Abstract
The aim of this paper is to investigate the boundary value problem of a fractional q-difference equation with ϕ-Laplacian, where ϕ-Laplacian is a generalized p-Laplacian operator. We obtain the existence and nonexistence of positive solutions in terms of different eigenvalue intervals for this problem by means of the Green function and Guo–Krasnoselskii fixed point theorem on cones. Finally, we give some examples to illustrate the use of our results.
Similar content being viewed by others
1 Introduction
The theory of q-calculus has been developed for more than 100 years; see [1]. As a branch of q-calculus, fractional q-calculus was first proposed by Al-Salam and Agarwal in the 1960s; see [2, 3]. Fractional q-calculus has a wide range of applications in many fields, such as cosmic strings and black holes, quantum theory, aerospace dynamics, biology, economics, control theory, medicine, electricity, signal processing, image processing, biophysics, blood flow phenomenon, and so on; see [4–10] and the references therein. The fractional q-difference equations are very important, and their basic theory has been continuously developed. Recently, as a new research direction, the solvability of boundary value problems (BVPs) of fractional q-difference equations have been widely concerned by scholars at home and abroad, and some conclusions have been obtained; see [11–14]. However, there are a few studies of eigenvalue problems for fractional q-difference equations with ϕ-Laplacian operator, and lots of work should be done.
In 2013, Li et al. [15] studied some positive solutions for a class of nonlinear fractional q-difference equations with parameters involving the Riemann–Liouville fractional derivative by means of a fixed theorem in cones,
where \(2<\alpha \leq 3\), and \(f: [0,1]\times R \rightarrow R\) is a nonnegative continuous function.
In 2015, Wang et al. [16] obtained the existence and uniqueness of solutions for a class of singular BVPs of nonlinear fractional q-difference equations by a fixed point theorem in partially ordered sets,
where \(2<\alpha \leq 3\), and \(f:[0,1]\times [0,+\infty ) \rightarrow [0,+\infty )\) is continuous with \(\lim_{t\rightarrow 0^{+}}f(t,\cdot )=\infty \).
In 2015, Han et al. [17] used the Green function and Guo–Krasnoselskii fixed-point theorem on cones to study solutions for eigenvalue problems of fractional differential equations with generalized p-Laplacian
where \(0< q<1\), \(2<\alpha \leq 3\), \(1<\beta \leq 2\), \(\lambda >0\) is a parameter, and \(D_{0^{+}}^{\beta }\) and \(D_{0^{+}}^{\alpha }\) are the standard Riemann–Liouville fractional derivatives.
Motivated by the work above, in this paper, we investigate the following BVP of fractional q-difference equation with ϕ-Laplacian:
where \(0< q<1\), \(2<\alpha \leq 3\), \(1<\beta \leq 2\), \(\lambda >0\) is a parameter, and \(D_{q}^{\beta }\), \(D_{q}^{\alpha }\) are the standard Riemann–Liouville fractional q-derivatives. As \(q\to 1^{-}\), problem (1.2) reduces to problem (1.1).
In this paper, we always assume that
-
(A1)
\(\phi :R\rightarrow R\) is an odd increasing homeomorphism, and there exist two increasing homeomorphisms \(\psi _{1},~\psi _{2}:(0,\infty )\rightarrow (0,\infty )\) such that
$$\begin{aligned} \psi _{1}(x)\phi (y)\leq \phi (xy)\leq \psi _{2}(x) \phi (y),\quad x,y>0; \end{aligned}$$ -
(A2)
\(f:(0,+\infty )\rightarrow (0,+\infty )\) is a continuous function.
A function ϕ satisfying (A1) is called a generalized p-Laplacian operator. Two important cases are \(\phi (u)=u\) and \(\phi (u)=|u|^{p-2}u\ (p>1)\); see [18] and the references therein.
