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Existence and multiplicity of positive solutions for singular fractional differential equations with integral boundary value conditions
Advances in Difference Equations volumeÂ 2016, ArticleÂ number:Â 31 (2016)
Abstract
In this paper, we discuss the existence and multiplicity of positive solutions for singular fractional differential equations with integral boundary value conditions
where \(3<\alpha<4\), \(0<\eta< 2\), \({}^{C} D^{\alpha}\) is the Caputo fractional derivative and f may be singular at \(u=0\). Our results are based on the LeraySchauder nonlinear alternative and a fixedpoint theorem in cones.
1 Introduction
In this paper, we discuss the following nonlinear fractional differential equations with integral boundary conditions:
where \(3<\alpha<4\), \(0<\eta< 2\), \({}^{C} D^{\alpha}\) is the Caputo fractional derivative, and f may be singular at \(u=0\).
Differential equations with fractional derivative have been proved to be strong tools in the modeling of many physical phenomena. In consequence the subject of fractional differential equations is gaining much importance and attention [1â€“3]. Some recent investigations have shown that many physical systems can be represented more accurately using fractional derivative formulations. For details, see [4â€“10].
Cabada and Wang [11] investigated the existence of positive solutions for fractional differential equations with integral boundary value conditions
by the GuoKrasnoselskii fixed point theorem, where \(2<\alpha<3\), \(0<\lambda< 2\), \({}^{C} D^{\alpha}\) is the Caputo fractional derivative, and f is continuous on \([0,1]\times[0,\infty)\).
Zhang et al. [12] considered the fractional boundary value problem with a pLaplacian operator as below:
where \(D^{\beta}_{t}\) and \(D^{\alpha}_{t}\) are the standard RiemannLiouville derivatives with \(1<\alpha\leq2\), \(0<\beta\leq1\), \(\varphi _{p}(s)=s^{p2}s\), \(p>1\), and f may be singular at \(t=0,1\) and \(x=0\). A is a function of the bounded variation and \(\int_{0}^{1}x(s)\,dA(s)\) denotes the RiemannStieltjes integral of x with respect to A. By using the method of upper and lower solutions and the Schauder fixed point theorem, the existence of positive solutions was established.
Zhou et al. [13] investigated the multiplicity of positive solutions of the nonlinear fractional differential equation boundary value problem
by means of the LeraySchauder nonlinear alternative, a fixedpoint theorem on cones, and a mixed monotone method, where \(1< q\leq2\), \(D^{q}_{0^{+}}\) is the standard RiemannLiouville derivative. The function f is a given function satisfying some assumptions.
But up to now, there are few papers that have considered the multiplicity of positive solutions with two integral boundary conditions and a nonlinear term f possessing a singularity at \(u=0\). Motivated by the results mentioned above, the aim of this paper is to establish the multiplicity of positive solutions for singular fractional differential equations with two integral boundary value conditions (1.1).
In this paper, in analogy with boundary value problems for differential equations of integer order, we first of all derive the corresponding Greenâ€™s function known as the fractional Greenâ€™s function. Here we give some properties that relate the expressions of \(G(t,s)\) and \(G(1,s)\). It is well known that cones play an important role in applying the Greenâ€™s function in research areas. Consequently problem (1.1) is reduced to an equivalent Fredholm integral equation. Finally, by using the LeraySchauder nonlinear alternative and a fixedpoint theorem in cones, the existence and multiplicity of positive solutions are obtained.
2 Background materials and Greenâ€™s function
For the readerâ€™s convenience, we present some necessary definitions from fractional calculus, both theory and lemmas. These definitions can be found in the recent literature such as [14].
Definition 2.1
[14]
For a function \(f:[0,\infty ]\rightarrow R\), the Caputo derivative of fractional order Î± is defined as
where \([\alpha]\) denotes the integer part of the real number Î±.
Definition 2.2
[14]
The RiemannLiouville fractional integral of order Î± for a function f is defined as
provided that such an integral exists.
Lemma 2.1
[14]
Let \(\alpha>0\), then the fractional differential equation
has a solution
where \(C_{i}\in R\), \(i=0,1,2,\ldots, n1 \), \(n=[\alpha]+1\).
Lemma 2.2
[14]
Let \(\alpha>0\), then
where \(C_{i}\in R\), \(i=0,1,2,\ldots, n1\), \(n=[\alpha]+1\).
In the following we present the Greenâ€™s function of a fractional differential equation with integral boundary conditions.
Lemma 2.3
Given \(y\in C(0,1)\cap L(0,1)\), \(3<\alpha <4\), and \(0<\eta< 2\), the unique solution of
is
where
Proof
By means of the Lemma 2.2, we can reduce (2.1) to the equivalent integral equation
From \(u''(0)=u'''(0)=0\), we have \(C_{2}=C_{3}=0\). Then
and by the condition \(u'(0)=u(1)=\eta\int_{0}^{1}u(s)\,ds\), we have
Then,
From the previous equality, we deduce that
Integrating the equation from 0 to 1, we have
So equation (2.4) implies that
Hence, we have
Therefore, the unique solution of (2.1) is
â€ƒâ–¡
Lemma 2.4
The function \(G(t,s)\) defined by (2.2) has the following properties:

(1)
\(G(1,s)=0\), for \(s\in[0,1]\) if and only if \(\eta=0\);

(2)
\(G(1,s)>0\), for \(s\in(0,1)\) and \(\eta\in(0,2)\);

(3)
\(tG(1,s)\leq G(t,s)\leq M_{0} G(1,s)\), for \(3<\alpha<4\), \(s\in(0,1)\) and \(\eta\in(0,2)\) where \(M_{0}=\frac{\alpha(\eta+2)}{2\eta(\alpha1)}\);

(4)
\(G(t,s)>0\), for \(t,s\in(0,1)\) and \(\eta\in(0,2)\).
Proof
Observing the expression of \(G(1,s)\), it is clear that (1) and (2) hold.
Here
In the following we will only prove (3), as (4) can be deduced directly from (3).
When \(0< t\leq s<1\), we have
Now, it is immediate to verify the following inequalities:
When \(0< s\leq t<1\), we have
and since \(s\geq ts\), we deduce that
On the other hand, we have
then the inequalities (3) are fulfilled.â€ƒâ–¡
Theorem 2.1
(LeraySchauder alternative)
Let E be a Banach space, X a closed, convex subset of E, U an open subset of X, and \(p\in U\). Suppose that \(A:\overline {U}\rightarrow X\) is a continuous, compact map, then either

(i)
A has a fixed point in UÌ…, or

(ii)
there is a \(u\in\partial U\) and \(\lambda\in(0,1) \) with \(u=\lambda AU+(1\lambda)p\).
Theorem 2.2
Let X be a Banach space, and let \(P\subset X\) be a cone. Assume \(\Omega_{1}\), \(\Omega_{2}\) are open and bounded subsets of X with \(0\in\Omega_{1}\in\overline{\Omega_{1}}\subset\Omega _{2}\), and let \(T:P\cap(\overline{\Omega_{2}}\setminus\Omega_{1})\rightarrow P\) be a completely continuous operator such that either

(i)
\(\Tu\\geq\u\\), \(u\in P\cap\partial\Omega_{1}\), and \(\Tu\\leq\ u\\), \(u\in P\cap\partial\Omega_{2}\); or

(ii)
\(\Tu\\leq\u\\), \(u\in P\cap\partial\Omega_{1}\), and \(\Tu\\geq \u\\), \(u\in P\cap\partial\Omega_{2}\).
Then the operator T has at least one fixed point in \(P\cap(\overline {\Omega_{2}}\setminus\Omega_{1})\).
3 Main results
In this section, we consider the existence and multiplicity of positive solutions of nonlinear fractional different equation
where \(3<\alpha<4\), \(0<\eta< 2\), \({}^{C} D^{\alpha}\) is the Caputo fractional derivative, and f may be singular at \(u=0\).
Theorem 3.1
Suppose that the following hypotheses hold:
 (H_{1}):

