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Maximum and minimum solutions for a nonlocal pLaplacian fractional differential system from ecoeconomical processes
Boundary Value Problemsvolume 2017, Article number: 118 (2017)
Abstract
This paper focuses on the maximum and minimum solutions for a fractional order differential system, involving a pLaplacian operator and nonlocal boundary conditions, which arises from many complex processes such as ecological economy phenomena and diffusive interaction. By introducing new type growth conditions and using the monotone iterative technique, some new results about the existence of maximal and minimal solutions for a fractional order differential system is established, and the estimation of the lower and upper bounds of the maximum and minimum solutions is also derived. In addition, the iterative schemes starting from some explicit initial values and converging to the exact maximum and minimum solutions are also constructed.
Introduction
The interest in using fractional differential equations in modeling ecological economy and diffusive processes has wide literature. Especially when one wants to model longrange ecological economy phenomena and diffusive interaction, fractional differential operator has higher accuracy than integer order differential model in depicting the coevolution process of economic, social and ecological subsystems and the transport of solute in highly heterogeneous porous media. Recently this interest has also been activated by recent progress in the mathematical theory and psychosocioeconomical dynamics, see [1, 2]. On the other hand, since fractional order derivative, which exhibits a long time memory behavior, is nonlocal, thus except for diffusive processes in porous medium flow and ecological economy phenomena, the differential equation with fractional derivative can also describe many other physical phenomena in natural sciences and engineering, such as earthquake, traffic flow, measurement of viscoelastic material properties, polymer rheology and various material processes [3–37].
In this paper, we study the existence of maximum and minimum solutions for the following nonlocal fractional differential system:
where \(\mathscr {D}_{t}^{\alpha_{i}}\), \(\mathscr {D}_{t}^{\beta_{i}}\) are the standard RiemannLiouville derivatives satisfying \(1<\alpha _{i}, \beta_{i}\le 2\), \(\int^{1}_{0}x(s)\,dA_{i}(s)\) denotes a RiemannStieltjes integral and \(A_{i}\) is a function of bounded variation, \(\varphi_{p_{i}}\) is the pLaplacian operator defined by \(\varphi_{p_{i}}(s)= s^{p_{i}2}s\), \(p_{i}>1\), where \(i=1,2\). Obviously, \(\varphi_{p_{i}}(s)\) is invertible and its inverse operator is \(\varphi_{q_{i}}(s)\), where \(q_{i}=\frac{p_{i}}{p_{i}1}\), \(i=1,2\), are conjugate indices of \(p_{i}\).
Recently, some interesting results about the existence of positive solutions for nonlinear fractional equation with pLaplacian operator have been reported [21–26, 38]. In [23], Chen and Liu investigated the existence of solutions for the antiperiodic fractional order pLaplacian boundary value problem with the following form:
where \(0 < \alpha, \beta\le1\), \(1 < \alpha+ \beta\le2\), \(\mathscr {D}_{t}^{\beta}\) is a Caputo fractional derivative, and \(f : [0, 1]\times R^{2} \to R\) is continuous. Under certain nonlinear growth conditions of the nonlinearity, a new existence result is obtained by using Schaefer’s fixed point theorem. By means of upper and lower solutions method, Wang et al. [26] studied the existence of positive solutions for the following nonlocal fractional pLaplacian equation:
where \(1<\alpha, \beta\le2\), \(0 \le a,b\le1\), \(0<\xi,\eta<1\). More recently, Zhang et al. [38] considered the uniqueness of positive solution for the following fractional order differential equation:
where \(\mathscr {D}_{t}^{\alpha }\), \(\mathscr {D}_{t}^{\beta }\), \(\mathscr {D}_{t}^{\gamma}\) are the standard RiemannLiouville derivatives, \(\int^{1}_{0}x(s)\,dA(s)\) denotes a RiemannStieltjes integral and \(0<\gamma\le1<\alpha \le2<\beta<3\), \(\alpha\gamma>1\), A is a function of bounded variation and dA can be a signed measure. Under the case where the nonlinearity \(f(u,v)\) may be singular at both \(u=0\) and \(v=0\), the uniqueness of positive solution for equation (1.4) was established via the fixed point theorem of the mixed monotone operator.
However, to the best of our knowledge, there are relatively few results on a fractional order differential system involving the pLaplacian operator and nonlocal RiemannStieltjes integral boundary conditions, and no work has been done concerning the maximal and minimal solutions of system (1.1). Thus, motivated by the above work, in this paper, we consider the maximum and minimum solutions for a fractional order pLaplacian system subject to a nonlocal RiemannStieltjes integral boundary condition. Difference from the above mentioned work is that in this paper we introduce new type growth condition of nonlinearity which covers a large number of nonlinear functions; at the same time, the existence, estimation of the lower and upper bounds and the convergent iterative scheme of minimal and maximal solutions for system (1.1) are also established.
Preliminaries and lemmas
A number of definitions for the fractional derivative have emerged over the years, and in this paper, we carry out our work base on the sense of RiemannLiouville fractional derivatives; for details, see [15, 39, 40]. Here we only recall a famous semigroup property for RiemannLiouville fractional calculus.
Proposition 2.1
Let \(\alpha> 0\), and \(f(t)\) is integrable, then
where \(c_{i}\in\mathbb{R}\) (\(i=1,2,\ldots,n\)), n is the smallest integer greater than or equal to α.
Let \(i=1,2\), we firstly focus on the following linear fractional differential equation subject to the nonlocal RiemannStieltjes integral boundary condition:
In order to establish the existence of positive solutions for system (1.1), it is necessary to find Green’s function of BVP (2.1). The following result has been given in [38].
Lemma 2.1
see [38]
Given \(h_{i}\in L^{1}(0, 1)\) and \(1<\alpha _{i}\le2\), then the following boundary value problem
has the unique solution
where
By Proposition 2.1, it is easy to get that the unique solution of the following boundary value problem
is \(t^{\alpha_{i}1}\). Thus, by [16] and [17], we have the following lemma.
Lemma 2.2
see [16]
If \(1<\alpha _{i}\le2\) and \(h_{i}\in L^{1}[0,1]\), then the boundary value problem (2.1) has the unique solution
where
and \(G_{\alpha_{i}}(t,s)\) is defined by (2.3).
Now we introduce the following necessary condition to ensure the nonnegativity of Green’s function.

