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Multiple monotone positive solutions of integral BVPs for a higherorder fractional differential equation with monotone homomorphism
Advances in Difference Equations volume 2016, Article number: 20 (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.
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]).
Recently, when ϕ is pLaplacian operator, that is, \(\phi(u)=\varphi_{p}(u)=u^{p2}u\) (\(p>1\)) and the nonlinear term does not depend on the firstorder derivative, the existence problems of positive solutions of boundary value problems have attracted much attention. It is worth to notice that the oddness of pLaplacian operator play a key role dealing with these problems. The existence of positive solutions for fractional differential equations with pLaplacian operator have been studied by several authors (see [18–22] and the references therein). In fact, the oddness of some operators does not hold and is not necessary. As the improvement and generalization of pLaplacian operator, some scholars have put forward the new operator ϕ satisfying the properties of homomorphism but being not the oddness. There have been many papers involving the integer order differential equations with a homomorphism ϕ (see [23–25] and references therein). However, to the best of our knowledge, there are no results concerning the higherorder fractional differential equations with a homomorphism ϕ. In this article, motivated by the above mentioned discussion, we study the existence of at least two monotone and concave positive solutions of integral boundary value problem for the nonlinear fractional differential equation with signchanging nonlinearity and delayed or advanced arguments as follows (abbreviated by BVPs (1.1) throughout this paper):
here \(\phi:\mathbb{R}\rightarrow \mathbb{R}\) is an increasing and positive homomorphism with \(\phi(0)=0\), and \(2\leq n1<\nu\leq n\). \(D_{0^{+}}^{\nu}\) is the Caputo fractional derivative and \(f:[0,1]\times[0,+\infty)\rightarrow (\infty,+\infty)\) is continuous function. \(h\in C([0,1],(0,+\infty))\) and λ denotes linear functional on \(C([0,1])\) given by \(\lambda[u]=\int_{0}^{1}u(t)\, dA(t)\) involving Stieltjes integrals with the suitable bounded variation \(A(\cdot)\) on \([0,1]\).
Throughout this paper, it is worth to notice that \(\lambda[u]\) is not assumed to be positive to all positive u because \(dA(t)\) could be a signchanging measure. In addition, a, b, and θ will be divided into two cases as follows:
 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.
A projection \(\phi:\mathbb{R}\rightarrow \mathbb{R}\) is called an increasing and positive homomorphism if the following conditions are satisfied:

(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
The RiemannLiouville fractional integral of order \(\nu>0\) of a function \(h:(0,\infty)\rightarrow \mathbb{R}\) is given by
provided that the righthand side is pointwise defined on \((0,\infty)\).
Definition 2.2
The Caputo fractional derivative of order \(\nu>0\) of a continuous function \(h:(0,\infty)\rightarrow \mathbb{R}\) is given by
where \(n1<\nu\leq n\), provided that the righthand side is pointwise defined on \((0,\infty)\).
Lemma 2.1
(see [26])
Suppose that \(\nu>0\), \(\nu\notin\mathbb{N}\). If \(u\in C(0,1]\) and \(D_{0^{+}}^{\nu}u\in L^{1}[0,1]\), then
where n is the smallest integer greater than or equal to ν.
Now, we present the necessary definitions from the theory of cones in Banach spaces.
Definition 2.3
Let E be a real Banach space. A nonempty closed convex set \(P\subset E\) is called a cone of E if it satisfies the following two conditions:

(1)
\(u\in P\), \(k\geq0\) implies \(k u\in P\);

(2)
\(u\in P\), \(u\in P\) implies \(u=0\).
Note that every cone \(P\subset E\) induces an ordering in E given by \(u\leq v\) if \(vu\in P\).
Definition 2.4
A mapping α is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E if \(\alpha :P\rightarrow [0,+\infty)\) is continuous and
Similarly, a mapping β is called to be a nonnegative continuous convex functional on a cone P of a real Banach space E if \(\beta:P\rightarrow [0,+\infty)\) is continuous and
A mapping γ is called a sublinear functional if and only if
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.
Property \(A_{1}\). Let P be a cone in a real Banach space E and Ω be a bounded open subset of E with \(0\in\Omega\). Then a continuous functional \(\beta: P\rightarrow [0,+\infty)\) is said to satisfy Property \(A_{1}\) if one of the following conditions holds:

(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.
