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Positive solutions of fractional differential equation nonlocal boundary value problems
Advances in Difference Equations volume 2015, Article number: 256 (2015)
Abstract
In this paper, we study the existence and uniqueness of positive solutions for a class of higherorder nonlocal fractional differential equations with RiemannStieltjes integral boundary conditions. We firstly convert the problem to an equivalent integral equation, and then by applying a fixed point theorem of a sum operator, the existence and uniqueness of positive solutions is established. Furthermore, an iterative scheme to approximate the solution is constructed and an example is given to illuminate the application of the main results.
Introduction
In this paper, we are interested in the existence and uniqueness of positive solutions for a fractional differential equation nonlocal boundary value problem (BVP for short):
where \(D_{0+}^{\alpha}\) is the standard RiemannLiouville fractional derivative, \(f: [0, 1]\times[0, +\infty)\times[0, +\infty)\rightarrow[0, +\infty )\) and \(g: [0, 1]\times[0, +\infty)\rightarrow[0, +\infty)\) are continuous, \(\int_{0}^{1}x(s)\,dA(s)\) denotes the RiemannStieltjes integral of x with respect to A, \(A: [0, 1]\rightarrow R\) is a function of bounded variation and dA can be a signed measure. Here we also recall that the idea using a RiemannStieltjes integral with a signed measures is due to Webb and Infante in [1, 2], which can cover the multipoint boundary conditions and the integral boundary conditions in a single framework as special cases.
Differential equations have recently been proved to be a valuable tool in modeling many phenomena arising from various fields of science and engineering. In consequence, the subject of differential equations has received much attention and many results on boundary value problems of differential equations have been reported. In particular, since many phenomena arising in a variety of different areas of applied mathematics and physics, such as heat conduction, polymer rheology, chemistry physics, fluid flows and electrical networks can be reduced to nonlocal RiemannStieltjes integral boundary value problems, a lot of work has been carried out to deal with the existence of solutions of nonlocal boundary value problems by using techniques of functional analysis (see [3–13]). In [14], the existence and uniqueness of positive solutions for the following nonlocal BVP:
is investigated by using the monotone iterative technique, where dA is allowed to be a signed measure. The same problem was studied by Webb and Zima [15] and the existence of multiple positive solutions under suitable conditions on \(f(t, x)\) was established.
On the other hand, fractional differential operator is nonlocal and thus fractional differential equations serve as an excellent tool for the description of hereditary properties of various materials and processes and many physical phenomena in natural sciences and engineering, such as earthquake, traffic flow, measurement of viscoelastic material properties, electrodynamics of a complex medium, polymer rheology (see [16–24]). Recently, Ahmad and Nieto [25] discussed the nonlinear Dirichlet boundary value problems of sequential fractional integrodifferential equations in the sense of the Caputo fractional derivative, and the existence results are established by means of some standard tools of fixed point theory. Some special cases of the BVP (1) were also studied, for example, Salm [26] studied the case of multipoint boundary vale problems when \(x(1)=\sum_{i = 1}^{m2}\zeta_{i}x(\eta_{i})\) and \(g(t, x(t))\equiv0\), and Zhang and Han [27] considered a singular \((n1, n)\) conjugatetype fractional differential equation
and the existence and uniqueness of the positive solutions was obtained provided that \(f(t, x)\) satisfies some growth conditions.
Motivated by the work mentioned above, we focus on the existence and uniqueness of positive solutions for the nonlocal BVP (1) based on a fixed point theorem of a sum operator. Our work presented in this work has the following new features. Firstly, the existence and uniqueness of positive solutions are obtained, which possess a nice estimate, i.e., there exist \(\lambda>\mu >0\) such that \(\mu t^{\alpha1} \leq x^{*}(t)\leq\lambda t^{\alpha1}\); secondly, the boundary conditions are nonlocal which involve the RiemannStieltjes integral of x with respect to A, moreover, dA can be a signed measure, this implies that it can cover the multipoint and integral boundary value problems as special cases; thirdly, we also construct an iterative sequence to approximate the positive solution.
The rest of this paper is organized as follows. In Section 2, we recall some definitions and facts. In Section 3, the main results are discussed by using the properties of the Green function and a fixed point theorem of a sum operator. Finally, in Section 4, an illustrative example is also presented.
Preliminaries
We use the following notations in this paper:
Now we begin this section with some preliminaries of cone and fractional calculus. Recall that a nonempty closed convex set \(P\subset E\) is called a cone if it satisfies

