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Extremal solutions for some periodic fractional differential equations
Advances in Difference Equations volumeÂ 2016, ArticleÂ number:Â 179 (2016)
Abstract
By using the lower and upper solution method, the existence of an iterative solution for a class of fractional periodic boundary value problems,
is discussed, where \(0< h<+\infty\), \(f\in C([0, h]\times R, R)\), \(D_{0+}^{\alpha}u (t) \) is the RiemannLiouville fractional derivative, \(0<\alpha< 1\). Different from other wellknown results, a new condition on the nonlinear term is given to guarantee the equivalence between the solution of the periodic boundary value problem and the fixed point of the corresponding operator. Moreover, the existence of extremal solutions for the problem is given.
1 Introduction
Differential equations of fractional order have played a significant role in engineering, science, and pure and applied mathematics in recent years. Some researchers paid attention to the existence results of the solution of the periodic boundary value problem for fractional differential equations, such as [1â€“17]. Some recent contributions to the theory of fractional differential equations initial value problems can be found in [4, 9].
In [4], by using the fixed point theorem of Schaeffer and the Banach contraction principle, Belmekki et al. obtained the Greenâ€™s function and gave some existence results for the nonlinear fractional periodic problem
where \(f: [0, 1] \times R \to R\) is continuous and the following assumptions hold:

(1)
there exists a constant \(M >0\) such that
$$\bigl f(t, u) \bigr\le M, \quad\mbox{for each } t \in(0, 1), u \in R, $$ 
(2)
there exists a constant \(k > 0\) such that
$$\biglf(t, u)  f(t, v)\bigr \le k uv, \quad\mbox{for each } t \in(0, 1), u, v \in R. $$
The above conditions (see LemmaÂ 4.2 of [4]) are very strong.
In [13], Wei et al. discussed the properties of the wellknown MittagLeffler function, and consider the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a RiemannLiouville fractional derivative
by using the monotone iterative method. In this result, the bounded demand of f in [13] and the monotone demand of f in [9] were removed. However, the application of LemmaÂ 1.1 in the proof of TheoremÂ 3.1 was not correct, due to \(\sigma(\eta)(t) \notin C[0, T]\). In other words, the definition of operator A may be not appropriate. Consequently, while the uniqueness result was correct, the existence of an extremal result was maybe wrong.
In [14], Wei and Dong studied the existence of solutions of the following periodic boundary value problem:
where \(D_{0+}^{\alpha}\) is the standard RiemannLiouville fractional derivative, \(D_{0+}^{2\alpha}u = D_{0+}^{\alpha}(D_{0+}^{\alpha}u)\) is the sequential RiemannLiouville fractional derivative, \(0 < T < \infty\), and f defined on \([0, T] \times R^{2}\) is continuous. The methods used in [14] are monotone iterative techniques and the Schauder fixed point theorem under the assumptions that there the upper and lower solutions exist.
In this paper, we will focus our attention on the following problem:
where \(f\in C([0, h]\times R, R)\), \(D_{0+}^{\alpha}u (t) \) is the RiemannLiouville fractional derivative, \(0<\alpha< 1\). The existence of the solution is obtained by the use of the upper and lower solution method which has been used by authors to deal with the fractional initial value problems [2].
The remainder of this paper is as follows. In SectionÂ 2, we recall some notions and the theory of the fractional calculus. SectionÂ 3 is devoted to the study of the existence of a solution utilizing the method of upper and lower solutions. The existence of extremal solutions is given. An example is given to illustrate the main result.
2 Preliminaries
Given \(0 \le a < b <+\infty\) and \(r>0\), define
Clearly, \(C_{r}[a, b]\) is a linear space with the normal multiplication and addition. Given \(u \in C_{r}[a, b]\), define
then \((C_{r}[a, b], \\cdot\)\) is a Banach space.
Lemma 2.1
([13])
For \(0 < \alpha\le1\), \(\lambda\ge0\), the MittagLeffler type function \(E_{\alpha, \alpha}(\lambda t^{\alpha}) \) satisfies
Lemma 2.2
The linear periodic problem
where \(\lambda\ge0\) is a constant and \(q \in L(0, h)\), has the following integral representation of the solution:
Proof
According to [8], for every initial condition
the unique solution of equation (2.1) is given by
Specially, choose \(u_{0}\) as
then \(u(t)\) satisfies the periodic boundary condition (2.2). That is to say that the linear periodic problem (2.1), (2.2) has the following integral representation of the solution:
The proof is complete.â€ƒâ–¡
Lemma 2.3
([18])
Suppose that E is an ordered Banach space, \(x_{0}, y_{0} \in E\), \(x_{0} \le y_{0}\), \(D=[x_{0}, y_{0}]\), \(T: D \to E\) is an increasing completely continuous operator and \(x_{0} \le Tx_{0}\), \(y_{0} \ge Ty_{0}\). Then the operator T has a minimal fixed point \(x^{*}\) and a maximal fixed point \(y^{*}\). If we let
then
Definition 2.1
A function \(v(t) \in C_{1\alpha}[0, h]\) is called a lower solution of problem (1.1), (1.2), if it satisfies
Definition 2.2
A function \(w(t) \in C_{1\alpha}[0, h]\) is called an upper solution of problem (1.1), (1.2), if it satisfies
3 The main results
The following assumptions will be used in this section:

(S1)
\(f: [0, h] \times R \to R\) is continuous and there exist constants \(A, B \ge0\) and \(0 < r_{1} \le1 < r_{2}<1/(1\alpha)\) such that for \(t \in[0, h]\)
$$ \biglf(t, u)  f(t, v)\bigr \le Auv^{r_{1}} + B uv^{r_{2}},\quad u, v \in R. $$(3.1)
Theorem 3.1
Suppose (S1) holds. Then u solves problem (1.1), (1.2) if and only if it is a fixed point of the operator \(T_{\lambda}: C_{1\alpha} [0, h] \to C_{1\alpha} [0, h]\) defined by
where \(\lambda\geq0\) is a constant.