Property \(A_{2}\). Let P be a cone in a real Banach space E and Ω be a bounded open subset of E with \(0\in\Omega\). Then a continuous functional \(\beta: P\rightarrow [0,+\infty)\) is said to satisfy Property \(A_{2}\) if one of the following conditions holds:

(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])
Let E be a Banach space, \(P\subset E\) be a cone, and \(\Omega_{1}\), \(\Omega_{2}\) be two bounded open balls of E centered at the origin with \(0\in\Omega_{1}\) and \(\overline{\Omega}_{1}\subset\Omega_{2}\). Suppose that \(T:P\cap(\overline{\Omega}_{2}\setminus\Omega _{1})\rightarrow P\) is a completely continuous operator, α and γ are nonnegative continuous functional on P conditions such that either
 (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 \(u\in P\cap\partial\Omega_{2}\)
holds. Then T has at least one fixed point in \(P\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\).
Now we present the Green’s functions for BVPs (1.1).
Lemma 2.3
Denote \(\rho(t)\triangleq\frac{a+(1a)t}{(1a)(1b)}\). If \(\lambda[\rho]=\int_{0}^{1}\rho(t)\,dA(t)\neq1\) and any \(y\in C[0,1]\), then the unique solution of
is given by
where
and
Proof
Applying Lemma 2.1 and noting that the boundary conditions \(u''(0)=u'''(0)=\cdots=u^{(n1)}(0)=0\), BVPs (2.1) can be expressed as
Thus, by the boundary value condition \(u(0)=au(1)\) and \(u'(1)=bu'(0)+\lambda[u]\), we get
which imply that
It follows from the definition of the RiemannLiouville fractional integral that
where \(G(t,s)\) is defined by (2.2). Integrating on both sides of (2.5) from 0 to 1, we have
which implies
Thus, the solutions of BVPs (2.1) are formulated by
where \(K(t,s)\) and \(H(t,s)\) are defined by (2.3) and (2.2), respectively.
Now, we will prove the uniqueness of solution for BVPs (2.1). In fact, let \(u_{1}(t)\) and \(u_{2}(t)\) be any two solutions of (2.1). Denote \(w(t)=u_{1}(t)u_{2}(t)\), then (2.1) is changed into the following system:
Similar to the above arguments, we get \(w(t)=0\), that is, \(u_{1}(t)=u_{2}(t)\), which means that the solution for BVPs (2.1) is unique. The proof is complete. □
In the rest of this paper, we always assume that the conditions are fulfilled as follows:
 (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
Assume that the condition (H_{1}) holds and any \(x\in C[0,1]\), then the unique solution of
is given by
Lemma 2.5
Assume that Case I and the condition (H_{1}) hold. Then the Green’s function \(H(t,s)\) defined by (2.2) has the following properties:

(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
When \(0\leq t\leq s\leq1\), we have \(\frac{\partial G_{2}(t,s)}{\partial t}=\frac{(1s)^{\nu2}}{(1b)\Gamma(\nu1)}\geq0\). When \(0\leq s\leq t\leq1\), we get
Therefore, \(\frac{\partial G(t,s)}{\partial t}\geq0\), that is, \(G(t,s)\) is increasing with respect to t, which implies that
for \(s\in[0,1]\). Hence, for \(t, s\in[0,1]\),
Therefore, \(H(t,s)\) is increasing with respect to t, which implies that
for \(s\in[0,1]\). The proof is complete. □
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
Assume that Case I and the condition (H_{1}) hold. If \(x\in C([0,1],[0,+\infty))\), then the unique solution \(u(t)\) of BVPs (2.7) satisfies

(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
In view of \(x\in C([0,1],[0,+\infty))\), we have \(\phi^{1} (\int_{0}^{s}x(\varsigma)\, d\varsigma )\geq0\), \(s\in[0,1]\). Together with \(H(t,s)\geq0\), \(\frac{\partial H(t,s)}{\partial t}\geq0\) and
we have
Thus we know that \(u(t)\) is increasing and concave on \([0,1]\), and \(u(0)\leq u(t)\leq u(1)=\u\\). According to the boundary condition \(u(0)=au(1)\), we obtain \(u(t)\geq u(0)=a\u\\) for any \(t\in[0,1]\). The proof is complete. □
Similar to the proof of Lemma 2.5 and Lemma 2.6, we have the following lemmas.
