials
Data sharing not applicable to this paper as no datasets were generated or analyzed during the current study.
Funding
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. GAUXKJS2018142).
Author information
Affiliations
Contributions
All authors contributed equally in writing this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
None of the authors have any competing interests in the manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Liang, Y., Yang, H. & Li, H. Existence of positive solutions for the fractional qdifference boundary value problem. Adv Differ Equ 2020, 416 (2020). https://doi.org/10.1186/s1366202002849w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366202002849w
MSC
 34B16
 34B18
Keywords
 Fractional qdifference equation
 Singularity
 Positive solutions
 Boundary value problems
 Fixed point theorem