- Research
- Open Access
- Published:
Existence of positive solutions for fractional differential equation involving integral boundary conditions with p-Laplacian operator
Advances in Difference Equations volume 2017, Article number: 135 (2017)
Abstract
The existence of positive solutions is considered for a fractional differential equation with p-Laplacian operator in this article. By employing the Avery-Henderson fixed point theorem, a new result is obtained for the boundary value problems. An example is also presented to illustrate the effectiveness of the main result.
1 Introduction
Fractional calculus is the extension of integer order calculus to arbitrary order calculus. With the development of fractional calculus, fractional differential equations have wide applications in the modeling of different physical and natural science fields, such as fluid mechanics, chemistry, control system, heat conduction, etc. There are many papers concerning fractional differential equations with the p-Laplacian operator [1–6] and fractional differential equations with integral boundary conditions [7–11].
By means of the Guo-Krasnosel’skii fixed point theorem on cones, Han et al. [5] investigate positive solutions for the following problems for the generalized p-Laplacian operator:
where \(1<\beta\leq2\), \(2<\alpha\leq3\), they obtain some new results of positive solutions for the aforementioned boundary value problem.
By means of the Avery-Henderson fixed point theorem and the Leggett-Williams fixed point theorem, Günendi and Yaslan [11] investigate positive solutions for the following problem with integral boundary conditions:
where \(n-1<\eta\leq n\), \(n\geq3\), \(\alpha, \beta, \gamma, \delta >0\), \(a_{p}, b_{p}\geq0\) are given constants. They show the existence of multiple positive solutions for the aforementioned boundary value problems.
Motivated by the aforementioned work, this work discusses the existence of positive solutions for this fractional differential equation:
where \(2<\alpha\leq 3\), \(2<\beta\leq3 \) and \(5<\alpha+\beta\leq6\). \(\phi_{p}(u)=|u|^{p-2}u\), \(p>1\). \({}^{\mathrm{c}}D_{0^{+}}^{\alpha}\) is the Caputo fractional derivative, \(D_{0^{+}}^{\beta}\) is the Riemann-Liouville fractional derivative.
We will always suppose the following conditions are satisfied:
- (H1):
-
\(g(t):[0,1]\rightarrow[0,+\infty)\) with \(g(t)\in L^{1}[0,1]\), \(\int_{0}^{1}g(t) \,\mathrm{d}t>0\) and \(\int_{0}^{1}tg(t) \,\mathrm {d}t>0\);
- (H2):
-
\(a, b\in(0,+\infty)\), \(a>\int_{0}^{1}g(t) \,\mathrm{d}t\) and \(b>a\);
- (H3):
-
\(f(t,u):[0,1]\times(0,\infty)\rightarrow(0,\infty)\) is continuous.
2 Background and definitions
To show the main result of this work, we give in the following some basic definitions and a theorem, which can be found in [12, 13].
Definition 2.1
The fractional integral of order \(\alpha>0\) of a function \(y:(0,+\infty)\rightarrow\mathbb{R}\) is given by
provided that the right side is pointwise defined on \((0,+\infty)\), where
Definition 2.2
For a continuous function \(y:(0,+\infty)\rightarrow\mathbb{R}\), the Riemann-Liouville derivative of fractional order \(\alpha>0\) is defined as
where \(n=[\alpha]+1\), provided that the right side is pointwise defined on \((0,+\infty)\).
Definition 2.3
For a continuous function \(y:(0,+\infty)\rightarrow\mathbb{R}\), the Caputo derivative of fractional order \(\alpha>0\) is defined as
where \(n=[\alpha]+1\), provided that the right side is pointwise defined on \((0,+\infty)\).