(i)
\(x\in P\), \(\lambda\geq0\Rightarrow\lambda x\in P \), and

(ii)
\(x\in P\), \(x\in P\Rightarrow x=\theta\),
where \((E,\Vert \cdot \Vert )\) is a real Banach space with partially ordered by a cone \(P \subset E\), i.e., \(x\leq y\) if and only if \(yx\in P\). Cone P is called normal if there exists a constant \(N>0\) such that, for all \(x, y \in E\), \(\theta\leq x\leq y\) implies \(\Vert x\Vert \leq N\Vert y\Vert \), and N is called the normal constant. If \(x, y\in E\), the set \([x, y]=\{z\in E\mid x\leq z\leq y\}\) is called the order interval, and denote \(x\sim y\) if there exist \(\lambda>0\) and \(\mu>0\) such that \(\lambda x\leq y\leq\mu x\). Clearly, ∼ is an equivalence relation. Given \(h> \theta\) (i.e., \(h\geq\theta\) and \(h\neq \theta\)), let \(P_{h}=\{x\in E\mid x\sim h\}\).
We say that an operator \(A: E\rightarrow E\) is increasing (decreasing) if \(x\leq y\) implies \(Ax\leq Ay\) (\(Ax\geq Ay\)).
Definition 2.1
([28])
Let \(x:[a, \infty)\rightarrow R\) and \(\alpha>0\) with \(\alpha\in R\). Then the RiemannLiouville fractional integral is defined to be
whenever the right side is defined. Similarly, \(\alpha>0\) with \(\alpha\in R\), we define the αth RiemannLiouville fractional derivative to be
where \(n\in N\) is the unique positive integer satisfying \(n1\leq\alpha< n\) and \(t>a\).
Proposition 2.1
([28])
The equality
holds for \(f\in L^{1}(a, b)\).
In [29], the authors obtained the following results.
Lemma 2.1
([29])
Given \(y\in C[0, 1]\). Then the BVP:
has a unique solution
where
is the Green function of BVP (2).
Lemma 2.2
([29])
The Green function \(G(t, s)\) satisfies the following properties:

(1)
\(G(t, s)>0\), for all \(t, s\in(0, 1)\);

(2)
$$ (1t)t^{\alpha1}s(1s)^{\alpha1} \leq\Gamma(\alpha)G(t, s) \leq (\alpha1) (1t)t^{\alpha1}, \quad \textit{for } t, s\in[0, 1]. $$
The following lemmas are obtained by Zhang and Han [27].
Lemma 2.3
([27])
Given \(y\in L^{1}[0, 1]\). Then the BVP:
has a unique solution
where \(H(t, s)\) is the Green function of BVP (3) and is given by
Lemma 2.4
Let \(0\leq\Lambda<1\) and \(\mathscr{G}_{A}(s)\geq0\) for \(s\in[0, 1]\). Then the Green function defined by (4) satisfies the following properties:

(1)
\(H(t, s)>0\), for all \(t, s\in(0, 1)\);

(2)
$$ \frac{1}{1\Lambda}t^{\alpha1}\mathscr{G}_{A}(s) \leq H(t, s)\leq \biggl(\frac{\Vert \mathscr{G}_{A}(s)\Vert }{1\Lambda} +\frac{1}{\Gamma(\alpha1)} \biggr)t^{\alpha1}, \quad \textit{for } t, s\in[0, 1]. $$
We recall the following lemmas and definitions which are important to prove our main results.
Definition 2.2
([30])
An operator \(A: E\rightarrow E\) is said to be positive homogeneous if it satisfies \(A(tx)=tAx\), \(\forall t>0\), \(x\in E\). An operator \(A: P\rightarrow P\) is said to be subhomogeneous if it satisfies
Definition 2.3
([30])
Let r be a real number with \(0\leq r<1\). An operator \(A: P\rightarrow P\) is said to be rconcave if it satisfies
Lemma 2.5
([31])
Let \(h>\theta \) and \(\beta\in(0, 1)\), \(A: P\times P\rightarrow P\) is a mixed monotone operator satisfying
and \(B: P \rightarrow P\) is an increasing subhomogeneous operator. Assume that

