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 Open Access
Multiple monotone positive solutions of integral BVPs for a higherorder fractional differential equation with monotone homomorphism
 Kaihong Zhao^{1}Email author and
 Juqing Liu^{2}
https://doi.org/10.1186/s1366201607434
© Zhao and Liu 2016
 Received: 30 April 2015
 Accepted: 10 January 2016
 Published: 26 January 2016
Abstract
This paper is concerned with the integral boundary value problems of higherorder fractional differential equation with. In the sense of a monotone homomorphism, some sufficient criteria are established to guarantee the existence of at least two monotone positive solutions by employing the fixed point theorem of cone expansion and compression of functional type proposed by Avery, Henderson and O’Regan. As applications, some examples are provided to illustrate the validity of our main results.
Keywords
 integral boundary value problems
 fractional differential equation
 monotone positive solutions
 monotone homomorphism
 fixed point theorem
MSC
 34B15
 34B18
 34B27
1 Introduction
In recent years, the fractional order differential equation has aroused great attention due to both the further development of fractional order calculus theory and the important applications for the theory of fractional order calculus in the fields of science and engineering such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electronanalytical chemistry, biology, control theory, fitting of experimental data, and so forth. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. Especially, the boundary value problems with RiemannStieltjes integral boundary conditions arise in a variety of different areas of applied mathematics and physics. For example, blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, and so on can be reduced to integral boundary problems. In a consequence, the subject of fractional differential equations is gaining much importance and attention. There have been many papers focused on boundary value problems of fractional ordinary differential equations (see [1–17]).
 Case I::

\(a,b\in(0,1)\) are two constants and \(\theta\in C([0,1],[0,1])\) with \(\theta(t)\geq t\) on \([0,1]\), it means \(\theta(t)\) is an advanced argument.
 Case II::

\(a,b\in(1,\infty)\) are two constants and \(\theta\in C([0,1],[0,1])\) with \(\theta(t)\leq t\) on \([0,1]\), it means \(\theta(t)\) is a delayed argument.
 (1)
if \(x\leq y\), then \(\phi(x)\leq\phi(y)\) for all \(x,y\in \mathbb{R}\);
 (2)
ϕ is a continuous bijection and its inverse mapping \(\phi^{1}\) is also continuous;
 (3)
\(\phi(xy)=\phi(x)\phi(y)\), for all \(x,y\in\mathbb{R}^{+}\), where \(\mathbb{R}^{+}=[0,+\infty)\).
Remark 1.1
It is easy to see that the pLaplacian operator \(\varphi_{p}(x)=x^{p2}x\) (\(p>1\)) satisfy the conditions (1)(3), that is, \(\varphi_{p}\) is an increasing and positive homomorphism. Therefore, the operator ϕ of BVPs (1.1) is regarded as the improvement and generalization of the classical pLaplacian operator \(\varphi_{p}(x)=x^{p2}x\) (\(p>1\)).
The remainder of this paper is organized as follows. In Section 2, we recall some useful definitions and properties and present the properties of the Green’s functions. In Section 3 and Section 4, we give some sufficient conditions for the existence of at least two monotone positive solutions of BVPs (1.1) in Case I and Case II, respectively. Finally, some examples are also provided to illustrate our main results in Section 5.
2 Preliminaries
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the recent literature.
Definition 2.1
Definition 2.2
Lemma 2.1
(see [26])
Now, we present the necessary definitions from the theory of cones in Banach spaces.
Definition 2.3
 (1)
\(u\in P\), \(k\geq0\) implies \(k u\in P\);
 (2)
\(u\in P\), \(u\in P\) implies \(u=0\).
Definition 2.4
Definition 2.5
Let E be a real Banach space. An operator \(T: E\rightarrow E\) is said to be completely continuous if it is continuous and maps any bounded sets into the precompact sets.
 (a)
β is convex, \(\beta(0)=0\), and \(\beta(u)\neq0\) if \(u\neq0\) and \(\inf_{u\in P\cap\partial\Omega}\beta(u)>0\),
 (b)
β is sublinear, \(\beta(0)=0\), and \(\beta(u)\neq0\) if \(u\neq0\) and \(\inf_{u\in P\cap\partial\Omega}\beta(u)>0\),
 (c)
β is concave and unbounded.
 (d)
β is convex, \(\beta(0)=0\), and \(\beta(u)\neq0\) if \(u\neq0\),
 (e)
β is sublinear, \(\beta(0)=0\), and \(\beta(u)\neq0\) if \(u\neq0\),
 (f)
\(\beta(u+v)\geq\beta(u)+\beta(v)\) for all \(u, v\in P\), \(\beta(0)=0\), and \(\beta(u)\neq0\) if \(u\neq0\).
We will establish the existence of multiple monotone positive solutions to BVPs (1.1) by applying the following fixed point theorem of cone expansion and compression of functional type by Avery, Henderson and O’Regan.
Lemma 2.2
(see [28])
 (B_{1}):

