Abstract
In this article, we study the existence of positive solutions to a class of two-term fractional nonlocal boundary value problems. The existence and multiplicity of positive solutions are established by means of fixed point index theory. The nonlinearity \(f(t,x)\) permits a singularity at \(t = 0,1\) and \(x=0\).
Similar content being viewed by others
1 Introduction
In this paper, we consider the following two-term fractional differential equation boundary value problems:
where \(D^{\alpha }_{0+}\) is the standard Riemann–Liouville derivative, \(1 < \alpha \leq 2\), \(b> 0\), \(\eta _{i}>0\), \(0 < \xi _{1}< \cdots < \xi _{m-2} < 1\), \(\sum^{m-2}_{i=1}\eta _{i}\xi _{i}^{\alpha -1}\leq 1\), \(f:(0,1)\times (0,+\infty )\rightarrow [0,+\infty )\) is continuous and may be singular at \(t = 0,1\) and \(x=0\).
Fractional differential equation boundary value problems (FBVPs) have attracted a great deal of attention during the past decades. The literature on boundary value problems of fractional differential equations is now much enriched (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]). Multi-term fractional differential equations appear in the mathematical models of many real world problems. For example, multi-term fractional differential equations have been used to model various types of visco-elastic damping ([20,21,22]). In [20], the authors introduced the Bagley–Torvik equation:
to describe model for the motion of thin plate in Newtonian fluid. In [22], the authors investigated the endolymph equation:
which can be used to describe model for the response of the semicircular canals to the angular acceleration. The existing literature on multi-term fractional differential equations equipped with initial conditions is quite wide. However, the boundary value problems of multi-term fractional differential equations needs to be investigated. For some recent developments on Caputo type multi-term FBVPs, we mention the papers [23, 24] and the references cited therein.
In [9], by using the technique of [25], we rewrite the original resonant problems as an equivalent non-resonant two-term FBVPs. Combining with the properties of the Green’s function we derived, the existence and uniqueness results of positive solutions are obtained by using of the fixed point index theory and iterative technique. It is well known that the suitable cone plays an important role in seeking positive solutions, which is usually depended on the positive properties of the Green function. However, there are much more difficulties in dealing with the Green functions of fractional-order boundary value problems than ordinary-order problems, especially for the case that \(1< \alpha <2\). In [26], we established some new positive properties of the Green function for a class of two-term fractional differential equation with Dirichlet-type boundary value conditions. It should be noted that the properties of the Green function derived in [9] is not suitable for some methods of nonlinear analysis to be used. By employing height functions of the nonlinear term on special bounded sets together with Leggett–Williams and Krasnosel’skii fixed point theorems, Zhang and Zhong [27] established the existence of triple positive solutions for a class of fractional differential equations with integral conditions.
Motivated by the above work, in this paper we aim to establish the existence of positive solutions to the FBVP (1.1). Our work presented in this paper has the following features. Firstly, we consider few cases of Riemann–Liouville type two-term FBVPs which has been studied before. Secondly, some new properties of the Green function for the case that \(1< \alpha <2\) have been discovered to deal with the difficulties related to the Green function for this case. Thirdly, the FBVP (1.1) possesses a singularity, that is, \(f(t,x)\) may be singular at \(t = 0,1\) and \(x =0\).
2 Basic definitions and preliminaries
For the convenience of the reader, we present some preliminaries and lemmas.
Definition 2.1
([28])
The fractional integral of order \(\alpha > 0\) of a function \(u:(0,+\infty )\rightarrow R\) is given by
provided that the right-hand side is point-wise defined on \((0,+ \infty )\).
Definition 2.2
([28])
The Riemann–Liouville fractional derivative of order \(\alpha > 0\) of a function \(u:(0,+\infty )\rightarrow R\) is given by
where \(n=[\alpha ]+1\), \([\alpha ]\) denotes the integer part of number α, provided that the right-hand side is point-wise defined on \((0,+\infty )\).
In [9], we have proved that the function
has a unique positive root \(b^{\ast }\). Throughout this paper, we assume that the following conditions hold:
- \((A_{1})\):
\(b\in (0, b^{\ast }]\) is a constant.
- \((A_{2})\):
\(f:(0,1)\times (0,+\infty )\rightarrow [0,+\infty )\) is continuous. In addition, for any \(R\geq r>0\), there exists \(\varPsi _{r,R} \in L^{1}[0,1]\cap C(0,1)\) such that
$$ f(t,x)\leq \varPsi _{r,R}(t),\quad \forall t \in (0,1), x\in \bigl[ rt ^{\alpha -1},R\bigr]. $$
Lemma 2.1
The unique solution of the problem
can be expressed by
where
here
is the Mittag-Leffler function.