(i)
there is a \(h_{0}\in P_{h}\) such that \(A(h_{0}, h_{0})\in P_{h}\) and \(Bh_{0}\in P_{h}\);

(ii)
there exists a constant \(\delta_{0}>0\) such that \(A(x, y)\geq\delta_{0}Bx\), \(\forall x, y\in P\).
Then

(1)
\(A: P_{h}\times P_{h}\rightarrow P_{h}\), \(B: P_{h}\rightarrow P_{h}\);

(2)
there exist \(u_{0}, v_{0}\in P_{h}\) and \(\gamma\in(0, 1)\) such that
$$\gamma v_{0}\leq u_{0}< v_{0},\qquad u_{0} \leq A(u_{0}, v_{0})+ Bu_{0}\leq A(v_{0}, u_{0})+Bv_{0}\leq v_{0}; $$ 
(3)
the operator equation \(A(x, x)+Bx=x\) has a unique solution \(x^{*}\) in \(P_{h}\);

(4)
for any initial values \(x_{0}, y_{0} \in P_{h}\), constructing successively sequences
$$x_{n}= A(x_{n1}, y_{n1})+Bx_{n1}, y_{n}=A(y_{n1}, x_{n1})+By_{n1},\quad n=1, 2, \ldots, $$then \(x_{n}\rightarrow x^{*}\) and \(y_{n}\rightarrow x^{*}\) as \(n\rightarrow \infty\).
Lemma 2.6
([32])
Let \(A: P\rightarrow P\) be an increasing γconcave operator and \(B: P \rightarrow P\) is an increasing subhomogeneous operator. Assume that

(i)
there exists a \(h>\theta\) such that \(Ah\in P_{h}\) and \(Bh\in P_{h}\);

(ii)
there exists a constant \(\delta_{0}>0\) such that \(Ax\geq\delta_{0}Bx\), \(\forall x\in P\).
Then the operator equation \(Ax+Bx=x\) has a unique solution \(x^{*}\) in \(P_{h}\). Moreover, for any initial value \(y_{0} \in P_{h}\), constructing successively sequences \(y_{n}=Ay_{n1}+By_{n1}\), \(n=1, 2, \ldots\) , then \(y_{n}\rightarrow x^{*}\) as \(n\rightarrow\infty\).
Remark 2.1
If operator \(B\equiv0\), Lemma 2.5, and Lemma 2.6 still hold.
Lemma 2.7
([31])
Let \(h>\theta\) and \(\alpha\in(0, 1)\). \(A: P\times P\rightarrow P\) is a mixed monotone operator and satisfies
\(B: P\rightarrow P\) is an increasing αconcave operator. Assume that

(i)
there is a \(h_{0}\in P_{h}\) such that \(A(h_{0}, h_{0})\in P_{h}\) and \(Bh_{0}\in P_{h}\);

(ii)
there exists a constant \(\delta_{0}>0\) such that \(A(x, y)\leq\delta_{0}Bx\), \(\forall x, y\in P\).
Then

(1)
\(A: P_{h}\times P_{h}\rightarrow P_{h}\) and \(B: P_{h}\rightarrow P_{h}\);

(2)
there exist \(u_{0}, v_{0}\in P_{h}\) and \(\gamma\in(0, 1)\) such that
$$\gamma v_{0}\leq u_{0}< v_{0}, \qquad u_{0}\leq A(u_{0}, v_{0})+ Bu_{0}\leq A(v_{0}, u_{0})+Bv_{0}\leq v_{0}; $$ 
(3)
the operator equation \(A(x, x)+Bx=x\) has a unique solution \(x^{*}\) in \(P_{h}\);