We aim to obtain the existence of at least one or two positive solutions in terms of different eigenvalue intervals using the Green function and Guo–Krasnoselskii fixed point theorem on cones. We also consider the nonexistence of positive solutions in terms of the parameter λ. Finally, we give some examples to illustrate our main results.
2 Preliminary results
In this section, we cite some definitions and fundamental results of the q-calculus and fractional q-calculus.
Definition 2.1
([1])
For \(0< q<1\), we define the q-derivative of a real-valued function f as
Note that \(\lim_{q\rightarrow 1^{-}}D_{q}f(x)=f'(x)\).
Definition 2.2
([1])
The q-integral of a function f in the interval \([0,b]\) is given by
If \(a\in [0,b]\) and f is defined in the interval \([0,b]\), then its integral from a to b is defined as
More definitions and properties of q-calculus can be found in [1]. In recent years, some results of q-calculus have been obtained; see [18–20] and the references therein.
Definition 2.3
([10])
Let \(\alpha \geq 0\), and let f be a function on \([0,1]\). The fractional q-integral of the Riemann–Liouville type is defined by \((I_{q}^{0}f)(x)=f(x)\) and
Definition 2.4
([21])
The fractional q-derivative of the Riemann–Liouville type of order \(\alpha \geq 0\) is defined by \((D_{q}^{0}f)(x)=f(x)\) and
where m is the smallest integer greater than or equal to α.
Lemma 2.1
Let \(\alpha ,\beta \geq 0\), and let f be a continuous differentiable function on \([0,1]\). Then we have the following formulas:
-
1.
\((I_{q}^{\beta }I_{q}^{\alpha }f)(x)=(I_{q}^{\alpha +\beta }f)(x)\),
-
2.
\((D_{q}^{\alpha }I_{q}^{\alpha }f)(x)=f(x)\).
Lemma 2.2
([22])
Let \(f:[0,1]\rightarrow R\) be differentiable, let p be a positive integer, and let \(\alpha >p-1\). Then
Lemma 2.3
([23])
Let \(\alpha \in R^{+}\), \(n:=\lceil \alpha \rceil \). Then
Lemma 2.4
([22])
Let \(\alpha \geq 0\). Then we have the following three formulas:
Remark 2.1
([22])
Note that if \(\alpha \geq 0\) and \(a\leq b\leq t\), then \((t-a)^{(\alpha )}\geq (t-b)^{(\alpha )}\).
Lemma 2.5
Let \(y\in C_{q}[0,1]\), \(2<\alpha \leq 3\), \(1<\beta \leq 2\). Then the BVP
has a unique solution
where
Proof
By Lemma 2.3 we have
Using Lemma 2.4 and the boundary conditions \(\phi (D_{q}^{\alpha }u(0))=D_{q}(\phi (D_{q}^{\alpha }u(1)))=0\), we get that
So we can obtain
Further, from
by Lemma 2.3 we have
Using the boundary condition \(u(0)=(D_{q}u)(0)=D_{q}u(1)=0\), we get
so we have
□
Lemma 2.6
Let \(2<\alpha \leq 3\). The functions \(G(t,qs)\) and \(H(s,q\tau )\) defined by (2.2) and (2.3), respectively, are continuous on \([0,1]\times [0,1]\) and satisfy
-
(i)
\(G(t,qs)\geq 0\), \(H(s,q\tau )\geq 0\) for \(t,s,\tau \in [0,1]\);
-
(ii)
\(G(t,qs)\leq G(1,qs)\), \(H(s,q\tau )\leq H(q\tau ,q\tau )\) for \(t,s,\tau \in [0,1]\);
-
(iii)
\(G(t,qs)\geq k(t)G(1,qs)\), \(H(s,q\tau )\geq s^{\beta -1} H(1,q\tau )\), where \(k(t)=t^{\alpha -1}\), for \(t,s,\tau \in [0,1]\).