Proof
Noticing that \(\eta _{i}>0 \), \(0 < \xi _{i} < 1\) and \(\sum_{i=1}^{m-2} \eta _{i} \xi _{i}^{\alpha -1}\leq 1\), we have
The proof is similar to Lemma 2.1 in [9], we omit it here. □
Lemma 2.2
The function \(K(t,s)\) satisfies the following properties:
- (1)
\(K(t,s) > 0\), \(\forall t,s\in (0,1)\);
- (2)
\(h_{2}(s)t^{\alpha -1}\leq K(t,s) \leq h_{1}(s)t^{\alpha -1}\), \(\forall t,s\in [0,1]\), where
$$ h_{1}(s)=(1-s)^{\alpha -1}E_{\alpha ,\alpha } \bigl(b(1-s)^{\alpha }\bigr)+E_{ \alpha ,\alpha }(b)q(s),\qquad h_{2}(s)=\frac{q(s)}{\varGamma (\alpha )}. $$
Proof
The proof is similar to Lemma 2.2 in [9], we omit it here. □
Lemma 2.3
([26])
The function \(K_{1}(t,s)\) has the following properties:
- (1)
\(K_{1}(t,s) > 0\), \(\forall t,s\in (0,1)\);
- (2)
\(K_{1}(t,s)=K_{1}(1-s,1-t)\), \(\forall t,s\in [0,1]\);
- (3)
\(K_{1}(t,s) \leq E_{\alpha ,\alpha }(b) s (1-s)^{\alpha -1}t ^{\alpha -2}\), \(\forall s\in [0,1]\), \(t\in (0,1]\);
- (4)
\(K_{1}(t,s) \geq M_{1} s (1-s)^{\alpha -1}(1-t)t^{\alpha -1}\), \(\forall t,s\in [0,1]\), where
$$ M_{1}=\min \biggl\{ \frac{1}{[\varGamma (\alpha )]^{2}E_{\alpha ,\alpha }(b)}, (\alpha -1)^{2}E_{\alpha ,\alpha }(b) \biggr\} . $$
Lemma 2.4
The function \(K(t,s)\) has the following properties:
- (1)
\(K(t,s) \leq t^{\alpha -2}E_{\alpha ,\alpha }(b) [s (1-s)^{ \alpha -1}+q(s)]\), \(\forall s\in [0,1]\), \(t\in (0,1]\);
- (2)
\(K(t,s) \geq M t^{\alpha -1}[s (1-s)^{\alpha -1}+q(s)]\), \(\forall t,s\in [0,1]\), where
$$ M=\min \biggl\{ \frac{1}{2\varGamma (\alpha )}, M_{1}, \frac{M _{1}}{2\varGamma (\alpha )}\times \frac{\sum_{i=1}^{m-2}\eta _{i}\xi _{i} ^{\alpha -1}-\sum_{i=1}^{m-2}\eta _{i} \xi _{i}^{\alpha }}{ E_{\alpha , \alpha }(b)-\sum_{i=1}^{m-2}\eta _{i}\xi _{i}^{\alpha -1}E_{\alpha , \alpha }(b\xi _{i}^{\alpha })} \biggr\} . $$
Proof
From (3) of Lemma 2.3, we have
Therefore (1) holds.
On the other hand, it follows from (4) of Lemma 2.3 that
Denote
Then
So (2) holds. □
By Lemma 2.2 and Lemma 2.4, we have the following lemma.
Lemma 2.5
The function \(K^{\ast }(t,s)=:t^{2-\alpha }K(t,s)\) satisfies:
- (1)
\(K^{\ast }(t,s) > 0\), \(\forall t,s\in (0,1)\);
- (2)
\(K^{\ast }(t,s)\leq h_{1}(s)t\), \(\forall t,s\in [0,1]\);
- (3)
\(K^{\ast }(t,s)\leq E_{\alpha ,\alpha }(b) [s (1-s)^{ \alpha -1}+q(s)]\), \(\forall s\in [0,1]\), \(t\in (0,1]\);
- (4)
\(K^{\ast }(t,s)\geq Mt [s (1-s)^{\alpha -1}+q(s) ]\), \(\forall t,s\in [0,1]\).