α satisfies Property \(A_{1}\) with \(\alpha (Tu)\geq\alpha(u)\), for all \(u\in P\cap\partial\Omega_{1}\), and γ satisfies Property \(A_{2}\) with \(\gamma(Tu)\leq\gamma(u)\), for all \(u\in P\cap\partial\Omega_{2}\), or
 (B_{2}):

γ satisfies Property \(A_{2}\) with \(\gamma (Tu)\leq\gamma(u)\), for all \(u\in P\cap\partial\Omega_{1}\), and α satisfies Property \(A_{1}\) with \(\alpha(Tu)\geq\alpha(u)\), for all \(u\in P\cap\partial\Omega_{2}\)
Now we present the Green’s functions for BVPs (1.1).
Lemma 2.3
Proof
 (H_{1}):

\(0\leq\lambda[\rho]<1\), \(\kappa(s)\geq0\) for \(s\in[0,1]\), where \(\kappa(s)\triangleq\int_{0}^{1} G(t,s)\,dA(t)\).
 (H_{2}):

\(f\in C([0,1]\times[0,+\infty),(\infty,+\infty))\).
By Lemma 2.3, it is easy to obtain Lemma 2.4.
Lemma 2.4
Lemma 2.5
 (i)
\(H(t,s)\geq0\) for any \(t,s\in[0,1]\);
 (ii)
\(\frac{\partial H(t,s)}{\partial t}\geq0\) for any \(t,s\in[0,1]\).
Proof
Let \(E=C[0, 1]\). Then E is a real Banach space with the norm \(\\cdot\\) defined by \(\u\=\max_{t\in[0,1]}u(t)\).
Lemma 2.6
 (1)
\(u(t)\geq0\) for any \(t\in[0,1]\);
 (2)
\(u(t)\) is increasing and concave on \([0,1]\);
 (3)
\(u(t)\geq a\u\\) for any \(t\in[0,1]\).
Proof
Similar to the proof of Lemma 2.5 and Lemma 2.6, we have the following lemmas.
Lemma 2.7
 (i)
\(H(t,s)\geq0\) for any \(t,s\in[0,1]\);
 (ii)
\(\frac{\partial H(t,s)}{\partial t}\leq0\) for any \(t,s\in[0,1]\).
Lemma 2.8
 (1)
\(u(t)\geq0\) for any \(t\in[0,1]\);
 (2)
\(u(t)\) is decreasing and concave on \([0,1]\);
 (3)
\(u(t)\geq\frac{1}{a}\u\\) for any \(t\in[0,1]\).
From Lemma 2.4, we can obtain the following lemma.
Lemma 2.9
Lemma 2.10
Assume that (H_{1}) and (H_{2}) hold, then \(F, T: E\rightarrow E\) defined by (2.8) are completely continuous.
Proof
First, we shall show that \(T: E\rightarrow E\) is completely continuous through three steps.
Step 1. Let \(u\in E\), in view of the nonnegativity and continuity of functions \(H(t,s)\), \(h(t)\), \(\theta(t)\), and \(f^{+}(t,u(\theta(t)))\), we conclude that \(T: E\rightarrow E\) is continuous.
3 Multiple monotone increasing positive solutions of BVPs (1.1) in Case I
In this section, we will discuss the existence of at least two monotone increasing positive solutions to BVPs (1.1) in Case I.
Theorem 3.1
 (C_{1}):