Proof
(i) Let
It is clear that \(g_{2}(t,qs)\geq 0\) for \(t,s\in [0,1]\). If \(s\leq t\), then in view of Remark 2.1, for \(t\neq 0\),
Therefore \(G(t,qs)\geq 0\). In the same way, we can obtain that \(H(s,q\tau )\geq 0\).
(ii) Fix \(s\in [0,1]\). For \(t\neq 0\), we have
and
Therefore \(g_{1}(t,qs)\), \(g_{2}(t,qs)\) are increasing functions of t for \(s\in [0,1]\). Thus \(G(t,qs)\leq G(1,qs)\). In the same way, we get that \(H(s,q\tau )\leq H(q\tau ,q\tau )\).
(iii) Suppose that \(s\leq t\). Then
For the other circumstance, we also get \(G(t,qs)\geq t^{\alpha -1}G(1,qs)\). In the same way, we get that \(H(s,q\tau )\geq s^{\beta -1} H(1,q\tau )\). The proof is completed. □
Lemma 2.7
([17])
Assume that (A1) holds. Then
Theorem 2.1
([24] (Krasnoselskii))
Let E be a Banach space, and let \(K\in E\) be a cone in E. Let \(\Omega _{1}\) and \(\Omega _{2}\) be open subsets of E with \(0\in \Omega _{1}\) and \(\overline{\Omega }_{1}\subset \Omega _{2}\). Let \(T:K\cap (\overline{\Omega }_{2}\backslash \Omega _{1})\rightarrow K\) be a completely continuous operator. In addition, suppose that either
- \((\mathrm {H_{1}})\):
-
\(\|Tu\|\leq \|u\|\), \(\forall u\in K\cap \partial \Omega _{1}\) and \(\|Tu\|\geq \|u\|\), \(\forall u\in K\cap \partial \Omega _{2}\) or
- \((\mathrm {H_{2}})\):
-
\(\|Tu\|\leq \|u\|\), \(\forall u\in K\cap \partial \Omega _{2}\) and \(\|Tu\|\geq \|u\|\), \(\forall u\in K\cap \partial \Omega _{1}\).
Then T has a fixed point in \(K\cap (\overline{\Omega }_{2}\backslash \Omega _{1})\).
3 Main results
In this section, we consider the existence of at least one or two positive solutions or no positive solution for the BVP (1.1).
Let the Banach space \(E=C_{q}[0,1]\) be endowed with norm \(\|u\|=\max_{0\leq t\leq 1}|u(t)|\). Define the cone
Let \(T_{\lambda }:P\rightarrow P\) be the operator defined by
Lemma 3.1
Assume that (A2) holds. Then \(T_{\lambda }:P\rightarrow P\) is completely continuous.
Proof
By Lemma 2.6 we have
Thus \(T_{\lambda }(P)\subset P\). In view of the nonnegativeness and continuity of \(G(t,qs)\), \(H(s,q\tau )\), and \(f(u(\tau ))\), we have that \(T_{\lambda }:P\rightarrow P\) is continuous.
Next, we prove that \(T_{\lambda }\) is uniformly bounded.
Let \(\Omega \subset P\) be bounded, that is, there exists a positive constant \(M>0\) such that \(\| u\| \leq M\) for all \(u\in \Omega \). Set \(L=\max_{0\leq u\leq M}|f(u)|+1\). Then, for \(u\in \Omega \) and all \(t\in [0,1]\), we have
Hence \(T_{\lambda }(\Omega )\) is uniformly bounded.
On the other hand, we prove that \(T_{\lambda }\) is equicontinuous.
