Abstract
After establishing a comparison result of the nonlinear RiemannLiouville fractional differential equation of order \({p\in(2, 3]}\), we obtain the existence of maximal and minimal solutions, and the uniqueness result for fractional differential equations. As an application, an example is presented to illustrate the main results.
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1 Introduction
In the past years, much attention has been devoted to the study of fractional differential equations due to the fact that they have many applications in a broad range of areas such as physics, chemistry, aerodynamics, electrodynamics of complex medium and polymer rheology. Many existence results of solutions to initial value problems and boundary value problems for fractional differential equations have been established in terms of all sorts of methods; see, e.g., [1–17] and the references therein. Generally speaking, it is difficult to get the exact solution for fractional differential equations. To obtain approximate solutions of nonlinear fractional differential problems, we can use the monotone iterative technique and the lower and upper solutions. This technique is well known and can be used for both initial value problems and boundary value problems for differential equations [18–20]. Recently, this method has also been applied to initial value problems and boundary value problems for fractional differential equations; see [21–33]. To the best of our knowledge, there is still little utilization of the monotone iterative method to a fractional differential equation of order \(p\in(2,3]\).
Consider the following nonlinear fractional differential equations:
where \(D^{p}\) is the standard RiemannLiouville derivative and \(p\in(2,3]\). In this paper, we give some sufficient conditions, under which such problems have extremal solutions. To formulate such theorems, we need the corresponding comparison results for fractional differential inequalities by use of the Banach fixed point theorem and the method of successive approximations. An example is added to verify assumptions and theoretical results.
For convenience, let us set the following notations:
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; see [1–4].
Definition 2.1
[1]
The RiemannLiouville fractional integral of order \(p>0\) of a function \(f:(0,\infty)\to\mathbb{R}\) is given by
provided that the righthand side is pointwise defined on \((0,\infty)\).
Definition 2.2
[1]
The RiemannLiouville fractional derivative of order \(p>0\) of a continuous function \(f:(0,\infty)\to\mathbb{R}\) is given by
where \({n1\leq p< n}\), provided that the righthand side is pointwise defined on \((0,\infty)\).
Lemma 2.1
[1]
Assume that \(u\in C(0,1)\cap L(0,1)\) with a fractional derivative of order \(p>0\) that belongs to \(C(0,1)\cap L(0,1)\). Then
for some \(c_{i}\in\mathbb{R}\), \(i=1,\ldots,N\), \(N=[p]\).
For brevity, let us take \(E=\{u: D^{p}u(t)\in C(0,1)\cap L(0,1)\}\). In the Banach space \(C[0,1]\), in which the norm is defined by \(\x\=\max_{t\in[0,1]}x(t)\), we set \(P=\{ x\in C[0,1] \mid x(t)\geq0, \forall t\in[0,1]\} \). P is a positive cone in \(C[0,1]\). Throughout this paper, the partial ordering is always given by P.
The following are the existence and uniqueness results of a solution for a linear boundary value problem, which is important for us in the following analysis.
Lemma 2.2
[6]
Let \(a\in\mathbb{R}\), \(\sigma\in C(0,1)\cap L(0,1)\) and \(2< p\leq3\), then the unique solution of
is given by
where \(G(t,s)\) is Green’s function given by
The following properties of Green’s function play an important part in this paper.
Lemma 2.3
[10]
The function \(G(t,s)\) defined by (2.2) satisfies the following conditions:

(1)
\(t^{p1}(1t)s(1s)^{p1}\leq\Gamma(p)G(t,s)\leq(p1)s(1s)^{p1}\), \(t,s\in(0,1) \),

(2)
\(t^{p1}(1t)s(1s)^{p1}\leq\Gamma(p)G(t,s)\leq(p1)t^{p1}(1t)\), \(t,s\in(0,1) \).
Lemma 2.4
Suppose that \(\sigma\in C(0,1)\cap L(0,1)\), and there exists \(M>0\) satisfying
then the linear boundary value problem
has exactly one solution given by
where
and
Proof
Using Lemma 2.2, it is easy to show that problem (2.4) is equivalent to the following integral equation:
i.e.,
where
We write in the form \(x=Tx\), where T is defined by the righthand side of (2.6). Clearly, T is an operator from \(C[0,1]\) into \(C[0,1]\). Now, we have to show that the operator T has a unique fixed point. To do this, we will prove that T is a contraction map. In fact, by Lemma 2.3, for \(x, y \in C[0,1]\), we obtain
This and condition (2.3) prove that problem (2.4) has a unique solution \(x(t)\) given by
where
Applying the method of successive approximations, it is easy to see that
Substituting (2.7) into (2.8), we get (2.5) and the proof is complete. □
Lemma 2.5
Suppose that the constant M given in Lemma 2.4 satisfies inequality (2.3) and
Then the function \(H(t,s)\) has the following properties:
Proof
It follows from the expression of \(G_{n}(t,s)\) that \(G_{n}(t,s)\leq0\) when n is odd and \(G_{n}(t,s)\geq0\) when n is even. By Lemma 2.3, we obtain that
and
Consequently, we have
(notice that (2.9) is equivalent to the inequality \(K_{1}>0\)) and
This completes the proof. □
Let
It is obvious that \(P_{1}\) is a cone and \(P_{1}\subset P\). We define the operator \(S:C[0,1]\rightarrow C[0,1]\) by
It is clear that S is a linear operator, and the operator equation \(x=S\sigma\) is equivalent to the existence of a solution for the problem
Lemma 2.6
S is a completely continuous operator and \(S(P)\subset P_{1}\).