Theorem 2.1
Avery-Henderson fixed point theorem [14]
Let \((E,\|\cdot\|)\) be a Banach space, and \(P\subset E\) be a cone. Let ψ and φ be increasing non-negative, continuous functionals on P, and ω be a non-negative continuous functional on P with \(\omega(0)=0\), such that, for some \(r_{3}>0\) and \(M>0\), \(\varphi(u)\leq\omega(u)\leq\psi (u)\), and \(\|u\|\leq M\varphi(u)\), for all \(u\in\overline{P(\varphi ,r_{3})}\), where \(P(\varphi,r_{3})=\{u\in P:\varphi(u)< r_{3}\}\). Suppose that there exist positive numbers \(r_{1}< r_{2}< r_{3}\), such that
If \(T:\overline{P(\varphi,r_{3})}\rightarrow P\) is a completely continuous operator satisfying:
-
(C1)
\(\varphi(Tu)>r_{3}\) for all \(u\in\partial P(\varphi,r_{3})\);
-
(C2)
\(\omega(Tu)< r_{2}\) for all \(u\in\partial P(\omega,r_{2})\);
-
(C3)
\(P(\psi,r_{1})\neq\emptyset\), and \(\psi(Tu)>r_{1}\) for all \(u\in \partial P(\psi,r_{1})\),
then T has at least two fixed points \(u_{1}\) and \(u_{2}\) such that \(r_{1}<\psi(u_{1}) \) with \(\omega(u_{1})< r_{2}\) and \(r_{2}<\omega(u_{2}) \) with \(\varphi(u_{2})< r_{3}\).
3 Preliminary lemmas
Lemma 3.1
The boundary value problem (1.1) is equivalent to the following equation:
where
\(\phi_{q}(s)\) is the inverse function of \(\phi_{p}(s)\), a.e., \(\phi_{q}(s)=|s|^{q-2}s\), \(\frac{1}{p}+\frac{1}{q}=1\).
Proof
From \(D_{0^{+}}^{\beta}[\phi_{p}({}^{\mathrm{c}}D_{0^{+}}^{\alpha} u(t))]+ f(t,u(t))=0\), we get
In view of \(\phi_{p}({}^{\mathrm{c}}D_{0^{+}}^{\alpha} u(0))=[\phi_{p}({}^{\mathrm{c}}D_{0^{+}}^{\alpha} u(0))]'=0\), we get \(c_{2}=c_{3}=0\), i.e.,
Conditions \(\phi_{p}({}^{\mathrm{c}}D_{0^{+}}^{\alpha} u(1))=0\) imply that
By use of (3.6) and (3.7), we get
In view of (3.8), we obtain
Let
by use of (3.9), we get
Conditions \(u''(0)=0\) imply that \(d_{2}=0\), i.e.,
then we have
Conditions \(u'(1)=0\) imply that
From \(au(0)+bu'(0)=\int_{0}^{1}g(t)u(t) \,\mathrm{d}t\), we get
Therefore, we can obtain
The proof is complete. □
Lemma 3.2
[15]
The function \(H(s,\tau)\) defined by (3.5) is continuous on \([0,1]\times[0,1]\) and satisfy
Let E be the real Banach space \(C[0,1]\) with the maximum norm, define the operator \(T:E\rightarrow E\) by
Lemma 3.3
For \(u\in C[0,1]\) with \(u(t)\geq0\), \((Tu)(t)\) is non-increasing and non-negative.
Proof
Since
so we get
So \(Tu(t)\) is non-increasing, then we have \(\min_{ t\in [0,1]}Tu(t)=Tu(1)\). We have
The proof is complete. □
4 Main results
Theorem 4.1
Suppose that there exist numbers \(0< r_{1}< r_{2}< r_{3}\) such that f satisfies the following conditions:
-
(H1)
\(f(t,u)> M_{3}\), for \(t\in[0,1]\), \(u\in[r_{3},\frac{r_{3}}{k}]\);
-
(H2)
\(f(t,u)< M_{2}\), for \(t\in[0,1]\), \(u\in[0,r_{2}]\);
-
(H3)
\(f(t,u)> M_{1}\), for \(t\in[0,1]\), \(u\in[0,r_{1}]\),
where
Then the problem (1.1) has at least two positive solutions \(u_{1}\) and \(u_{2}\) such that \(r_{1}<\psi(u_{1}) \) with \(\omega(u_{1})< r_{2}\) and \(r_{2}<\omega(u_{2}) \) with \(\varphi(u_{2})< r_{3}\).
Proof
Define the cone \(P\subset E\) by
where
For any \(u\in P\), in view of Lemma 3.3, we get
Therefore, \(T:P\rightarrow P\). In view of the Arzela-Ascoli theorem, we have \(T:P\rightarrow P\) is completely continuous.
We define the functions on the cone P:
Obviously, we have \(\omega(0)=0\), \(\varphi(u)\leq\omega(u)\leq\psi(u)\).