Since \(G(t,qs)\) is continuous on \([0,1]\times [0,1]\), it is uniformly continuous on \([0,1]\times [0,1]\). Thus, for any \(\varepsilon >0\), there exists a constant \(\delta >0\) such that \(t_{1},t_{2}\in [0,1]\) with \(| t_{1}-t_{2}| <\delta \) imply
Then, for all \(u\in \Omega \),
Hence \(T_{\lambda }(\Omega )\) is equicontinuous. By the Arzelà–Ascoli theorem we have that \(T_{\lambda }:P\rightarrow P\) is completely continuous. The proof is completed. □
For convenience, we denote
Theorem 3.1
Assume that (A1), (A2), and \(f_{\infty }\psi _{1}(A_{1}^{-1})>F_{0}\psi _{2}(A_{2}^{-1})\) hold. Then for each
the BVP of fractional q-difference Eq. (1.1) has at least one positive solution. Here we impose that \(f_{\infty }^{-1}=0\) if \(f_{\infty }=+\infty \) and \(F_{0}^{-1}=+\infty \) if \(F_{0}=0\).
Proof
Let λ satisfy (3.1), and let \(\varepsilon >0\) be such that
We separate the proof into two steps.
(1) By the definition of \(F_{0}\) there exists \(r_{1}>0\) such that
If \(u\in P\) with \(\| u\| =r_{1}\), then from (3.2) and (3.3) we obtain
Let \(\Omega _{1}=\{u\in E\mid \| u\| < r_{1}\}\). Then
(2) By the definition of \(f_{\infty }\) there exists \(r_{3}>0\) such that
If \(u\in P\) with \(\| u\| =r_{2}=\max \{2r_{1},r_{3}\}\), then from (3.2) and (3.5) we obtain
Let \(\Omega _{2}=\{u\in E\mid \| u\| < r_{2}\}\). Then
From (3.4) and (3.6) and from Theorem 2.1 we have that \(T_{\lambda }\) has a fixed point \(u\in P\cap (\overline{\Omega }_{2}\setminus \Omega _{1})\) with \(r_{1}\leq \| u\| \leq r_{2}\) and that u is a positive solution for the BVP of fractional q-difference Eq. (1.1). The proof is completed. □
Theorem 3.2
Assume that (A1), (A2), and \(f_{0}\psi _{1}(A_{1}^{-1})>F_{\infty }\psi _{2}(A_{2}^{-1})\) hold. Then for each
the BVP of fractional q-difference Eq. (1.1) has at least one positive solution. Here we impose that \(f_{0}^{-1}=0\) if \(f_{0}=+\infty \) and \(F_{\infty }^{-1}=+\infty \) if \(F_{\infty }=0\).
Proof
Let λ satisfy (3.7), and let \(\varepsilon >0\) be such that
We separate the proof into two steps.
(1) By the definition of \(f_{0}\) there exists \(r_{1}>0\) such that
If \(u\in P\) with \(\| u\| =r_{1}\), then similarly to the second part of the proof of Theorem 3.1, let \(\Omega _{1}=\{u\in E\mid \| u\| < r_{1}\}\). Then
(2) By the definition \(F_{\infty }\) there exists \(R_{1}>0\) such that
We consider two cases:
Case 1: When f is bounded, then there exists \(N>0\), such that \(|f(u)|\leq N\) for \(u\in (0,+\infty )\). If \(u\in P\) with \(\|u\|=r_{3}\), where \(r_{3}=\max \{2r_{1},\phi ^{-1}(\lambda N)A_{1}\}\), then
So let \(\Omega _{3}=\{u\in E\mid \| u\| < r_{3}\}\). Then
Case 2: Suppose f is unbounded. Then there exists \(r_{4}>\max \{2r_{1},R_{1}\}\) such that \(f(u)\leq f(r_{4})\) for \(0< u< r_{4}\). If \(u\in P\) with \(\| u\| =r_{4}\), then by (3.7) and (3.11) we have
Thus we suppose \(\Omega _{4}=\{u\in E\mid \| u\| < r_{4}\}\). Then
In view of Cases 1 and 2, we let \(\Omega _{2}=\{u\in E\mid \| u\| < r_{2}\}\), where \(r_{2}=\max \{r_{3},r_{4}\}\). Then
From (3.10) and (3.14) and from Theorem 2.1 we obtain that \(T_{\lambda }\) has a fixed point \(u\in P\cap (\overline{\Omega }_{2}\backslash \Omega _{1})\) with \(r_{1}\leq \| u\| \leq r_{2}\). Obviously, u is a positive solution of the BVP of fractional q-difference Eq. (1.1). The proof is completed. □
Theorem 3.3
Assume that (A1) and (A2) hold and there exist \(r_{2}>r_{1}>0\) such that
Then the BVP of fractional q-difference Eq. (1.1) has a positive solution \(u\in P\) with \(r_{1}\leq \| u\| \leq r_{2}\).