\(f(t,u)>\phi(L_{1}r)\), for all \((t,u)\in[0,1]\times[r,R]\);
 (C_{2}):

\(f(t,u)\leq\phi(L_{2}R)\), for all \((t,u)\in[0,1]\times[aR,R]\);
 (C_{3}):

\(0\leq f^{0}\leq\phi(L_{2})\);
 (C_{4}):

\(f(t,u)\geq0\), for all \((t,u)\in[0,1]\times[0,R]\),
Proof
Similarly, we can get the following theorem.
Theorem 3.2
 (C_{5}):

\(f(t,u)<\phi(L_{2}r)\), for all \((t,u)\in[0,1]\times[ar,r]\);
 (C_{6}):

\(f(t,u)\geq\phi(L_{1}R)\), for all \((t,u)\in[0,1]\times[R,\frac{R}{a}]\);
 (C_{7}):

\(\phi(L_{1})\leq f_{0}\leq+\infty\);
 (C_{8}):

\(f(t,u)\geq0\), for all \((t,u)\in[0,1]\times[0,\frac{R}{a}]\),
4 Multiple monotone decreasing positive solutions of BVPs (1.1) in Case II
In this section, we will discuss the existence of at least two monotone decreasing positive solutions to BVPs (1.1) in Case II.
Theorem 4.1
 (D_{1}):

\(f(t,u)>\phi(L_{2}r)\), for all \((t,u)\in[0,1]\times[r,R]\);
 (D_{2}):

\(f(t,u)\leq\phi(L_{1}R)\), for all \((t,u)\in[0,1]\times[\frac{R}{a},R]\);
 (D_{3}):

\(0\leq f^{0}\leq\phi(L_{1})\);
 (D_{4}):

\(f(t,u)\geq0\), for all \((t,u)\in[0,1]\times[0,R]\),
Proof
Similar to the above arguments, we obtain the following theorem.
Theorem 4.2
 (D_{5}):

\(f(t,u)<\phi(L_{1}r)\), for all \((t,u)\in[0,1]\times[\frac{r}{a},r]\);
 (D_{6}):

\(f(t,u)\geq\phi(L_{2}R)\), for all \((t,u)\in[0,1]\times[R,aR]\);
 (D_{7}):

\(\phi(L_{2})\leq f_{0}\leq+\infty\);
 (D_{8}):

\(f(t,u)\geq0\), for all \((t,u)\in[0,1]\times[0,aR]\),
5 Illustrative examples
In this section, we give some examples to illustrate our main results.
Example 5.1
Example 5.2
Next, we provide an example when ϕ is pLaplacian operator, that is, \(\phi(u)=\varphi_{p}(u)=u^{p2}u\) (\(p>1\)). Meanwhile, we compare with the previous wellknown results of the literature [18, 19].
Example 5.3
Remark 5.1
When \(\theta(t)=t\), \(h(t)=1\), and \(A(t)\equiv0\), the system of Example 5.3 is changed into the equations of [18, 19]. In this paper, we consider the effect of timedelays and the integral boundary value conditions. Meanwhile, the operator ϕ includes the pLaplacian operator. Therefore, our study improves and extends the previous wellknown results.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their useful and valuable suggestions. This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant (No. 11161025; No. 11326101) and the Yunnan Province natural scientific research fund project (No. 2011FZ058).
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.
Authors’ Affiliations
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