Lemma 2.7
Assume that Case II and the condition (H_{1}) hold. Then the Green’s function \(H(t,s)\) defined by (2.2) has the following properties:

(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
Assume that Case II and the condition (H_{1}) hold. If \(x\in C([0,1],[0,+\infty))\), then the unique solution \(u(t)\) of BVPs (2.7) satisfies

(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 \(t\in[0,1]\).
For simplicity, we introduce some important notations as follows:
It follows from \(u(0)=au(1)\) that \(H(0,s)=aH(1,s)\) and \(L_{2}=aL_{1}\).
From Lemma 2.4, we can obtain the following lemma.
Lemma 2.9
Suppose that conditions (H_{1}) and (H_{2}) hold, then \(u\in E\) is a solution of BVPs (1.1) if and only if \(u\in E\) is a solution of the integral equation
Define \(F, T: E\rightarrow E\) to be the operators defined as
where \(f^{+}(t,u(\theta(t)))=\max\{f(t,u(\theta(t))),0\}\). Then, by Lemma 2.9, the existence of solutions for BVPs (1.1) is translated into the existence of the fixed point for \(u = Fu\), where F is given by (2.8). Thus, the fixed point of the operator F coincides with the solution of problem (1.1).
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.
Step 2. We will prove that T maps bounded sets into bounded sets. Indeed, it is enough to show that for any \(\rho>0\) there exists a positive constant M such that, for each \(u\in\Omega_{\rho}=\{u\in E: \u\\leq\rho\}\), \(\Tu\\leq M\) when \(h(t)\leq l_{1}\), \(f^{+}(t,u(\theta(t)))\leq l_{2}\), where \(l_{i}\) (\(i=1,2\)) are some fixed positive constants. In fact, for each \(t\in[0,1]\), \(u\in\Omega_{\rho}\), by Lemma 2.5, we have
which imply that \(\Tu\\leq M\).
Step 3. T is equicontinuous. In fact, since \(H(t,s)\) are continuous on \([0,1]\times[0,1]\), it is uniformly continuous on \([0,1]\times[0,1]\). Thus, for fixed \(s\in[0,1]\) and for any \(\epsilon>0\), there exists a constant \(\delta>0\) such that, for any \(t_{1}, t_{2}\in[0,1]\) with \(t_{2}t_{1}<\delta\), we have \(H(t_{2},s)H(t_{1},s)<\frac{\epsilon}{\phi^{1}(l_{1}l_{2})}\). Then
which means that \(T(\Omega_{\rho})\) is equicontinuous on \([0,1]\). Thus, by means of the ArzelaAscoli theorem, we see that \(T: E\rightarrow E\) is completely continuous. Similarly, it is easy to show that \(F: E\rightarrow E\) is also completely continuous. The proof is complete. □
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.
Let the cone \(P\subset E\) be defined by
The operators \(F, T: E\rightarrow E\) are defined as (2.8). It follows from Lemma 2.10 that the operators \(T, F: P\rightarrow P\) are all completely continuous. Define two continuous functionals α and γ on the cone P as follows:
It is clear that \(\alpha(u)\leq\gamma(u)\) for all \(u\in P\).
Theorem 3.1
Assume that (H_{1}) and (H_{2}) hold. If there exist constants \(r, R\in(0,\infty)\) with \(r\leq aR\) such that the following conditions are satisfied:
 (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 \((t,u)\in[0,1]\times[0,R]\),
then BVPs (1.1) have at least two increasing and concave positive solutions \(u_{1}(t)\), \(u_{2}(t)\), which satisfy \(0\leq u_{2}(t)< r< u_{1}(t)\leq R\) for \(t\in[0,1]\).