Proof
Applying the ArzelaAscoli theorem and a standard argument, we can prove that S is a completely continuous operator. In the following, we prove that \(S(P)\subset P_{1}\). In fact, for any \(x\in P\), it follows from Lemma 2.5 that
which implies that
On the other hand, by Lemma 2.5 again, we have
which together with (2.11) implies
Therefore, \(S(P)\subset P_{1}\). This completes the proof. □
Lemma 2.7
Suppose that \(x\in E\) satisfies
where M satisfies (2.3), (2.9) and
Then \(x(t)\geq0\) for \(t\in[0,1]\).
Proof
Let \(\sigma(t)=D^{p} x(t)+Mx(t)\) and \(a=x(1)\). Then
By Lemma 2.4, (2.5) holds. By the proof of Lemma 2.5, we have
Thus, by (2.5) and Lemma 2.5, we have that \(x(t)\geq0\) for \(t\in[0,1]\), and the lemma is proved. □
Lemma 2.8
[34]
Suppose that \(S:C[0,1]\rightarrow C[0,1]\) is a completely continuous linear operator and \(S(P)\subset P\). If there exist \(\psi\in C[0,1]\setminus(P)\) and a constant \(c>0\) such that \(cS\psi\geq\psi\), then the spectral radius \(r(S)\neq0\) and S has a positive eigenfunction corresponding to its first eigenvalue \(\lambda_{1}=(r(S))^{1}\), i.e., \(\varphi=\lambda_{1}S\varphi\).
Lemma 2.9
Suppose that S is defined by (2.10), then the spectral radius \(r(S)\neq0\) and S has a positive eigenfunction \(\varphi^{*}(t)\) corresponding to its first eigenvalue \(\lambda_{1}=(r(S))^{1}\).
Proof
By Lemma 2.5, \(H(t,s)>0\) for all \(t,s\in(0,1)\). Take \(\psi(t)=t^{p1}(1t)\). Then, for \(t\in[0,1]\), by Lemma 2.5 we have
So there exists a constant \(c>0\) such that \(c(S\psi)(t)\geq\psi(t)\), \(\forall t\in[0,1]\). From Lemma 2.8, we know that the spectral radius \(r(S)\neq0\) and S has a positive eigenfunction corresponding to its first eigenvalue \(\lambda_{1}=(r(S))^{1}\). This completes the proof. □
3 Main results
In this section, we prove the existence of extremal solutions and the uniqueness result of differential equation (1.1). We list the following assumptions for convenience.
 \((H_{1})\) :

There exist \(\alpha_{0}, \beta_{0}\in E\) with \(\alpha_{0}(t)\leq\beta _{0}(t)\) such that
$$\begin{gathered} D^{p} \alpha_{0}(t)+f \bigl(t, \alpha_{0}(t) \bigr) \geq0,\quad t\in(0,1), \alpha_{0}(0)= \alpha_{0}'(0)=0, \alpha_{0}(1)\leq0, \\ D^{p} \beta_{0}(t)+f \bigl(t, \beta_{0}(t) \bigr) \leq0, \quad t\in(0,1), \beta_{0}(0)= \beta_{0}'(0)=0, \beta_{0}(1)\geq0. \end{gathered} $$  \((H_{2})\) :

\(f\in C([0,1]\times\mathbb{R}, \mathbb{R})\) and there exists \(M>0\) such that
$$f(t,x)f(t,y)\geqM(xy), $$where \(\alpha_{0}(t)\leq y\leq x\leq \beta_{0}(t)\) and M satisfies (2.3), (2.9) and (2.12).
Theorem 3.1
Suppose that \((H_{1})\) and \((H_{2})\) hold. Then there exist monotone iterative sequences \(\{\alpha_{n}(t)\}, \{\beta_{n}(t)\}\) which converge uniformly on \([0,1]\) to the extremal solutions of problem (1.1) in the sector \(\Omega=\{v\in C[0,1]: \alpha_{0}(t)\leq v(t) \leq\beta_{0}(t), t\in[0,1]\}\).