(4)
for any initial values \(x_{0}, y_{0} \in P_{h}\), constructing successively the sequences
$$x_{n}= A(x_{n1}, y_{n1})+Bx_{n1},\qquad y_{n}=A(y_{n1}, x_{n1})+By_{n1}, \quad n=1, 2, \ldots, $$we have \(x_{n}\rightarrow x^{*}\) and \(y_{n}\rightarrow x^{*}\) as \(n\rightarrow\infty\).
Lemma 2.8
Assume that \(f[0, 1]\times[0, \infty)\times[0, \infty) \rightarrow[0, \infty)\) and \(g: [0, 1]\times[0, \infty) \rightarrow[0, \infty) \) are continuous. Then the BVP (1) has a unique solution
where \(H(t, s)\) is defined by (4).
Proof
By using similar method to Lemma 2.3 and standard arguments, we can show the conclusion. □
Main results
The basic space used in this paper is the space \(C[0, 1]\), it is a Banach space if it is endowed with the norm \(\Vert x\Vert ={\sup}\{\vert x(t)\vert : t\in[0, 1]\}\) for any \(x\in C[0, 1]\), and E can equip with a partial order \(x, y\in C[0, 1]\), \(x\leq y\Longleftrightarrow x(t)\leq y(t)\) for \(t\in[0, 1]\). Let \(P=\{x\in C[0, 1]\mid x(t)\geq0, t\in [0, 1]\}\). Clear P is a normal cone in \(C[0, 1]\) and the normality constant is 1.
First, we give the existence and uniqueness of positive solutions to the BVP (1).
Theorem 3.1
Assume that
 (H_{1}):

A is a function of bounded variation such that \(\mathscr{G}_{A}(s)\geq0\) for \(s\in[0, 1]\) and \(\Lambda\in[0, 1)\);
 (H_{2}):

\(f(t, x, y): [0, 1]\times[0, +\infty)\times[0, +\infty )\rightarrow[0, +\infty)\) is continuous and increasing in x and y decreasing, and there exists a constant \(\gamma\in(0, 1) \) such that
$$f\bigl(t, \lambda x, \lambda^{1}y\bigr)\geq\lambda^{\gamma}f(t, x, y),\quad \forall t\in[0, 1], x, y\in[0, +\infty); $$  (H_{3}):

\(g(t, x): [0, 1]\times[0, +\infty)\rightarrow[0, +\infty)\) is continuous and increasing in \(x\in[0, +\infty)\), \(g(t, \lambda x)\geq\lambda g(t, x)\) for \(\lambda\in(0, 1)\), \((t,x)\in [0, 1]\times[0, +\infty)\), and \(g(t, 0) \not\equiv0\);
 (H_{4}):

there exists a constant \(\delta_{0}>0\) such that \(f(t, x, y)\geq\delta_{0}g(t, x)\), \(t\in[0, 1]\), \(x, y\geq0\).
Then the BVP (1) has a unique positive solution \(x^{*}\) in \(P_{h}\), where \(h(t)=t^{\alpha1}\), \(t\in[0, 1]\). And for any initial value \(x_{0}, y_{0}\in P_{h}\), constructing successively the sequences
we have \(x_{n}(t)\rightarrow x^{*}(t)\) and \(y_{n}(t)\rightarrow x^{*}(t)\) as \(n\rightarrow\infty\).
Proof
Applying Lemma 2.8, BVP (1) is equivalent to the integral equation
Let \(A: P\times P\rightarrow E\) be the operator defined by
and \(B: P\rightarrow E\) be the operator defined by
It is simple to show that x is the solution of BVP (1) if and only if x solves the operator equation \(x=A(x, x)+ Bx\). From (H_{1}) and (H_{2}) we know that \(A: P\times P\rightarrow P\) and \(B: P\rightarrow P\). We shall prove the theorem through two steps.
Step 1. We assert that A is a mixed monotone operator and satisfies (5) and B is an increasing subhomogeneous operator. In fact, for \(x_{i}\), \(y_{i}\in P\), \(i= 1, 2\) with \(x_{1}\geq x_{2}\), \(y_{1}\leq y_{2}\), we know that \(x_{1}(t)\geq x_{2}(t)\) and \(y_{1}(t)\leq y_{2}(t)\) for all \(t\in [0, 1]\). It follows from (H_{1}), (H_{2}), and Lemma 2.4 that
which implies that \(A(x_{1}, y_{1})\geq A(x_{2}, y_{2})\). Similar to the argument of (7), we get \(Bx_{1}\geq Bx_{2}\). For any \(\lambda\in(0, 1)\) and \(x, y \in P\), together with (H_{2}), we obtain
This means \(A(\lambda x, \lambda^{1}y)\geq\lambda^{\gamma}A(x, y)\) holds for \(\lambda\in(0, 1)\), \(x, y\in P\). Therefore the operator A satisfies (5). Also, for any \(\lambda\in(0, 1)\), \(x\in P\), by (H_{3}), we get
that is, \(B(\lambda x)\geq\lambda B(x)\) for any \(\lambda\in(0, 1)\), \(x\in P\). Hence the operator B is subhomogeneous.
Step 2. Now we verify that conditions (i) and (ii) of Lemma 2.5. First, we prove that \(A(h, h)\in P_{h}\) and \(Bh\in P_{h}\). It is enough to address the following conclusions:

(a)
there exist \(a_{1}, a_{2}>0\) such that \(a_{2}h(t)\leq A_{1}(h, h)(t)\leq a_{1}h(t)\), \(t\in[0, 1]\);

(b)
there exist \(b_{1}, b_{2}>0\) such that \(b_{2}h(t)\leq B_{1}h(t)\leq b_{1}h(t)\), \(t\in[0, 1]\).
Let
It follows from (H_{2}) and Lemma 2.4 that, for any \(t\in[0, 1]\),
and
According to (H_{2})(H_{4}), we get
Due to \(g(t, 0) \not\equiv0\), we obtain
and in consequence, \(a_{1}>0\) and \(a_{2}>0\). Thus, \(a_{2}h(t)\leq A(h, h)(t)\leq a_{1}h(t)\), \(t\in[0, 1]\), and hence we get (a). An argument similar to the one used in (a) shows that (b) holds with
Hence the condition (i) of Lemma 2.5 is proved. It remains to show that the condition (ii) of Lemma 2.5 is satisfied. For \(x\in P\), and for any \(t\in[0, 1]\), taking (H_{4}) into consideration, we get
in other words, \(A(x, y)\geq\delta_{0}Bx\), \(\forall x\in P\). Therefore, an application of Lemma 2.5 implies: the operator equation \(x=A(x, x)+Bx\) has a unique positive solution \(x^{*}(t)\) in \(P_{h}\). Consequently, BVP (1) has a unique positive solution \(x^{*}(t)\) in \(P_{h}\). Moreover, for any initial value \(x_{0}, y_{0}\in P_{h}\), constructing successively the sequence
we have \(x_{n}(t)\rightarrow x^{*}(t)\), \(y_{n}(t)\rightarrow x^{*}(t)\) as \(n\rightarrow\infty\). □
Remark 3.1
In Theorem 3.1, we cannot only obtain the existence of unique positive solution, but also construct an iterative sequence for approximate the unique positive solution for any initial value in \(P_{h}\). Moreover, the estimate of unique positive solution is derived with \(\mu t^{\alpha1}\leq x^{*}(t)\leq\lambda t^{\alpha1}\) for some \(\lambda> \mu>0\). Thus the property of the unique positive solution is more clear.
If \(g(t, x(t))\equiv0\), from Remark 2.1, we have the following corollary.
Corollary 3.1
Assume that (H_{1}) and (H_{2}) hold. If \(f(t, 0) \not\equiv0\).
Then the problem
has a unique positive solution \(x^{*}\) in \(P_{h}\), where \(h(t)=t^{\alpha1}\), \(t\in[0, 1]\). Moreover, constructing successively the sequence
for any initial value \(x_{0}, y_{0}\in P_{h}\), we have \(x_{n}(t)\rightarrow x^{*}(t)\), \(y_{n}(t)\rightarrow x^{*}(t)\) as \(n\rightarrow\infty\).
If the nonlinear term \(f(t, x, x)\) is replaced by \(f(t, x)\), we can get the following results.
Theorem 3.2
Assume that (H_{1}) and (H_{3}) hold and
 (H_{5}):