Proof
Set \(\Omega_{1}=\{u\in E:\alpha(u)< r\}\). For any \(u\in P\cap\partial\Omega_{1}\), we have \(r=\alpha(u)=u(0)\leq u(t)\leq u(1)=\frac{u(0)}{a}=\frac{\alpha(u)}{a}=\frac{r}{a}\leq R\) for \(t\in[0,1]\) and \(r\leq u(\theta(t))\leq R\) for \(0\leq t\leq\theta(t)\leq1\). It follows from condition (C_{1}) that
By (2.8), (3.2), and (3) of Lemma 2.6, we get
Therefore,
On the other hand, taking \(\Omega_{2}=\{u\in E:\gamma(u)< R\}\) and \(u\in\overline{\Omega}_{1}\), we have \(r\geq\alpha(u)=u(0)=au(1)=a\gamma(u)\). Thus \(R\geq \frac{r}{a}\geq\gamma(u)\), that is, \(u\in\Omega_{2}\) implies \(\overline{\Omega}_{1}\subseteq\Omega_{2}\). For any \(u\in P\cap\partial\Omega_{2}\), we obtain \(aR=a\gamma(u)=au(1)=u(0)\leq u(t)\leq u(1)=\gamma(u)=R\) for \(t\in[0,1]\) and \(aR\leq u(\theta(t))\leq R\) for \(0\leq t\leq\theta(t)\leq1\). It follows from condition (C_{2}) that
According to (2) of Lemma 2.6 and (3.5), we have
So
Noting that we have the condition (C_{3}), namely, \(f^{0}\in[0,\phi(L_{2})]\), for \(\varepsilon=\phi(L_{2})f^{0}\geq0\), we know that there exists a sufficiently small constant \(r_{0}\in(0,r)\) such that
Set \(\Omega_{3}=\{u\in E:\gamma(u)< r_{0}\}\). It is clear that \(\overline{\Omega}_{3}\subseteq\Omega_{1}\). For any \(u\in P\cap\partial\Omega_{3}\), we have \(ar_{0}=a\gamma(u)=au(1)=u(0)\leq u(t)\leq u(1)=\gamma(u)=r_{0}\) for \(t\in[0,1]\) and \(ar_{0}\leq u(\theta(t))\leq r_{0}\) for \(0\leq t\leq\theta(t)\leq1\). It follows from (3.2) and (3.8) that
Equation (3.9) and (2) of Lemma 2.6 give
which implies that
Clearly, α satisfies condition (c) of Property \(A_{1}\) and γ satisfies condition (d) of Property \(A_{2}\). By (3.4), (3.7) and condition (B_{1}) of Lemma 2.2, we know that T has a fixed point \(u_{1}\in P\cap(\overline{\Omega}_{2} \setminus\Omega_{1})\), that is, \(r\leq u_{1}\leq R\). Equations (3.4) and (3.11) together with condition (B_{2}) of Lemma 2.2 imply that T has a fixed point \(u_{2}\in P\cap(\overline{\Omega}_{1} \setminus\Omega_{3})\), namely, \(r_{0}\leq u_{2}\leq r\). It is worth noting that (3.4) is a strict inequality, that is to say, the operator T has not the fixed point on the boundary \(\partial\Omega_{1}\). Thus we obtain \(0\leq u_{2}(t)< r< u_{1}(t)\leq R\). By condition (C_{4}), we have \(f(t,u_{i})\geq0\), for \(t\in[0,1]\), that is, \(f^{+}(t,u_{i})=f(t,u_{i})\) (\(i=1,2\)). Hence, \(Fu_{i}=Tu_{i}\) (\(i=1,2\)). Consequently, BVPs (1.1) have at least two increasing and concave positive solutions with \(0\leq u_{2}(t)< r< u_{1}(t)\leq R\). The proof is complete. □
Similarly, we can get the following theorem.
Theorem 3.2
Assume that (H_{1}) and (H_{2}) hold. If there exist constants \(r, R\in(0,\infty)\) with \(r\leq R\) such that the following conditions are satisfied:
 (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 \((t,u)\in[0,1]\times[0,\frac{R}{a}]\),
then BVPs (1.1) have at least two increasing and concave positive solutions \(u_{1}(t)\), \(u_{2}(t)\) with \(0\leq u_{2}(t)< r< u_{1}(t)\leq\frac{R}{a}\) for \(t\in[0,1]\).
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.
Let the cone \(P\subset E\) be defined by
The operators \(F, T: P\rightarrow P\) are defined as (2.8). By Lemma 2.10, we know that the operators \(T, F: P\rightarrow P\) are all completely continuous. Define two continuous functionals α and γ on the cone P as follows:
Obviously, \(\alpha(u)\leq\gamma(u)\) for all \(u\in P\).