Let \(E=C[0,1]\) be endowed with the maximum norm \(\| u\| = \max_{0\leq t\leq 1}| u(t)|\), θ is the zero element of E, \(B_{r}=\{u\in E : \| u\|< r\}\). Define a cone P by
Define the height functions as follows:
For convenience, we list here some assumptions to be used later:
- \((H_{1})\):
there exist \(r_{1}>0\) and a nonnegative function \(b_{1} \in L^{1}[0,1]\) with \(\int _{0}^{1}b_{1}(s)\,ds>0\), such that
$$ f\bigl(t,t^{\alpha -2}x\bigr) \geq b_{1}(t)x , \quad \forall (t,x)\in (0,1) \times (0,r_{1}]; $$- \((H_{2})\):
there exist \(r_{2}>0\) and a nonnegative function \(b_{2} \in L^{1}[0,1]\) with \(\int _{0}^{1}b_{2}(s)\,ds>0\), such that
$$ f\bigl(t,t^{\alpha -2}x\bigr) \leq b_{2}(t)x , \quad \forall (t,x)\in (0,1) \times [r_{2},+\infty ); $$- \((H_{3})\):
there exists \(r_{3}>0\) such that
$$ \int _{0}^{1}\bigl[s (1-s)^{\alpha -1}+q(s) \bigr]\psi (t,r_{3})>\frac{r_{3}}{M}; $$- \((H_{4})\):
there exist \(r_{4}>0\) such that
$$ \int _{0}^{1}\bigl[s (1-s)^{\alpha -1}+q(s) \bigr]\varPsi (t,r_{4})< \frac{r_{4}}{E _{\alpha ,\alpha }(b)}. $$
Define operators A, \(L_{1}\) and \(L_{2}\) as follows:
Lemma 2.6
For any \(r>0\), \(A: P\setminus B_{r} \rightarrow P\) is completely continuous.
Proof
The proof is similar to Lemma 2.3 in [13], we omit it here. □
By the extension theorem of a completely continuous operator (see Theorem 2.7 of [29]), for any \(r>0\), there exists an extension operator \(\widetilde{A}:P \rightarrow P\), which is still completely continuous. Without loss of the generality, we still write it as A. By virtue of the Krein–Rutmann theorem and Lemma 2.5, we have the following lemma.
Lemma 2.7
\(L_{i}:P\rightarrow P\) (\(i=1,2\)) are completely continuous linear operator. Moreover, the spectral radius \(r(L_{i})> 0\) and \(L_{i}\) has a positive eigenfunction \(\varphi _{i}\) corresponding to its first eigenvalue \((r(L_{i}))^{-1}\), that is, \(L_{i}\varphi _{i}=r(L _{i})\varphi _{i}\).
Lemma 2.8
([29])
Let P be a cone in a Banach space E, and Ω be a bounded open set in E. Suppose that \(A: \overline{ \varOmega }\cap P\rightarrow P\) is a completely continuous operator. If there exists \(u_{0}\in P\) with \(u_{0}\neq \theta \) such that
then \(i(A,\varOmega \cap P,P)=0\).
Lemma 2.9
([29])
Let P be a cone in a Banach space E, and Ω be a bounded open set in E. Suppose that \(A: \overline{ \varOmega }\cap P\rightarrow P\) is a completely continuous operator. If
then \(i(A,\varOmega \cap P,P)=1\).
Lemma 2.10
([29])
Let P be a cone in a Banach space E, and Ω be a bounded open set in E. Suppose that \(A: \overline{ \varOmega }\cap P\rightarrow P\) is a completely continuous operator. If
then \(i(A,\varOmega \cap P,P)=0\).
3 Main results
Theorem 3.1
Assume that there exist \(r_{2}>r_{1}>0\) such that \((H_{1})\) and \((H_{2})\) hold, and
Then FBVP (1.1) has at least one positive solution.
Proof
For any \(u\in \partial B_{r_{1}}\cap P\), it follows from \((H_{1})\) that
Suppose that A has no fixed points on \(\partial B_{r_{1}}\cap P\) (otherwise, the proof is finished). In the following, we will show that
in which \(\varphi _{1}\) is the positive eigenfunction of \(L_{1}\) satisfying \(L_{1}\varphi _{1}=r(L_{1})\varphi _{1}\). If otherwise, there exist \(\mu _{0}> 0\) and \(u_{1}\in \partial B_{r_{1}}\cap P\) such that
Therefore,
Set
It is clear that \(\mu ^{\ast }\geq \mu _{0}\), and \(u_{1}\geq \mu ^{ \ast }\varphi _{1}\). Since \(L_{1}\) is nondecreasing linear operator, one has
Then
which contradicts the definition of \(\mu ^{\ast }\). So (3.1) holds. It follows from Lemma 2.8 that
Denote
Next, we will prove that W is bounded.