Proof
Let \(\Omega _{1}=\{u\in E|\|u\|< r_{1}\}\). Then for \(u\in P\cap \partial \Omega _{1}\), we obtain
Suppose \(\Omega _{2}=\{u\in E\mid \| u\| < r_{2}\}\). Then for \(u\in P\cap \partial \Omega _{2}\), we have
Thus by Theorem 2.1 the BVP of fractional q-difference Eq. (1.1) has a positive solution \(u\in P\) with \(r_{1}\leq \| u\| \leq r_{2}\). The proof is completed. □
Theorem 3.4
Assume that (A1) and (A2) hold. Let \(\lambda _{1}=\sup_{r>0} \frac{\phi (r)}{\phi (A_{1})\max_{0\leq u\leq r}f(u)}\). If \(f_{0}=+\infty \) and \(f_{\infty }=+\infty \), then the BVP of fractional q-difference Eq. (1.1) has at least two positive solutions for each \(\lambda \in (0,\lambda _{1})\).
Proof
We define \(x(r)=\frac{\phi (r)}{\psi _{2}(A_{1})\max_{0\leq u\leq r}f(u)}\). In view of the continuity of \(f,~f_{0}=+\infty \), and \(f_{\infty }=+\infty \), we obtain that \(x(r):(0,+\infty )\rightarrow (0,+\infty )\) is continuous and \(\lim_{r\rightarrow 0^{+}}x(r)=\lim_{r\rightarrow +\infty }x(r)=0\). So there exists \(r_{0}\in (0,+\infty )\) such that \(x(r_{0})=\sup_{r>0}x(r)=\lambda _{1}\). For all \(\lambda \in (0,\lambda _{1})\), there exist constants \(a_{1},~a_{2}>0\) such that \(x(a_{1})=x(a_{2})=\lambda \), where \(0< a_{1}< r_{0}< a_{2}<+\infty \). Thus
By the conditions \(f_{0}=+\infty \) and \(f_{\infty }=+\infty \) there exist constants \(b_{1},~b_{2}>0\), where \(0< b_{1}< a_{1}< r_{0}< a_{2}< b_{2}<+\infty \), such that
so that
By (3.15), (3.17), (3.16), and (3.18), combined with Theorems 2.1 and 3.3, the BVP of fractional q-difference Eq. (1.1) has at least two positive solutions for each \(\lambda \in (0,\lambda _{1})\). The proof is completed. □
Theorem 3.5
Assume that (A1) and (A2) hold. If \(F_{0}<+\infty \) and \(F_{\infty }<+\infty \), then there exists \(\lambda _{0}>0\) such that for all \(0<\lambda <\lambda _{0}\), the BVP of fractional q-difference Eq. (1.1) has no positive solution.