Theorem 4.1
Assume that (H_{1}) and (H_{2}) hold. If there exist constants \(r, R\in(0,\infty)\) with \(ar\leq R\) such that the following conditions are satisfied:
 (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 \((t,u)\in[0,1]\times[0,R]\),
then BVPs (1.1) have at least two decreasing and concave positive solutions \(u_{1}(t)\), \(u_{2}(t)\) with \(0\leq u_{2}(t)< r< u_{1}(t)\leq R\) for \(t\in[0,1]\).
Proof
Let \(\Omega_{1}=\{u\in E:\alpha(u)< r\}\). For any \(u\in P\cap\partial\Omega_{1}\), we have \(r=\alpha(u)=u(1)\leq u(t)\leq u(0)=au(1)=a\alpha(u)=ar\leq R\) for \(t\in[0,1]\) and \(r\leq u(\theta(t))\leq R\) for \(0\leq\theta(t)\leq t\leq1\). It follows from condition (D_{1}) that
By (2.8), (4.2), and (3) of Lemma 2.8, we get
Therefore,
Next, taking \(\Omega_{2}=\{u\in E:\gamma(u)< R\}\) and \(u\in\overline{\Omega}_{1}\), we have \(r\geq\alpha(u)=u(1)=\frac{u(0)}{a}=\frac{\gamma(u)}{a}\). Thus \(R\geq ar\geq\gamma(u)\), that is, \(u\in\Omega_{2}\) implies \(\overline{\Omega}_{1}\subseteq\Omega_{2}\). For any \(u\in P\cap\partial\Omega_{2}\), we obtain \(\frac{R}{a}=\frac{\gamma(u)}{a}=\frac{u(0)}{a}= u(1)\leq u(t)\leq u(0)=\gamma(u)=R\) for \(t\in[0,1]\) and \(\frac{R}{a}\leq u(\theta(t))\leq R\) for \(0\leq\theta(t)\leq t\leq1\). It follows from condition (D_{2}) that
In the light of (4.5) and (2) of Lemma 2.8, we have
which implies
Noting that the condition (D_{3}), namely, \(f^{0}\in[0,\phi(L_{1})]\), for \(\varepsilon=\phi(L_{1})f^{0}\geq0\), we know that there exists a sufficiently small constant \(r_{0}\in(0,r)\) such that
Set \(\Omega_{3}=\{u\in E:\gamma(u)< r_{0}\}\). Clearly, \(\overline{\Omega}_{3}\subseteq\Omega_{1}\). For any \(u\in P\cap\partial\Omega_{3}\), we have \(\frac{r_{0}}{a}=\frac{\gamma(u)}{a}=\frac{u(0)}{a}= u(1)\leq u(t)\leq u(0)=\gamma(u)=r_{0}\) for \(t\in[0,1]\) and \(\frac{r_{0}}{a}\leq u(\theta(t))\leq r_{0}\) for \(0\leq\theta(t)\leq t\leq1\). It follows from (4.2) and (4.8) that
Equation (4.9) together with (2) of Lemma 2.8 gives
So
Obviously, α satisfies condition (c) of Property \(A_{1}\) and γ satisfies condition (d) of Property \(A_{2}\). By (4.4), (4.7) and condition (B_{1}) of Lemma 2.2, we know that T has a fixed point \(u_{1}\in P\cap(\overline{\Omega}_{2} \setminus\Omega_{1})\), that is, \(r\leq u_{1}\leq R\). Equations (4.4) and (4.11) together with condition (B_{2}) of Lemma 2.2 imply that T has a fixed point \(u_{2}\in P\cap(\overline{\Omega}_{1} \setminus\Omega_{3})\), namely, \(r_{0}\leq u_{2}\leq r\). It is worth noting that (4.4) is a strict inequality, that is to say, the operator T has not the fixed point on the boundary \(\partial\Omega_{1}\). Thus we obtain \(0\leq u_{2}(t)< r< u_{1}(t)\leq R\). By condition (D_{4}), we have \(f(t,u_{i})\geq0\), for \(t\in[0,1]\), that is, \(f^{+}(t,u_{i})=f(t,u_{i})\) (\(i=1,2\)). Hence, \(Fu_{i}=Tu_{i}\) (\(i=1,2\)). Consequently, BVPs (1.1) have at least two decreasing and concave positive solutions with \(0\leq u_{2}(t)< r< u_{1}(t)\leq R\). The proof is complete. □
Similar to the above arguments, we obtain the following theorem.