For any \(u \in W\), one has
in which \(\tilde{u}(t)=\min \{u(t),r_{2}\}\). It is easy to see that
Therefore,
Then
here
Then
By \(r(L_{2})< 1\), the inverse operator of \((I-L_{2})\) can be expressed by
Therefore, (3.3) yields \(u(t)\leq (I-L_{2})^{-1}M_{3}\leq M_{3}\| (I-L _{2})^{-1} \|\), \(t\in [0,1]\), so W is bounded.
Choose \(R> \max \{r_{2},M_{3}\| (I-L_{2})^{-1} \|\}\). From Lemma 2.9, one has
It follows from (3.2) and (3.4) that
Then A has a fixed point \(u^{\ast }\in (B_{R}\backslash \bar{B}_{r _{1}})\cap P\), that is,
It is easy to check that \(t^{\alpha -2}u^{\ast }(t)\) is a positive solution of FBVP (1.1). □
Theorem 3.2
Assume that there exist \(r_{4}>r_{3}>0\) such that \((H_{3})\) and \((H_{4})\) hold. Then FBVP (1.1) has at least one positive solution.
Proof
For any \(u \in \partial B_{r_{3}}\cap P\), it follows from \((H_{3})\) and Lemma 2.5 that
Therefore,
\(\forall \lambda \in (0, 1]\), \(u\in \partial B_{r_{3}}\cap P\), we have \(\lambda u\leq u\leq r_{3}\). This with (3.5) implies
It follows from Lemma 2.10 that
Next, we prove that
If otherwise, there exists \(u_{1}\in \partial B_{r_{4}}\cap P\), \(\mu _{0} \geq 1\) such that \(Au_{1}= \mu _{0} u_{1}\). From \((H_{4})\) and Lemma 2.5, we have
which contradicts \(\|u_{1}\|=r_{4}\). Then, by Lemma 2.9, we have
Equations (3.6) and (3.7) yield
Then A has a fixed point \(u^{\ast }\in (B_{R}\backslash \bar{B}_{r _{1}})\cap P\). Clearly, \(t^{\alpha -2}u^{\ast }(t)\) is a positive solution of FBVP (1.1). □
Theorem 3.3
Assume that there exist \(r_{3}>r_{4}>r_{1}>0\) such that \((H_{1})\), \((H_{3})\) and \((H_{4})\) hold with \(r(L_{1})\geq 1 \). Then FBVP (1.1) has at least two positive solutions.
Proof
Suppose that \(\exists r'_{1}\in (0,r_{1})\) such that A has no fixed points on \(\partial B_{r'_{1}}\cap P\) (otherwise, the proof is finished). By the proof of Theorem 3.1 and Theorem 3.2, we have
Therefore,
Then FBVP (1.1) has at least two positive solutions. □
Theorem 3.4
Assume that there exist \(r_{2}>r_{3}>r_{4}>r_{1}>0\) such that \((H_{1})\)–\((H_{4})\) hold with \(r(L_{1})\geq 1 >r(L_{2})>0 \). Then FBVP (1.1) has at least three positive solutions.
Proof
By Theorem 3.3 and (3.4), we get
Therefore,
Then FBVP (1.1) has at least three positive solutions. □
4 Example
Example 4.1
Consider the following problem:
where
For any \(t\in [0,+\infty )\), noticing the function \(\varGamma (\cdot )\) is strictly increasing on \([2,+\infty )\), we have
Therefore \(g(\frac{1}{5})\leq -\frac{1}{2\sqrt{\pi }}+\frac{1}{5}e ^{\frac{1}{5}}\thickapprox -0.282+ 0.243= -0.039<0\), which implies \(\frac{1}{5}< b^{\ast }\).
For any \(R\geq r>0\), \(\forall t \in (0,1)\), \(x\in [ r\sqrt{t},R]\), we have
It is clear that
Therefore, we have
Denote
Let
It follows from Lemma 2.7 that \(r(L_{3}),r(L_{4})>0\).
Set
and
Then
It is easy to see that
So the assumptions of Theorem 3.1 are satisfied. Thus Theorem 3.1 ensures that FBVP (4.1) has at least one positive solution.
5 Conclusions
In this article, we consider a class of Riemann–Liouville type two-term fractional nonlocal boundary value problems for the case that \(1< \alpha <2\). Some new properties of the Green function have been discovered to construct an exact cone. By using fixed point index theory on the exact cone, the existence and multiplicity of positive solutions are established. The nonlinearity \(f(t,x)\) permits a singularity at \(t = 0,1\) and \(x=0\).