Proof
First of all, we show that the operator \(T_{\lambda}\) is well defined. Clearly \(t^{\alpha1} E_{\alpha, \alpha}(\lambda t^{\alpha}) \in C_{1\alpha}[0, h]\), so it is enough to prove that for every \(u \in C_{1\alpha}[0, h] \), the function
belongs to \(C_{1\alpha}[0, h]\). Taking into account that f is continuous on \([0, h] \times R\), for \(u \in C_{1\alpha}[0, h]\), we have
On the other hand, under the condition (S1), we have
where \(C= \max_{t \in[0, h]}f(t, 0)\).
By LemmaÂ 2.1, for \(u \in C_{1\alpha}[0, h]\), we have
That is to say that
The above inequalities and the assumption \(0< r_{1} \le1 < r_{2} < 1/(1\alpha)\) imply that
Combining with the fact that \(\lim_{t \to0+}E_{\alpha, \alpha }(\lambda t^{\alpha}) = E_{\alpha, \alpha}(0)=1/\Gamma(\alpha)\) yields
The above arguments combined with LemmaÂ 2.2 imply that the fixed point of the operator \(T_{\lambda}\) solves the periodic boundary value problem (1.1), (1.2), and vice versa. The proof is complete.â€ƒâ–¡
In the following, we consider the compactness of the set of the space \(C_{r}[0, h]\).
Let \(F \subset C_{r}[0, h]\) and \(E= \{g(t)= t^{r} h(t) \mid h(t)\in F\}\), then \(E \subset C[0, h]\). It is clear that F is a bounded set of \(C_{r}[0, h]\) if and only if E is a bounded set of \(C[0, h]\).
Therefore, to prove that \(F \subset C_{r}[0, h]\) is a compact set, it is enough to prove that \(E \subset C[0, h]\) is a bounded and equicontinuous set.
Theorem 3.2
Suppose (S1) holds. Then the operator \(T_{\lambda}: C_{1\alpha}[0, h] \to C_{1\alpha}[0, h]\) is completely continuous.
Proof
Given \(u_{n} \to u \in C_{1\alpha}[0, h]\), with the definition of \(T_{\lambda}\), the condition (S1), and LemmaÂ 2.1, one has
That is to say that \(T_{\lambda}\) is continuous.
Suppose that \(F \subset C_{1\alpha}[0, h]\) is a bounded set and there is a positive constant M such that \(\u\ \le M\) for \(u \in F\). The proof process of TheoremÂ 3.1 shows that \(T_{\lambda}(F) \subset C_{1\alpha}[0, h] \) is bounded.
We omit the proof details of the equicontinuity of \(T(F)\) here and refer the reader to [2] for a similar details. The proof is complete.â€ƒâ–¡
Theorem 3.3
Assume (S1) hold and \(v, w \in C_{1\alpha}[0, h]\) are lower and upper solutions of problem (1.1), (1.2), respectively, such that
Moreover, \(f: [0, h] \times R \to R\) satisfies
Then the fractional periodic boundary value problem (1.1), (1.2) has a minimal solution \(x^{*}\) and a maximal solution \(y^{*}\) such that
Proof
Clearly, if the functions v, w are lower and upper solutions (or strict) of problem (1.1), (1.2), then there are \(v \le T_{\lambda}v\), \(w \ge T_{\lambda}w\) (or the inequality is strict). In fact, by the definition of the lower solution, there exist \(q(t) \ge0\) and \(\epsilon\ge0\) such that
By the use of TheoremÂ 3.1 and LemmaÂ 2.1, one has
Similarly, we have \(w \ge T_{\lambda}w\).
By condition (3.3) and TheoremÂ 3.2, the operator \(T_{\lambda}: C_{1\alpha }[0, h]\to C_{1\alpha}[0, h]\) is an increasing completely continuous operator. Setting \(D:= [v, w]\), by the use of LemmaÂ 2.3, the existence of \(x^{*}\), \(y^{*}\) is obtained. The proof is complete.â€ƒâ–¡
Remark 3.1
The main result is a consequence of the classical monotone iterative technique [19, 20]. However, the periodic condition is not the same.
Example 3.1
Consider the following periodic fractional boundary value problem:
where \(\alpha=0.3\), \(h=0.7\), \(f(t,u)=\frac{t}{10} [1+u(t)]\). Obviously, the function \(f(t, u)\) satisfies condition (3.3) and (S1), \(f(t,0)\geq0\), and \(f(t,0) \not\equiv0\) for \(t \in[0, h]\). Thus, \(v(t) \equiv0\) is a lower solution of problem (3.4), (3.5). Choose \(u(t) = 2t^{\alpha1}\operatorname{Cos}[2t] +t^{\alpha}\), one can check that \(u\in C_{1\alpha}[0, h]\) is an upper solution of problem (3.4), (3.5), and \(v(t) \leq u(t)\) for \(t \in[0, h]\). By the use of TheoremÂ 3.3, problem (3.4), (3.5) has at least one solution.
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Acknowledgements
The authors express their sincere thanks to the anonymous reviews for their valuable suggestions and corrections for improving the quality of the paper. This work is supported by NSFC (11571207), the Taishan Scholar project.
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Zhang, W., Bai, Z. & Sun, S. Extremal solutions for some periodic fractional differential equations. Adv Differ Equ 2016, 179 (2016). https://doi.org/10.1186/s1366201608694
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DOI: https://doi.org/10.1186/s1366201608694