(H0)
\(A_{i}\) is functions of bounded variation satisfying \(\mathcal{G}_{A_{i}}(s)\ge0\) for \(s\in[0, 1]\) and \(0\le\mathcal{A}_{i}<1\).
Lemma 2.3
see [16]
Assume (H0) holds, then \(G_{\alpha_{i}}(t,s)\) and \(H_{i}(t,s)\) have the following properties:

(1)
\(G_{\alpha_{i}}(t,s)\) and \(H_{i}(t,s)\) are nonnegative and continuous for \((t,s)\in[0,1]\times[0,1]\).

(2)
For any \(t,s\in [0,1]\), \(G_{\alpha_{i}}(t,s)\) satisfies
$$ \frac{t^{\alpha_{i} 1}(1t)s(1s)^{\alpha_{i} 1}}{\Gamma(\alpha_{i} )}\leq G_{\alpha_{i}}(t,s) \leq \frac{\alpha_{i}1}{\Gamma(\alpha_{i} )}s(1s)^{\alpha_{i} 1}. $$(2.7) 
(3)
There exist two constants a, b such that
$$ a{t^{\alpha_{i}1}}\mathcal{G}_{A_{i}}(s)\le H_{i}(t,s)\le b t^{\alpha_{i}1}, \quad s,t\in[0,1]. $$(2.8)
For convenience of writing, for \(h_{i}\in L^{1}[0,1]\) and \(p_{i}>1\), \(h_{i}\ge0\), \(i=1,2\), we rewrite the following linear boundary value problems:
and
as
Lemma 2.4
The linear boundary value problem (2.9) has a unique positive solution
where \(q_{i}\) is conjugate indices of \(p_{i}\).
Proof
Let \(w_{i} =\mathscr {D}_{t}^{\alpha_{i}}x_{i}\), \(v_{i} = \varphi_{p_{i}}(w_{i})\), by Lemma 2.1, the unique solution of the boundary value problem
is
Noting that \(\mathscr {D}_{t}^{\alpha_{i}}x=w_{i}\), \(w_{i} = \varphi^{1}_{p_{i}}(v_{i})\) as well as (2.10), we get that the solution of (2.9) satisfies
which implies that
that is,
□
To establish the existence of positive solution of system (1.1), the following new growth condition for nonlinearity will be used in the rest of the paper.