For any \(u\in\overline{P(\varphi,r_{3})}\), we get \(\min_{ t\in [0,1]}u(t)\geq k\|u\|\), that is, \(\varphi(u)\geq k\|u\|\), therefore we obtain \(\|u\|\leq\frac{1}{k}\varphi(u)\). For any \(u\in\partial P(\omega,r_{2})\), we get \(\omega(lu)= l\omega (u)\) for \(0\leq l\leq1\).
In the following, we prove that the conditions of Theorem 2.1 hold.
Firstly, let \(u\in\partial P(\varphi,r_{3})\), that is, \(u\in[r_{3},\frac {r_{3}}{k}]\) for \(t\in[0,1]\). By means of (H1), we have
where \(B(2,\beta)=\int_{0}^{1}\tau(1-\tau)^{\beta-1}\,\mathrm {d}\tau\). So we get
Secondly, let \(u\in\partial P(\omega,r_{2})\), that is, \(u\in[0,r_{2}]\) for \(t\in[0,1]\). By means of (H2), we get
So we have
Finally, let \(u\in\partial P(\psi,r_{1})\), that is, \(u\in[0,r_{1}]\) for \(t\in[0,1]\). By means of (H3), we get
So we get
Therefore, in view of Theorem 2.1, we see that the problem (1.1) has at least two positive solutions \(u_{1}\) and \(u_{2}\) such that \(r_{1}<\psi(u_{1}) \) with \(\omega(u_{1})< r_{2}\) and \(r_{2}<\omega(u_{2}) \) with \(\varphi(u_{2})< r_{3}\). □
5 Example
In this section, we give a simple example to explain the main result.
Example 5.1
For the problem (1.1), Let \(\alpha=2.8\), \(\beta =2.3\), \(a=4\), \(b=10\), \(p=2\), \(g(t)=t\), then we get \(q=2\), \(\int^{1}_{0} g(t)\, \mathrm{d}t=\frac{1}{2}\), \(\int^{1}_{0} tg(t)\,\mathrm{d}t=\frac{1}{3}\),
Let
From a direct calculation, we get
In view of Theorem 4.1, we see that the aforementioned problem has at least two positive solutions \(u_{1}\) and \(u_{2}\) such that \(0.5<\psi(u_{1}) \) with \(\omega(u_{1})<9\) and \(9<\omega(u_{2}) \) with \(\varphi(u_{2})<10\).
References
Chen, T, Liu, W: An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator. Appl. Math. Lett. 25, 1671-1675 (2012)
Chai, G: Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator. Bound. Value Probl. 2012, 18 (2012)
Feng, X, Feng, H, Tan, H: Existence and iteration of positive solutions for third-order Sturm-Liouville boundary value problems with p-Laplacian. Appl. Math. Comput. 266, 634-641 (2015)
Li, Y, Qi, A: Positive solutions for multi-point boundary value problems of fractional differential equations with p-Laplacian. Math. Methods Appl. Sci. 39, 1425-1434 (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)
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)
Jankowski, T: Positive solutions to fractional differential equations involving Stieltjes integral conditions. Appl. Math. Comput. 241, 200-213 (2014)
Cabada, A, Dimitrijevic, S, Tomovic, T, Aleksic, S: The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions. Math. Methods Appl. Sci. (2016). doi:10.1002/mma.4105
Ntouyas, S, Etemad, S: On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions. Appl. Math. Comput. 266, 235-243 (2015)
Cabada, A, Hamdi, Z: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251-257 (2014)
Günendi, M, Yaslan, I: Positive solutions of higher-order nonlinear multi-point fractional equations with integral boundary conditions. Fract. Calc. Appl. Anal. 19, 989-1009 (2016)
Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999)
Samko, S, Kilbas, A, Marichev, O: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)
Avery, RI, Chyan, CJ, Henderson, J: Twin solutions of boundary value problems for ordinary differential equations and finite difference equations. Comput. Math. Appl. 42, 695-704 (2001)
Yuan, C: Multiple positive solutions for \((n-1, 1)\)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 2010, 36 (2010)
Acknowledgements
The author would like to thank the anonymous referees for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Li, Y. Existence of positive solutions for fractional differential equation involving integral boundary conditions with p-Laplacian operator. Adv Differ Equ 2017, 135 (2017). https://doi.org/10.1186/s13662-017-1172-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-017-1172-8
MSC
- 34B15
- 34B18
Keywords
- positive solutions
- Riemann-Liouville fractional derivatives
- Caputo fractional derivatives
- p-Laplacian
- Avery-Henderson fixed point theorem