\(f:[0,1]\times(0,\infty)\rightarrow[0,\infty)\) is continuous and
$$0\leq f(t,u)=g(u)+h(u),\quad\textit{for } (t,u)\in[0,1]\times(0,\infty), $$with \(g(u)>0\) is nonincreasing and \(h(u)/g(u)\) is nondecreasing in \(u\in (0,\infty)\);
 (H_{2}):

there exists a constant \(K_{0}>0\) such that \(g(ab)\leq K_{0}g(a)g(b)\) for all \(a,b\geq0\);
 (H_{3}):

\(\int_{0}^{1}g(s)\,ds<\infty\);
 (H_{4}):

there exists a positive number r such that
$$\biggl\{ 1+\frac{h(r)}{g(r)}\biggr\} M_{0}K_{0}g\biggl( \frac{r}{M_{0}}\biggr) \int_{0}^{1}G(1,s)g(s)\,ds< r; $$  (H_{5}):

there exists a positive number \(R>r\) with
$$\biggl(1\frac{2}{\alpha}\biggr)g(R) \int_{0}^{1}G(1,s) \biggl\{ 1+\frac{h(\frac {R}{M_{0}}s)}{g(\frac{R}{M_{0}}s)} \biggr\} \,ds\geq R. $$
Then problem (3.1) has a solution u with \(r\leq\u\\leq R\).
Proof
Let \(E=C[0,1]\) be endowed with the maximum norm, \(\u\ =\max_{0\leq t\leq1}u(t)\), define the cone \(K\subset E\) by
and let
Next let \(T:K\cap(\overline{\Omega_{2}}\setminus\Omega_{1})\rightarrow E\) be defined by
First we show T is well defined. To see this, notice that if \(u\in K\cap(\overline{\Omega_{2}}\setminus\Omega_{1}) \) then \(r\leq\u\\leq R\) and \(u(t)\geq\frac{t}{M_{0}}\u\\geq\frac {t}{M_{0}}r>0\), \(0< t<1\), from (H_{1}) we have
These inequalities with (H_{3}) guarantee that \(T:K\cap(\overline {\Omega_{2}}\setminus\Omega_{1})\rightarrow E\) is well defined. If \(u\in K\cap(\overline{\Omega_{2}}\setminus\Omega_{1})\), then we have
i.e., \(Tu\in K\) so \(T: K\cap(\overline{\Omega_{2}}\setminus \Omega_{1})\rightarrow K\).
Next we show \(T: K\cap(\overline{\Omega_{2}}\setminus\Omega _{1})\rightarrow K\) is continuous and compact. Let \(u_{n},u\in K\cap (\overline{\Omega_{2}}\setminus\Omega_{1})\) with \(\u_{n}u\\rightarrow0\) as \(n\rightarrow\infty\). Of course \(r\leq\u_{n}\\leq R\), \(r\leq\u\\leq R\), \(u_{n}(t)\geq\frac{tr}{M_{0}}>0 \) and \(u(t)\geq\frac{tr}{M_{0}}>0\), for \(0< t<1\). So
and
for \(s\in(0,1)\). Now this together with the Lebesgue dominated convergence theorem guarantees that
Therefore, \(T: K\cap(\overline{\Omega_{2}}\setminus\Omega _{1})\rightarrow K\) is continuous.
Next, we show that T is uniformly bounded. For \(u\in K\cap (\overline{\Omega_{2}}\setminus\Omega_{1})\) we have
Hence, \(T: K\cap(\overline{\Omega_{2}}\setminus\Omega_{1})\rightarrow K\) is bounded.
Finally, we show that T is equicontinuous.
For all \(\epsilon>0\), each \(u\in K\cap(\overline{\Omega_{2}}\setminus \Omega_{1})\), \(t,t'\in[0,1]\), \(t< t'\), since \(G(t,s)\) is uniformly continuous on \(t,s\in[0,1]\times[0,1]\), there exists \(\eta>0\), such that when \(t't<\eta\) we have
By means of the ArzelaAscoli theorem, \(T: K\cap(\overline{\Omega _{2}}\setminus\Omega_{1})\rightarrow K\) is compactly continuous.
Now we prove that
To see this, let \(u\in K\cap\partial\Omega_{1}\), then \(\u\=r\) and \(u(t)\geq\frac{tr}{M_{0}}>0\) for \(t\in(0,1)\), and we have
Therefore, \(\Tu\\leq\u\\), i.e. (3.3) holds.
Finally, we prove that
To see this, let \(u\in K\cap\partial\Omega_{2}\), then \(\u\=R\) and \(u(t)\geq\frac{tR}{M_{0}}>0\) for \(t\in(0,1)\),
This implies that (3.4) holds.
It follows from Theorem 2.2, (3.3), and (3.4) that T has a fixed point \(\widetilde{u}\in K\cap(\overline{\Omega_{2}}\setminus\Omega _{1})\). Clearly this fixed point is a positive solution of (3.1) satisfying \(r\leq\\widetilde{u}\\leq R \).â€ƒâ–¡
Theorem 3.2
Suppose the conditions (H_{1})(H_{4}) hold. In addition, assume that
 (H_{6}):