(H1)
\(f_{1},f_{2}: [0,+\infty)\times[0,+\infty)\to(0,+\infty )\) are continuous and nondecreasing in the first variable and second variable, and there exist positive constants \(\epsilon_{1}>\frac{1}{q_{1}1}\), \(\epsilon_{2}>\frac{1}{q_{2}1}\) and M such that
$$ \max \biggl\{ \sup_{\substack{{s,t\ge0}\\{s+t\neq0}}}\frac{f_{1} ( s,t ) }{(s+t)^{\epsilon_{1}}}, \sup _{\substack{{s,t\ge0}\\{s+t\neq0}}}\frac{f_{2} ( s,t ) }{(s+t)^{\epsilon_{2}}} \biggr\} \le M. $$(2.11)
Remark 2.1
In this work, we introduce the growth condition (2.11) for the first time which differs from previous work [10, 11, 16–26, 38]. In particular, it includes a large number of nonlinear functions, some basic examples of \(f_{1}\), \(f_{2}\) satisfying (H1) are

(1)
\(f(s,t)=a_{0}+\sum_{i=1}^{m}a_{i}(s+t)^{\gamma _{i}}\), where \(a_{i},\gamma_{i}>0\), \(i=0,1,2,\ldots,m\).

(2)
\(f(s,t)= [ a+\sum_{i=1}^{m}a_{i}(s+t)^{\mu_{i}} ] ^{\frac{1}{\mu }}\), where a, μ, \(a_{i},\mu_{i}\) (\(i=1,2,\ldots,m\)) are positive constants.

(3)
\(f(s,t)={(s+t+1)^{\mu+1}}\ln ( 1+\frac{1}{1+s+t} ) +(s+t)^{\mu}+a\), \(a, \mu>0\).

(4)
\(f(s,t)=\ln(2+s+t)\).
Proof
(1)(4) are obvious, we omit the proof. □
Remark 2.2
If \(f_{i}(s,t)=g_{i}(s+t)\), then the following interesting cases are also included by (2.11):