\(f(t, x): [0, 1]\times[0, +\infty)\rightarrow[0, +\infty)\) is continuous and increasing with respect to the second argument, and there exists a constant \(\gamma\in(0, 1)\) such that \(f(t, \lambda x)\geq\lambda^{\gamma} f(t, x)\), \(\forall t\in[0, 1]\), \(\lambda\in(0, 1)\), \(x\in[0, \infty)\);
 (H_{6}):

there exists a constant \(\delta_{0}>0\) such that \(f(t, x)\geq\delta_{0}g(t, x)\) for \(t\in[0, 1]\), \(x\geq0\).
Then the BVP
has a unique positive solution \(x^{*}\) in \(P_{h}\), where \(h(t)=t^{\alpha1}\), \(t\in[0, 1]\). Moreover, for any initial value \(y_{0}\in P_{h}\), constructing successively the sequences
we have \(y_{n}(t)\rightarrow x^{*}(t)\) as \(n\rightarrow\infty\).
Proof
Applying Lemma 2.3, BVP (8) is equivalent to the integral formulation given by
Let \(A: P\rightarrow E\) be the operator defined by
and \(B: P\rightarrow E\) be the operator defined by
It is simple to show that \(x^{*}\) is the solution of BVP (1) if and only if \(x^{*}\) solves the operator equation \(x=Ax+Bx\). Similar to the proof of Theorem 3.1, we know A is an increasing γconcave operator and B is an increasing subhomogeneous operator.
Take
Combining the proof of Theorem 3.1 with (H_{3}), (H_{5}), (H_{6}), and Lemma 2.4, the conditions (i) and (ii) of Lemma 2.6 are satisfied. Therefore, an application of Lemma 2.6 implies: the operator equation \(x=Ax+Bx\) has a unique positive solution \(x^{*}\) in \(P_{h}\). Consequently, BVP (8) has a unique positive solution \(x^{*}\) in \(P_{h}\). Moreover, constructing successively the sequence
for any initial value \(y_{0}\in P_{h}\), we have \(y_{n}(t)\rightarrow x^{*}(t)\) as \(n\rightarrow\infty\). □
Corollary 3.2
Assume that (H_{1}) and (H_{5}) hold.
Then the problem
has a unique positive solution \(x^{*}\) in \(P_{h}\), where \(h(t)=t^{\alpha1}\), \(t\in[0, 1]\). Moreover, for any initial value \(y_{0}\in P_{h}\), constructing successively the sequences
we have \(y_{n}(t)\rightarrow x^{*}(t)\) as \(n\rightarrow\infty\).
From the proof of Theorem 3.1 and using Lemma 2.7, we can prove the following conclusion.
Theorem 3.3
Assume that (H_{1}) holds and
 (H_{7}):

\(f(t, x, y): [0, 1]\times[0, +\infty)\times[0, +\infty )\rightarrow[0, +\infty)\) is continuous and increasing in \(x\in[0, +\infty)\) for fixed \(t\in[0, 1]\), \(y\in[0, +\infty)\) decreasing in \(y\in[0, +\infty)\) for fixed \(t\in[0, 1]\), \(x\in[0, +\infty)\), and \(f(t, \lambda x, \lambda^{1}y)\geq\lambda f(t, x, y)\), \(\forall t\in [0, 1]\), \(x, y\in[0, +\infty)\);
 (H_{8}):

\(g(t, x): [0, 1]\times[0, +\infty)\rightarrow[0, +\infty)\) is continuous and increasing in \(x\in[0, +\infty)\) for fixed \(t\in[0, 1]\), and there exists a constant \(\gamma\in(0, 1) \) such that \(g(t, \lambda x)\geq\lambda^{\gamma} g(t, x)\) for \(\lambda\in(0, 1)\), \(t\in[0, 1]\), \(u\in[0, +\infty)\) and \(g(t, 0) \not\equiv0\);
 (H_{9}):

