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Solvability for some class of multiorder nonlinear fractional systems
 Yige Zhao^{1}Email authorView ORCID ID profile,
 Xinyi Hou^{1},
 Yibing Sun^{1} and
 Zhanbing Bai^{2}
https://doi.org/10.1186/s1366201919702
© The Author(s) 2019
 Received: 22 August 2018
 Accepted: 15 January 2019
 Published: 23 January 2019
Abstract
The existence of some class of multiorder nonlinear fractional systems is investigated in this paper. Some sufficient conditions of solutions for multiorder nonlinear systems are obtained based on fixed point theorems. Our results in this paper improve some known results.
Keywords
 Nonlinear fractional differential system
 Fractional Green’s Function
 Fixed point theorem
 Multiorder
MSC
 34A08
 34B18
1 Introduction
Fractional calculus has drawn people’s attention extensively. This is because of its extensive development of the theory and its applications in various fields, such as physics, engineering, chemistry and biology; see [1–9]. To be compared with integer derivatives, fractional derivatives are used for a better description of considered material properties, and the design of mathematical models by the differential equations of fractional order can more accurately describe the characteristics of the realworld phenomena; see [4, 7, 8]. Recently, many papers about the solvability for fractional equations have appeared; see [10–18].
Furthermore, the study of fractional systems has also been a topic focused on; see [19–25]. Although the coupled systems of fractional boundary value problems have been considered by some authors, coupled systems with multiorder fractional orders are seldom discussed. The orders of the nonlinear fractional systems which are considered in the existing papers belong to the same interval \((n, n+1]\) (\(n\in \mathbb{N}^{+}\)); see [19–24].
Motivated by all the work above, in this paper we consider the existence of boundary value problem for multiorder nonlinear differential system (1) without the conditions \(f(t,0)\equiv 0\) and \(g(t,0)\equiv 0\). Our analysis relies on the Schauder fixed point theorem and the Banach contraction principle. Some sufficient conditions of the existence of boundary value problem for the multiorder nonlinear fractional differential systems are given. Our results in this paper improve some wellknown results in [25]. Finally, we present examples to demonstrate our results.
The plan of the paper is as follows. Section 2 gives some preliminaries to prove our main results. Section 3 considers the solvability of multiorder nonlinear system (1) by the Schauder fixed point theorem and the Banach contraction principle. Section 4 presents illustrative examples to verify our new results, which is followed by a brief conclusion in Sect. 5.
2 Preliminaries
In this section, we give some definitions and lemmas about fractional calculus; see [25–27].
Definition 2.1
([26])
Definition 2.2
([26])
For the solutions of fractional equations which are expressed based on Green’s function refer to Lemma 2.3 and Lemma 2.5 in [25].
Lemma 2.1
 \((C1)\) :

\(G_{1}(t,s)>0\), for \(t, s\in (0,1)\);
 \((C2)\) :

\(q_{1}(t)k_{1}(s)\leq {\varGamma (\alpha )}G_{1}(t,s) \leq (\alpha 1)k_{1}(s)\), for \(t, s\in (0,1)\), where \(q_{1}(t)=t^{\alpha 1}(1t)\), \(k_{1}(s)=s(1s)^{\alpha 1}\).
Lemma 2.2
 \((D1)\) :

\(G_{2}(t,s)>0\), for \(t, s\in (0,1)\);
 \((D2)\) :

\((\beta 2)q_{2}(t)k_{2}(s)\leq {\varGamma (\beta )}G_{2}(t,s) \leq M_{0} k_{2}(s)\), for \(t, s\in (0,1)\), where \(M_{0}=\max \{ \beta 1, (\beta 2)^{2}\}\), \(q_{2}(t)=t^{\beta 2}{(1t)^{2}}\), \(k _{2}(s)={s^{2}}(1s)^{\beta 2}\).
We recall the following fixed point theorem for our main results.
Lemma 2.3
([27])
 \((E1)\) :

A has a fixed point in U; or
 \((E2)\) :

there exist a \(u\in \partial U\), and a \(\lambda \in (0,1)\) with \(u=\lambda Au\).
3 Main results
In this section, we establish the existence of multiorder nonlinear fractional systems (1).
\(I=[0,1]\), and \(C(I)\) denotes the space of all continuous real functions defined on I. \(P=\{x(t)x\in C(I)\}\) denotes a Banach space endowed with the norm \(\x\_{P}=\max_{t\in I}x(t)\). We define the norm by \(\(x,y)\_{P\times P}=\max \{\x\_{P},\y\_{P}\}\) for \((x,y)\in P \times P\), then \((P\times P,\\cdot \_{P\times P})\) is a Banach space.
Lemma 3.1
Suppose that \(f, g: I\times [0,+\infty )\to [0,+\infty )\) are continuous. Then \((x,y)\in P\times P\) is a solution of (1) if and only if \((x,y)\in P\times P\) is a solution of system (2).
This proof can be referred to that of Lemma 3.3 in [24], so it is omitted.
Theorem 3.1
 \((H_{1})\) :