Proof
Since \(F_{0}<+\infty \) and \(F_{\infty }<+\infty \), there exist \(M_{1},M_{2}, r_{1},r_{2}>0\), such that \(r_{1}< r_{2}\) and
Let \(M_{0}=\max \{M_{1},M_{2},\max_{r_{1}\leq u\leq r_{2}}{ \frac{f(u)}{\phi (u)}}\}\). Then we have
Let v be a positive solution of the fractional q-difference equation boundary value problem (1.1). We will show that this leads to a contradiction for \(0<\lambda <\lambda _{0}\), where \(\lambda _{0}:=M_{0}^{-1}\psi _{1}(A_{1}^{-1})\). Indeed, since \(T_{\lambda }v(t)=v(t)\) for \(t\in [0,1]\), we have
a contradiction. Therefore the BVP of fractional q-difference Eq. (1.1) has no positive solution. The proof is completed. □
Theorem 3.6
Assume that (A1) and (A2) hold. If \(f_{0}>0\) and \(f_{\infty }>0\), then there exists \(\lambda _{0}'>0\) such that for all \(\lambda >\lambda _{0}'\), the BVP of fractional q-difference Eq. (1.1) has no positive solution.
Proof
Since \(f_{0}>0\) and \(f_{\infty }>0\), there exist \(m_{1},m_{2},r_{3},r_{4}>0\) such that \(r_{3}< r_{4}\) and
Let \(m_{0}=\max \{m_{1},m_{2},\max_{r_{3}\leq u\leq r_{4}}{ \frac{f(u)}{\phi (u)}}\}\). Then we have
Let v be a positive solution of the fractional q-difference equation BVP (1.1). We will show that this leads to a contradiction for \(\lambda >\lambda _{0}'\), where \(\lambda _{0}':=m_{0}^{-1}\psi _{2}(A_{2}^{-1})\). Indeed, since \(T_{\lambda }v(t)=v(t)\) for \(t\in [0,1]\), we have
a contradiction. Therefore the BVP of fractional q-difference Eq. (1.1) has no positive solution. The proof is completed. □
4 Some examples of application
Example 4.1
Consider the following fractional q-difference equation BVP:
Here \(q=\frac{1}{2}\), \(\alpha =\frac{5}{2}\), \(\beta =\frac{3}{2}\), \(\phi (u)=u\), \(f(u)=7u-6\sin u\). Take \(\psi _{1}(x)=\psi _{2}(x)=x\), \(\delta =0.9\).
By a simple calculation we obtain \(\Gamma _{\frac{1}{2}}(\frac{5}{2})\approx 1.1906\), \(\Gamma _{ \frac{1}{2}}(\frac{3}{2})\approx 0.9209\), \(A_{1}\approx 0.05523\), \(A_{2} \approx 0.00811\), \(F_{0}=1\), \(f_{\infty }=7\), \(f_{\infty }\psi _{1}(A_{1}^{-1})\approx 126.74271\), \(F_{0}\psi _{2}(A_{2}^{-1}) \approx 123.30456\), \(\psi _{2}(A_{2}^{-1})f_{\infty }^{-1}\approx 17.6194\) and \(\psi _{1}(A_{1}^{-1})F_{0}^{-1})\approx 18.10610\).
Obviously, \(f_{\infty }\psi _{1}(A_{1}^{-1})>F_{0}\psi _{2}(A_{2}^{-1})\). By Theorem 3.1 we obtain that BVP (4.1) has at least one positive solution for each \(\lambda \in (17.61494, 18.10610)\).
Example 4.2
Consider the following fractional q-difference equation BVP with ϕ-Laplacian:
Here \(q=\frac{1}{2}\), \(\alpha =\frac{5}{2}\), \(\beta =\frac{3}{2}\), \(\phi (u)=|u|u\), and \(f(u)=\frac{(u^{3}+u^{2})(\sin u+9)}{25u+1}\).
Take \(\psi _{1}(x)=\psi _{2}(x)=x^{2}\), \(\delta =0.9\). By calculating we get \(A_{1}\approx 0.07088\), \(A_{2}\approx 0.01550\), \(F_{\infty }=0.4\) and \(f_{0}=9 \). Thus, \(f_{0}\psi _{1}(A_{1}^{-1})>F_{\infty }\psi _{2}(A_{2}^{-1})\). By Theorem 3.1 we obtain that BVP (4.2) has at least one positive solution for each \(\lambda \in (462.24260, 497.6280425)\).