Abbreviations
- FBVP:
-
Fractional differential equations boundary value problems
References
Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389(1), 403–411 (2012)
Cabada, A., Hamdi, Z.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251–257 (2014)
Cui, Y.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)
Cui, Y., Ma, W., Sun, Q., Su, X.: New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal., Model. Control 23, 31–39 (2018)
Hao, X., Wang, H., Liu, L., Cui, Y.: Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator. Bound. Value Probl. 2017, 182 (2017)
Henderson, J., Luca, R.: Existence of positive solutions for a singular fractional boundary value problem. Nonlinear Anal., Model. Control 22(1), 99–114 (2017)
Wang, G., Ahmad, B., Zhang, L.: Existence results for nonlinear fractional differential equations with closed boundary conditions and impulses. Adv. Differ. Equ. 2012, 169 (2012)
Wang, Y., Liu, L., Wu, Y.: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 74(11), 3599–3605 (2011)
Wang, Y., Liu, L.: Positive solutions for a class of fractional 3-point boundary value problems at resonance. Adv. Differ. Equ. 2017, 7 (2017)
Wu, J., Zhang, X., Liu, L., Cui, Y.: The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. Bound. Value Probl. 2018, 82 (2018)
Zhang, X., Shao, Z., Zhong, Q.: Positive solutions for semipositone \((k, n-k)\) conjugate boundary value problems with singularities on space variables. Appl. Math. Lett. 72, 50–57 (2017)
Wang, Y., Liu, L.: Positive solutions for a class of fractional infinite-point boundary value problems. Bound. Value Probl. 2018, 118 (2018)
Wang, Y.: Existence and multiplicity of positive solutions for a class of singular fractional nonlocal boundary value problems. Bound. Value Probl. 2019, 92 (2019)
Wang, Y.: Necessary conditions for the existence of positive solutions to fractional boundary value problems at resonance. Appl. Math. Lett. (2019). https://doi.org/10.1016/j.aml.2019.05.007
Zhang, X., Wang, L., Sun, Q.: Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter. Appl. Math. Comput. 226, 708–718 (2014)
Zhang, X., Liu, L., Wu, Y.: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26–33 (2014)
Zhang, X., Liu, L., Wu, Y., Cui, Y.: New result on the critical exponent for solution of an ordinary fractional differential problem. J. Funct. Spaces 2017, Article ID 3976469 (2017)
Zou, Y., He, G.: On the uniqueness of solutions for a class of fractional differential equations. Appl. Math. Lett. 74, 68–73 (2017)
Zuo, M., Hao, X., Liu, L., Cui, Y.: Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Bound. Value Probl. 2017, 161 (2017)
Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51(2), 294–298 (1984)
Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Physica A 278(1–2), 107–125 (2000)
Elshehawey, E.F., Elbarbary, E.M.A., Afifi, N.A.S., El-Shahed, M.: On the solution of the endolymph equation using fractional calculus. Appl. Math. Comput. 124(3), 337–341 (2001)
Ahmad, B., Ntouyas, S.K.: Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions. Appl. Math. Comput. 266, 615–622 (2015)
Ahmad, B., Luca, R.: Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions. Appl. Math. Comput. 339, 516–534 (2018)
Webb, J.R.L., Zima, M.: Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems. Nonlinear Anal. 71(3–4), 1369–1378 (2009)
Wang, Y., Liu, L.: Positive properties of the Green function for two-term fractional differential equations and its application. J. Nonlinear Sci. Appl. 10, 2094–2102 (2017)
Zhang, X., Zhong, Q.: Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables. Appl. Math. Lett. 80, 12–19 (2018)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Differential Equations. Elsevier, Amsterdam (2006)
Guo, D.: Nonlinear Functional Analysis. Shandong Science and Technology Press, Jinan (1985) (in Chinese)
Acknowledgements
The author would like to thank the referees for their pertinent comments and valuable suggestions.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
This work was supported by the Natural Science Foundation of Shandong Province of China (ZR2017MA036, ZR2014MA034), the National Natural Science Foundation of China (11571296, 11871302).
Author information
Authors and Affiliations
Contributions
The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Additional information
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
Wang, Y. Positive solutions for a class of two-term fractional differential equations with multipoint boundary value conditions. Adv Differ Equ 2019, 304 (2019). https://doi.org/10.1186/s13662-019-2250-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-019-2250-x