Proof
First, for any \(\alpha_{n1}\), \(\beta_{n1}\), \(n\geq1\), we define two sequences \(\{\alpha_{n}(t)\}\), \(\{\beta_{n}(t)\}\) by relations
By Lemma 2.4, \(\{\alpha_{n}(t)\}\), \(\{\beta_{n}(t)\}\) are well defined. Moreover, \(\{\alpha_{n}(t)\}\), \(\{\beta_{n}(t)\}\) can be rewritten as follows:
where \(\mathbf{F}:C[0,1]\rightarrow C[0,1]\) is given by
Next, we show that \(\{\alpha_{n}(t)\}\), \(\{\beta_{n}(t)\}\) satisfy the property
Let \(w(t)=\alpha_{1}\alpha_{0}\). By condition \((H_{1})\), we obtain
Thus, by Lemma 2.5, we have that \(w(t)\geq0\), \(t\in[0,1]\). By a similar way, we can show that \(\beta_{1}\leq \beta_{0}\).
Let \(w(t)=\beta_{1}\alpha_{1}\). From condition \((H_{2})\), we obtain
By Lemma 2.5, we obtain \(w(t)\geq0\), \(t\in[0,1]\). Hence, we have the relation \(\alpha_{0} \leq\alpha_{1} \leq \beta_{1} \leq \beta_{0}\). It follows from \((H_{1})\) that S F is nondecreasing in the sector Ω, and then
Thus, by induction, we have
Applying the standard arguments and Lemma 2.6, we have that
uniformly on \([0,1]\), and that \(\alpha^{*}\), \(\beta^{*}\) are the solutions of boundary value problem (1.1). Furthermore, \(\alpha^{*}\) and \(\beta^{*}\) are a minimal solution and a maximal solution of (1.1) in Ω, respectively. □
The uniqueness results of a solution to problem (1.1) are established in the following theorem.
Theorem 3.2
Assume that conditions \((H_{1})\) and \((H_{2})\) hold, and that there exists \(M_{1}>0\) such that
where \(\alpha_{0}(t)\leq y\leq x\leq \beta_{0}(t)\) and \(M_{1}\) satisfies
Then BVP (1.1) has a unique solution in Ω, i.e., \(\alpha^{*}=\beta^{*}\).
Proof
It follows from the proof of Theorem 3.1 that
Let \(u(t)=\beta^{*}(t)\alpha^{*}(t)\). Then, by (3.2), we have \(u\in P\) and
Applying mathematical induction, for \(n\in\mathbb{N}\), we get
The above inequality guarantees \(\u\\leq(M+M_{1})^{n}\S\^{n}\u\\) for \(n\in\mathbb{N}\). It is easy to see that
In fact, \(\u\\neq0\) implies that \(1\leq(M+M_{1})^{n}\S\^{n}\) for \(n\in N\), and consequently, \(1\leq\lim_{n\rightarrow\infty}\sqrt[n]{(M+M_{1})^{n}\S\ ^{n}}=(M+M_{1})r(S)\), in contradiction to (3.3). □
Remark 3.1
From the point of view of differential equation, we know that (3.3) is equivalent to the inequality \(M_{1}r(S_{1})<1 \), where \(S_{1}:C[0,1]\rightarrow C[0,1]\) is defined by
At the end of this section, we give a rough estimate for \(r(S_{1})\). For \(x\in C[0,1]\), we have
which implies \(\S_{1}\\leq\frac{(p1)}{\Gamma(p+2)}\), hence \(r(S_{1})\leq\S_{1}\\leq\frac{(p1)}{\Gamma(p+2)}\). On the other hand, take \(\psi(t)=t^{p1}(1t)\), by Lemma 2.3, we have
Thus \(r(S_{1})=\lim_{n\rightarrow\infty}\sqrt[n]{\S_{1}\^{n}}\geq\frac {B(p+1,p+1)}{\Gamma(p)}\). So we have
4 Example
Consider the following problem:
Obviously, \(f(t,x)=\frac{\sqrt{\pi}}{112} (t^{2}x )^{3}\frac {\sqrt{\pi}}{112}t^{2}x^{2}\). Take \(\alpha_{0}(t)=t^{2}\), \(\beta_{0}(t)=t^{2}\), then
It shows that condition \((H_{1})\) of Theorem 3.2 holds. On the other hand, it is easy to verify that condition \((H_{2})\) and (3.2) hold for \(M=M_{1}=\frac{\sqrt{\pi}}{8}\) and \(r(S_{1})\leq\S_{1}\\leq\frac{4}{35\sqrt{\pi}}\).
Therefore, by Theorem 3.2, there exist iterative sequences \(\{\alpha_{n}\} \), \(\{\beta_{n}\}\) which converge uniformly to the unique solution in \([\alpha_{0},\beta _{0}]\), respectively.
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Acknowledgements
The Project supported by the National Natural Science Foundation of China (11371221, 11371364, 11571207), and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.
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Cui, Y., Sun, Q. & Su, X. Monotone iterative technique for nonlinear boundary value problems of fractional order \(p\in(2,3]\) . Adv Differ Equ 2017, 248 (2017). https://doi.org/10.1186/s136620171314z
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DOI: https://doi.org/10.1186/s136620171314z