Example 4.3
Consider the following fractional q-difference equation BVP:
Here \(q=\frac{1}{2}\), \(\alpha =\frac{5}{2}\), \(\beta =\frac{3}{2}\), \(\phi (u)=u\), and \(f(u)=\frac{(20u^{2}+u)(\sin u+2)}{u+1}\). Take \(\psi _{1}(x)=\psi _{2}(x)=x\), \(\delta =0.9\). By calculating we have \(A_{1}\approx 0.05523\), \(A_{2}\approx 0.00811\), \(F_{0}=f_{0}=2\), \(F_{\infty }=60\), \(f_{ \infty } =20\), and \(u< f(u)<60u\) for \(u>0\).
-
(i)
By Theorem 3.1 we obtain that BVP (4.3) has at least one positive solution for each \(\lambda \in (6.1653,9.05305)\).
-
(ii)
By Theorem 3.5 we obtain that BVP (4.3) has no positive solution for each \(\lambda \in (0, 0.30177)\).
-
(iii)
By Theorem 3.6 we obtain that BVP (4.3) has no positive solution for each \(\lambda \in (123.30456, +\infty )\).
Example 4.4
Consider the following fractional q-difference equation BVP with ϕ-Laplacian:
Here \(q=\frac{1}{2}\), \(\alpha =\frac{5}{2}\), \(\beta =\frac{3}{2}\), \(\phi (u)=|u|u\), and \(f(u)=\frac{(u^{3}+u^{2})(\arctan u+8))}{40u+1}\). Take \(\psi _{1}(x)=\psi _{2}(x)=x^{2}\), \(\delta =0.9\). Then we get \(A_{1}\approx 0.07088\), \(A_{2}\approx 0.01550\), \(f_{0}=8\), \(F_{\infty }=0.23927\), and \(\frac{1}{5}\phi (u)< f(u)<9.57080\phi (u)\) for \(u>0\).
-
(i)
By Theorem 3.2 we obtain that BVP (4.4) has at least one positive solution for each \(\lambda \in (520.02293, 831.91078)\).
-
(ii)
By Theorem 3.5 we obtain that BVP (4.3) has no positive solution for each \(\lambda \in (0, 20.79777)\).
-
(iii)
By Theorem 3.6 we obtain that BVP (4.3) has no positive solution for each \(\lambda \in (20800.91719, +\infty )\).
5 Conclusions
This research establishes the existence of at least one or two positive solutions in terms of different eigenvalue intervals for the BVP of ϕ-Laplacian fractional q-difference equation, by applying the Green function and Guo–Krasnoselskii fixed point theorem on cones. This enriches the theories for fractional q-difference equations and provides the theoretical guarantee for the application of fractional q-difference equations in such fields as aerodynamics, electrodynamics of complex medium, capacitor theory, electrical circuits, control theory, and so on. At the same time, we also consider the nonexistence of a positive solution in terms of the parameter λ. In the future, we will use bifurcation theory, critical point theory, variational method, and other methods to continue our works in this area.
Availability of data and materials
Not applicable.