there exists a constant \(\delta_{0}>0\) such that \(f(t, x, y)\leq\delta_{0}g(t, x)\), \(t\in[0, 1]\), \(x, y\geq0\).
Then BVP (1) has a unique positive solution \(x^{*}\) in \(P_{h}\) and for any \(x_{0}, y_{0}\in P_{h}\), constructing successively the sequences
we have \(x_{n}(t)\rightarrow x^{*}(t)\) and \(y_{n}(t)\rightarrow x^{*}(t)\) as \(n\rightarrow\infty\).
If the nonlocal boundary condition \(x(1)=\int_{0}^{1}x(s)\,dA(s)\) replace by local boundary condition \(u(1)=0\), we can obtain the following results.
Corollary 3.3
Assume that (H_{2})(H_{4}) hold. Then the problem
has a unique positive solution \(x^{*}\) in \(P_{h}\), where \(h(t)=(1t)t^{\alpha1}\), \(t\in[0, 1]\). Moreover, for any initial value \(x_{0}, y_{0}\in P_{h}\), constructing successively the sequences
we have \(x_{n}(t)\rightarrow x^{*}(t)\) and \(y_{n}(t)\rightarrow x^{*}(t)\) as \(n\rightarrow\infty\).
Corollary 3.4
Assume that (H_{2}) holds. Then the problem
has a unique positive solution \(x^{*}\) in \(P_{h}\), where \(h(t)=(1t)t^{\alpha1}\), \(t\in[0, 1]\). Moreover, for any initial value \(x_{0}, y_{0}\in P_{h}\), constructing successively the sequences
we have \(x_{n}(t)\rightarrow x^{*}(t)\) and \(y_{n}(t)\rightarrow x^{*}(t)\) as \(n\rightarrow\infty\).
Corollary 3.5
Assume that (H_{3}), (H_{5}), and (H_{6}) hold. Then the problem
has a unique positive solution \(x^{*}\) in \(P_{h}\), where \(h(t)=(1t)t^{\alpha1}\), \(t\in[0, 1]\). Moreover, for any initial value \(y_{0}\in P_{h}\), constructing successively the sequences
we have \(y_{n}(t)\rightarrow x^{*}(t) \) as \(n\rightarrow\infty\).
Corollary 3.6
Assume that (H_{5}) holds. Then the problem
has a unique positive solution \(x^{*}\) in \(P_{h}\), where \(h(t)=(1t)t^{\alpha1}\), \(t\in[0, 1]\). Moreover, for any initial value \(y_{0}\in P_{h}\), constructing successively the sequences
we have \(y_{n}(t)\rightarrow x^{*}(t) \) as \(n\rightarrow\infty\).
Example
Consider the following boundary value problem:
In this case, \(\alpha=\frac{5}{2}\). Problem (9) can be regard as a boundary value problem of form (1) with
and
Now we verify that conditions (H_{1})(H_{4}) are satisfied. By a simple computation, we have
and
Then \(\Lambda\approx0.0576\) and \(\mathscr{G}_{A}(s)\geq0\) for all \(s\in [0, 1]\). This implies that (H_{1}) holds. From (10) and (11) we have f and g are continuous and increasing in \(x\in[0, \infty)\) for fixed \(t\in[0, 1]\). Moreover, for any \(\lambda\in(0, 1)\), \(t\in[0,1]\), \(x\in(0, \infty )\), we get \(\arctan(\lambda x)\geq\lambda\arctan x\). Therefore
and
where \(\gamma=\frac{1}{2}\). Thus (H_{2}) and (H_{3}) are proved and \(g(t, 0)=t^{3}\not\equiv0\). It remains to show that (H_{4}) holds. Take \(\delta_{0}\in(0, 1]\), and we obtain
Therefore, all of the conditions in Theorem 3.1 are satisfied. By using Theorem 3.1, we know that the BVP (9) has a unique positive solution in \(P_{h}\) with \(h(t)=t^{\frac{3}{2}}\).
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Tan, J., Cheng, C. & Zhang, X. Positive solutions of fractional differential equation nonlocal boundary value problems. Adv Differ Equ 2015, 256 (2015). https://doi.org/10.1186/s1366201505828
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DOI: https://doi.org/10.1186/s1366201505828
Keywords
 fractional differential equation
 boundary value problem
 fixed point theorem
 mixed monotone operators
 RiemannStieltjes integral