for each \(L>0\), there exists a function \(\varphi_{L}\in C[0,1]\), \(\varphi_{L}>0\), for \(t\in(0,1)\), such that \(f(t,u)>\varphi_{L}(t)\), for \((t,u)\in(0,1)\times(0,L]\). Then (3.1) has a solution u with \(0<\u\<r\).
Proof
The existence is proved using Theorem 2.1, together with a truncation technique. The idea is that we first show (3.1) has a positive solution u satisfying \(u(t)>0\) for \(t\in(0,1)\). Similarly to the proof of Theorem 3.1, we show
has a positive solution.
Since (H_{4}) holds, we can choose \(n_{0}\in\{ 1,2,\ldots\}\) such that
Let \(N_{0}\in\{ n_{0},n_{0}+1,\ldots\}\). Consider the family of integral equations
where \(n\in N_{0}\) and
Similarly to the proof of Theorem 3.1, we can easily verify that \(A_{n}\) is well defined and maps E to K. Moreover, \(A_{n} \) is continuous and completely continuous. By the LeraySchauder alternative principle, we need to consider
i.e.
where \(\lambda\in(0,1)\). We claim that any fixed point u of (3.7) for any \(\lambda\in(0,1)\) must satisfy \(\u\\neq r\). Otherwise, assume that u is a fixed point of (3.7) for some \(\lambda \in(0,1)\) such that \(\u\= r\). Then \(u(t)\geq\frac{1}{n}\) for \(t\in [0,1]\). Note that
Hence, for all \(t\in[0,1]\), we have
Thus we have the condition (H_{1}), for all \(t\in[0,1]\),
Therefore,
This a contradiction to the choice of \(n_{0}\) and the claim is proved.
Now the LeraySchauder alternative (Theorem 2.1) guarantees \(A_{n}\) has a fixed point, denoted by \(u_{n}\), \(u_{n}(t)\geq\frac{1}{n}\) in \(\overline {B_{r}}=\{u\in E: \u\< r\}\), and it satisfies
Next we claim that \(u_{n}(t)\) have a uniform positive lower bound, i.e., there exists a constant \(\delta>0\), independent of \(n\in N_{0}\), such that
for all \(n\in N_{0}\). Since (H_{6}) holds, there exists a continuous function \(\varphi_{r}(t)>0\) such that \(f(t,u(t))>\varphi_{r}(t)\) for all \((t,u)\in(0,1)\times(0,r]\). Since \(u_{n}(t)< r\), we have
In order to pass the fixed point \(u_{n}\) of the truncation equations (3.6) to that of the original equation (3.5) we need the following fact:
In fact, for all \(\epsilon>0\), each \(u_{n}\in B_{r}\), \(t,t'\in[0,1]\), \(t< t'\), since \(G(t,s)\) is uniformly continuous on \((t,s)\in[0,1]\times[0,1]\), there exists \(\tau>0\), such that when \(t't<\tau\) we have
then
The facts \(\u_{n}\< r\) and (3.9) show that \(\{u_{n}\}_{n\in N_{0}}\) is a bounded and equicontinuous family on \([0,1]\). Now the ArzelaAscoli theorem guarantees that \(\{u_{n}\} _{n\in N_{0}}\) has a subsequence \(\{u_{n_{k}}\}_{n_{k}\in N_{0}}\), converging uniformly on \([0,1]\) to a function \(u\in E\). From the facts \(\u_{n}\< r\) and (3.8), u satisfies \(\delta t< u(t)< r\) for all \(t\in[0,1]\). Moreover, \(u_{n_{k}}\) satisfies the integral equation
Letting \(k\rightarrow\infty\), we arrive at
Therefore, u is a positive solution of (3.1) and satisfies \(0<\u\<r\).â€ƒâ–¡
Theorem 3.3
Suppose that (H_{1})(H_{6}) are satisfied. Then problem (3.1) has two positive solutions u, Å© with \(0<\u\<r\leq\\widetilde{u}\\leq R\).
Proof
From the proof of Theorem 3.1, we see that (3.1) has a positive solution \(\widetilde{u(t)}\) with \(r\leq\\widetilde{u}\\leq R\), and by Theorem 3.2, we see that (3.1) has another positive solution \(u(t)\) with \(0<\ u\<r\). Thus (3.1) has at least two positive solutions.â€ƒâ–¡
Example 3.1
Consider the boundary value problem
where \(3<\alpha<4\), \(0<\eta<2\), \(0< a<1\), \(b\geq0\) and \(\omega>0 \) is a given parameter. Then:

(i)
if \(b<1\), then (3.10) has at least one nonnegative solution for each \(\omega>0\);

(ii)
if \(b>1\), then (3.10) has at least one nonnegative solution for each \(0<\omega<\omega_{1}\), where \(\omega_{1}\) is some positive constant;

(iii)
if \(b>1\), then (3.10) has at least two nonnegative solutions for each \(0<\omega<\omega_{1}\).
Proof
We apply Theorem 3.3. Note that (H_{6}) holds with \(\phi _{L}(t)=L^{\alpha}\). Let
Then (H_{1}) and (H_{2}) are satisfied. Since \(0< a<1\), condition (H_{3}) is also satisfied. Now for (H_{4}) to be satisfied we need
for some \(r>0\), where
Therefore (3.10) has at least one nonnegative solution for
Note that \(\omega_{1}=\infty\) if \(b<1\), and if \(b>1\) set
The function \(l(r)\) possesses a maximum at
then \(\omega_{1}=l(r_{0})>0\). We have the desired results (i) and (ii).
If \(b>1\), condition (H_{5}) becomes
for some \(R>0\), where
Since \(b>1\), the righthand side goes to 0 as \(R\rightarrow+\infty\). Thus, for any given \(0<\omega<\omega_{1}\), it is always possible to find an \(R\gg r\) such that (3.11) is satisfied. Thus, (3.10) has an additional nonnegative solution Å©. This implies that (iii) holds.â€ƒâ–¡
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He, Y. Existence and multiplicity of positive solutions for singular fractional differential equations with integral boundary value conditions. Adv Differ Equ 2016, 31 (2016). https://doi.org/10.1186/s1366201507297
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DOI: https://doi.org/10.1186/s1366201507297