References
Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)
Al-Salam, W.A.: Some fractional q-integrals and q-derivatives. Proc. Edinb. Math. Soc. 15(2), 135–140 (1966)
Agarwal, R.P.: Certain fractional q-integrals and q-derivatives. Math. Proc. Camb. Philos. Soc. 66, 365–370 (1969)
Han, Z., Lu, H., Zhang, C.: Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian. Appl. Math. Comput. 257, 526–536 (2015)
Xu, X., Zhang, H.: Multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations. Bound. Value Probl. 34, 1–18 (2018)
Jiang, W., Kosmatov, N.: Solvability of a third-order differential equation with functional boundary conditions at resonance. Bound. Value Probl. 81, 1–20 (2017)
Yang, Y.Y., Wang, Q.R.: Multiple positive solutions for p-Laplacian equations with integral boundary conditions. J. Math. Anal. Appl. 453(1), 558–571 (2017)
Liu, X., Jia, M., Ge, W.: The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator. Appl. Math. Lett. 65, 56–62 (2017)
Jannelli, A., Ruggieri, M., Speciale, M.P.: Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation. Commun. Nonlinear Sci. Numer. Simul. 70, 89–101 (2019)
Jannelli, A., Rrggieri, M., Speciale, M.P.: Exact and numerical solutions of time-fractional advection–diffusion equation with a nonlinear source term by means of the Lie symmetries. Nonlinear Dyn. 92(2), 543–555 (2018)
Ma, K., Sun, S., Han, Z.: Existence of solutions of boundary value problems for singular fractional q-difference equations. J. Appl. Math. Comput. 54(1–2), 23–40 (2017)
Zhai, C., Ren, J.: Positive and negative solutions of a boundary value problem for a fractional q-difference equation. Adv. Differ. Equ. 2017, 82 (2017). https://doi.org/10.1186/s13662-017-1138-x
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
Araci, S., Sen, E., Acikgoz, M., Srivastava, H.M.: Existence and uniqueness of positive and nondecreasing solutions for a class fractional bounday value problems involving the p-Laplacian operator. Adv. Differ. Equ. 2015, 40 (2015). https://doi.org/10.1186/s13662-015-0375-0
Li, X., Han, Z., Sun, S.: Existence of positive solutions of nonlinear fractional q-difference equation with parameter. Adv. Differ. Equ. 2013, 260 (2013). https://doi.org/10.1186/1687-1847-2013-260
Wang, J., Yu, C., Guo, Y.: Positive solutions for a class of singular boundary value problems with fractional q-difference equations. J. Funct. Spaces 2015, 1–8 (2015)
Han, Z., Lu, H., Zhang, C.: Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian. Appl. Math. Comput. 257, 526–536 (2015)
Wang, H.: On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl. 281, 287–306 (2003)
Duran, U., Acikgoz, M., Araci, S.: A study on some new results arising from (p, q)-calculus. TWMS J. Pure Appl. Math. 1(11), 57–71 (2020)
Acikgoz, M., Ates, R., Duran, U., Araci, S.: Applications of q-umbral calculus to modified Apostol type q-Bernoulli polynomials. J. Math. Stat. 14, 7–15 (2018)
Atici, F.M., Eloe, P.W.: Fractional q-calculus on a time scale. J. Nonlinear Math. Phys. 14(3), 341–352 (2007)
Ferreira, R.A.C.: Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 70, 1 (2010)
Anna, M.H., Mansour, Z.S.: q-Fractional Calculus and Equations. Springer, Berlin (2012)
Krasnoselskii, M.A.: Positive Solution of Operator Equation. Noordhoff, Groningen (1964)
Acknowledgements
The authors wish to thank the reviewers for their comments and efforts toward improving this manuscript.
Funding
The author is very grateful to the referees for their very helpful comments and suggestions. The research project is supported by the National Natural Science Foundation of China (11772007), Beijing Natural Science Foundation (1172002, Z180005), the Natural Science Foundation of Hebei Province (A2015208114), and the Foundation of Hebei Education Department (QN2017063).
Author information
Authors and Affiliations
Contributions
Conceptualization, CY and JW; methodology, CY; data curation, JW and SW; original draft preparation, BZ; review and editing, JW. All authors have read and agreed with the published version of the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
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
Wang, J., Yu, C., Zhang, B. et al. Positive solutions for eigenvalue problems of fractional q-difference equation with ϕ-Laplacian. Adv Differ Equ 2021, 499 (2021). https://doi.org/10.1186/s13662-021-03652-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-021-03652-x