Theorem 4.2
Assume that (H_{1}) and (H_{2}) hold. If there exist constants \(r, R\in(0,\infty)\) with \(r\leq R\) such that the following conditions are satisfied:
 (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 \((t,u)\in[0,1]\times[0,aR]\),
then BVPs (1.1) have at least two decreasing and concave positive solutions \(u_{1}(t)\), \(u_{2}(t)\) with \(0\leq u_{2}(t)< r< u_{1}(t)\leq aR\) for \(t\in[0,1]\).
5 Illustrative examples
In this section, we give some examples to illustrate our main results.
Example 5.1
Consider the integral boundary value problem for the fractional differential equation
where \(\nu=\frac{5}{2}\), \(a=b=\frac{1}{2}\), \(h(t)=1\), \(\theta(t)=\sqrt{t}\), \(dA(t)=(2t1)\, dt\), \(\phi(u)=\bigl \{\scriptsize{\begin{array}{l@{\quad}l} u, & u<0, \\ u^{2},& u\geq0, \end{array}} \bigr.\) and
In view of \(dA(t)=(2t1)\, dt\), we have
Obviously, the conditions (H_{1}) and (H_{2}) hold. A simple calculation gives
Choose \(r=1\), \(R=4\), obviously, \(r< aR\). Thus we get
and \(f(t,u)\geq0\) for all \((t,u)\in[0,1]\times[0,4]\). Clearly, all the conditions of Theorem 3.1 are satisfied. Therefore, we know that BVP (5.1) has at least two increasing and concave positive solutions \(u_{1}(t)\), \(u_{2}(t)\) satisfying \(0\leq u_{2}(t)<1<u_{1}(t)\leq4\) for \(t\in[0,1]\).
Example 5.2
Consider the integral boundary value problem for the fractional differential equation
where \(\nu=\frac{7}{2}\), \(a=b=2\), \(h(t)=e^{t}\), \(\theta(t)=\sin t\), \(\phi(u)=u\), \(f(t,u)= \frac{1}{10} (u^{4}+(1+t)\sqrt{2u}+t+2 )\), \((t,u)\in[0,1]\times[0,+\infty)\), and
By simple calculation, we get
\(L_{1}\approx1.217724677\), and \(L_{2}=2L_{1}\approx2.435449353\). Obviously, the conditions (H_{1}) and (H_{2}) hold.
Taking \(r=2\), \(R=3\), we obtain
and \(f(t,u)\geq0\) for all \((t,u)\in[0,1]\times[0,6]\). Thus, all the conditions of Theorem 4.2 are satisfied. So we conclude that BVP (5.2) has least two decreasing and concave positive solutions \(u_{1}(t)\), \(u_{2}(t)\) satisfying \(0\leq u_{2}(t)<2<u_{1}(t)\leq6\) for \(t\in[0,1]\).
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
Consider the integral boundary value problem for the fractional differential equation with the pLaplacian
where \(\nu=\frac{5}{2}\), \(p=3\), \(q=\frac{3}{2}\), \(a=\frac{1}{2}\), \(b=\frac{1}{21}\), \(h(t)=1\), \(\theta(t)=\sqrt{t}\), \(dA(t)=\frac{1}{5}t\,dt\), and
In view of \(dA(t)=\frac{1}{5}t\,dt\), we have
Obviously, the conditions (H_{1}) and (H_{2}) hold. By simple calculations, we have
and
Choose \(r=1\), \(R=5\), obviously, \(r< aR\). Thus we get
and \(f(t,u)\geq0\) for all \((t,u)\in[0,1]\times[0,5]\). Clearly, all the conditions of Theorem 3.1 are satisfied. Therefore, we know that BVP (5.3) has at least two increasing and concave positive solutions \(u_{1}(t)\), \(u_{2}(t)\) satisfying \(0\leq u_{2}(t)<1<u_{1}(t)\leq5\) for \(t\in[0,1]\).
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.
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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).
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Zhao, K., Liu, J. Multiple monotone positive solutions of integral BVPs for a higherorder fractional differential equation with monotone homomorphism. Adv Differ Equ 2016, 20 (2016). https://doi.org/10.1186/s1366201607434
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DOI: https://doi.org/10.1186/s1366201607434