There exist two nonnegative functions \(a_{1}(t), b _{1}(t)\in L(0,1)\) and two nonnegative continuous functions \(p(x), q(x):[0,+\infty )\to [0,+\infty )\) such that \(f(t,x)\leq a _{1}(t)+p(x)\), \(g(t,x)\leq b_{1}(t)+q(x)\);
 \((H_{2})\) :

\(\lim_{x\to +\infty }\frac{p(x)}{x}<\bar{A}\), \(\lim_{x\to +\infty }\frac{q(x)}{x}<\bar{B}\).
Proof
Theorem 3.2
 \((H_{3})\) :

There exist two nonnegative functions \(a_{2}(t), b _{2}(t)\in L(0,1)\) such that \(f(t,x)\leq a_{2}(t)+d_{1}x^{\rho _{1}}\), \(g(t,x)\leq b_{2}(t)+d_{2}x^{\rho _{2}}\), where \(d_{i}\geq 0\), \(0< \rho _{i}<1\) for \(i=1, 2\);
 \((H_{4})\) :

\(f(t,x)\leq d_{1}x^{\rho _{1}}\), \(g(t,x)\leq d_{2}x^{ \rho _{2}}\) where \(d_{i}>0\), \(\rho _{i}>1\) for \(i=1, 2\).
Proof
Theorem 3.3
 \((H_{5})\) :

There exist two nonnegative functions \(a_{3}(t), b _{3}(t)\in L(0,1)\) such that \(f(t,x_{1} )f(t,x_{2} )\leq a_{3}(t)x _{1} x_{2} \), \(g(t,x_{1} )g(t,x_{2})\leq b_{3}(t)x_{1}x_{2}\), \(t\in [0,1]\) and f, g satisfies \(f(0,0)=0\), \(g(0,0)=0\).
 \((H_{6})\) :

Suppose that \(\lambda =\max \{\lambda _{1}, \lambda _{2}\}<1\), where$$ \lambda _{1}= \int _{0}^{1}\frac{(\alpha 1)k_{1}(s)a_{3}(s)}{\varGamma ( \alpha )}\,ds, \qquad \lambda _{2}= \int _{0}^{1} \frac{M_{0} k_{2}(s)b_{3}(s)}{\varGamma (\beta )}\,ds. $$
Proof
By the Banach contraction principle, T has a unique fixed point which is a solution of the system (1). □
4 Example
In this section, we will present examples to illustrate the main results.
Example 4.1
Choose \(a_{1}(t)=3t\), \(b_{1}(t)=2t^{2}\) and \(p(x)=7x\), \(q(x)=25x\). So \((H_{1})\) holds. Since \(\bar{A}=7.7545\), \(\bar{B}=26.1714\), thus \((H_{2})\) holds. By Theorem 3.1, the system (3) has a solution.
Remark 4.1
In Example 4.1 and Example 4.2 of [25], the systems (4.1) and (4.2) with conditions \(f(t,0)\equiv 0\) and \(g(t,0)\equiv 0\) are considered. However, in Example 4.1 of this paper, \(f(t,y)=2t+28 (t\frac{1}{2} )^{2}\), \(g(t,y)= t ^{2}+100 (t\frac{1}{2} )^{2}\), we can easily see that \(f(t,0)\not \equiv 0\) and \(g(t,0)\not \equiv 0\). Thus, it is clear that one cannot deal with the system (3) of this paper by the method presented [25].
Example 4.2
Note that \(a_{2}(t)=b_{2}(t)=0\) and \(d_{1}=d_{2}=\frac{81}{256}\). By Theorem 3.2, the system (4) has a solution.
5 Conclusion
We have considered the solvability of some class of multiorder nonlinear fractional differential systems in this paper. Some sufficient conditions for multiorder nonlinear differential systems have been established by fixed point theorems. Our results improve the work presented in [25].
In future work, one can study the stability and the stabilization problems for multiorder nonlinear fractional differential systems which concern the existence of solutions.
Declarations
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.
Funding
This research is supported by the National Natural Science Foundation of China (G61703180, G61773010, G61573215, G61601197), the Natural Science Foundation of Shandong Province (ZR201702200024, ZR2017LF012, ZR2016AL02), the Project of Shandong Province Higher Educational Science and Technology Program (J18KA230, J17KA157), and the Scientific Research Foundation of University of Jinan (1008399, 160100101).
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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