Case 1.
There exists a constant \(\epsilon_{i}>\frac{1}{q_{i}1}\) such that \(\frac{g_{i}(x)}{x^{\epsilon_{i}}}\) is increasing on x and
$$\lim_{x\to+\infty}{\frac{g_{i}(x)}{x^{\epsilon_{i}}}}=M>0. $$ 
Case 2.
There exists a constant \(\epsilon_{i}>\frac{1}{q_{i}1}\) such that \(\frac{g_{i}(x)}{x^{\epsilon_{i}}}\) is nonincreasing on x.
Case 1 and Case 2 indicate that \(g_{i}\) can be superlinear or sublinear or mixed cases of them; moreover, this shows that assumption (2.11) is very easy to be satisfied.
Let \(E=C[0,1]\times C[0,1]\) be the Banach space of all continuous functions with the norm
Define a cone P in E
and operators \(T_{1}\), \(T_{2}\), T,
then the fixed point of operator T in E is the solution of system (1.1).
Lemma 2.5
Assume that (H0)(H1) hold. Then \(T: P \to P\) is a continuous, compact operator.
Proof
For any \((x_{1},x_{2})\in P\), we can find four nonnegative numbers \(L_{x_{i}}>l_{x_{i}}\ge0\) such that
If \((x_{1},x_{2})=(0,0)\), notice that \(f_{i}(0,0)\neq0\), then it follows from (2.8) that \(T(x_{1},x_{2})\in P\). Otherwise, by (H1), we know that \(T_{1}\) and \(T_{2}\) are increasing with respect to \(x_{1}\), \(x_{2}\). Now we divide into two cases to prove the righthand side of inequality of (2.12) is valid for T.
Case 1. If \(x_{1}(t)+x_{2}(t)\ge1\), by (2.8), we get
Case 2. If \(x_{1}(t)+x_{2}(t)<1\), we have
where
On the other hand,
where
In the same way, there exist two constants \(L^{*}_{x_{2}}>l^{*}_{x_{2}}\ge0\) such that
Therefore T is well defined and uniformly bounded and \(T(P) \subset P\).
On the other hand, according to the ArzelaAscoli theorem and the Lebesgue dominated convergence theorem, we know that \(T:P \to P\) is completely continuous. □
Main results
Lemma 3.1
Suppose \(\epsilon_{i}(q_{i}1)>1\), \(i=1,2\), then the equation
has a unique positive solution \(r^{*}\) in \([0,\infty )\).
Proof
Let
then \(\varphi(x)\) is continuous in \([0,\infty )\), and
On the other hand, we also have
Thus by (3.2)(3.5), equation (3.1) has a unique positive solution \(r^{*}\) in \([0,\infty )\). □
Theorem 3.1
Suppose conditions (H0) and (H1) hold, and
Then system (1.1) has the minimal and maximal solutions, \(x^{*}=(x^{*}_{1},x^{*}_{2})\) and \(y^{*}=(y^{*}_{1},y^{*}_{2})\), which are positive; and there exist some nonnegative numbers \(m_{i}< n_{i}\), \(i=1,2,3,4\), such that
Moreover, for initial values \(u^{(0)}(t)=(0,0)\), \(w^{(0)}(t)=(r^{*},r^{*})\), let \(\{{u^{(n)}}\}\) and \(\{{w^{(n)}}\}\) be the iterative sequences generated by
then
uniformly for \(t\in[0,1]\).
Proof
Take \(P[0, r^{*}] = \{(x_{1},x_{2}) \in P : 0 \le\x_{1}\ \le r^{*}, 0 \le\x_{2}\ \le r^{*}\}\), we firstly prove \(T(P[0, r^{*}]) \subset P[0, r^{*}]\).
In fact, for any \((x_{1},x_{2}) \in P[0, r^{*}]\), if \((x_{1},x_{2})\equiv(0,0)\), it follows from (2.8) that
and
Otherwise, for any \(t\in(0,1)\), we have
So, by (H1), we get
Similar to (3.12), we have
which implies that \(T(P[0, r^{*}]) \subset P[0, r^{*}]\).
Let \(u^{(0)}(t) =(u^{(0)}_{1}(t),u^{(0)}_{2}(t))=(0,0)\) and
it follows from \(u^{(0)}(t)\in P([0,r^{*}])\) that \(u^{(1)}(t) \in T(P[0, r^{*}])\).
Denote
It follows from \(T(P[0, r^{*}]) \subset P[0, r^{*}]\) that \(u_{n}(t) \in P[0, r^{*}]\) for \(n\ge1\). Noticing that T is compact, we get that \(\{u^{(n)}\}\) is a sequentially compact set.
On the other hand, since \(u^{(1)}(t)\ge0=u^{(0)}(t)\), we have
By induction, we get
Consequently, there exists \(x^{*}=(x^{*}_{1},x^{*}_{2}) \in P[0, r]\) such that \(u^{(n)} \to x^{*}\). Letting \(n \to+\infty\), from the continuity of T and \(Tu^{(n)} =u^{(n1)}\), we obtain \(Tx^{*} = x^{*}\), which implies that \(x^{*}=(x^{*}_{1},x^{*}_{2})\) is a nonnegative solution of the nonlinear integral equation (1.1). Since \(x^{*}\in P\), there exist constants \(0< m_{1}< n_{1}\), \(0< m_{2}< n_{2}\) such that
and consequently \(x^{*}\) is a positive solution of system (1.1).
Now let \(w^{(0)}(t) =(w^{(0)}_{1}(t),w^{(0)}_{2}(t))=(r^{*},r^{*})\) and
Since \(w^{(0)}(t)=(r^{*},r^{*}) \in P[0, r^{*}]\), and then \(w^{(1)}(t)\in P[0, r^{*}]\). Thus denote
It follows from \(T(P[0, r^{*}]) \subset P[0, r^{*}]\) that
From Lemma 2.5, T is compact, consequently \(\{w^{(n)}\}\) is a sequentially compact set.
Now, since \(w^{(1)}(t) \in P[0, r^{*}]\), we get
It follows from (H1) that \(w^{(2)}(t) = Tw^{(1)}(t) \le Tw^{(0)}(t)=w^{(1)}(t)\). By induction, we obtain
Consequently, there exists \(y^{*}(t)\in P[0, r^{*}]\) such that \(w^{(n)}(t) \to y^{*}=(y^{*}_{1},y^{*}_{2})\). Letting \(n \to+\infty\), from the continuity of T and \(Tw^{(n)}(t) = w^{(n1)}(t)\), we have \(Ty^{*} = y^{*}\), which implies that \(y^{*}\) is another nonnegative solution of the boundary value problem (1.1) and \(y^{*}\) also satisfies (3.7) since \(y^{*}\in P\).
In the end, we prove that \(x^{*}\) and \(y^{*}\) are maximum and minimum solutions for system (1.1). Let x̃ be any positive solution of system (1.1), then \(u^{(0)}=0\le\tilde{x}\le r^{*}=w^{(0)}\), and \(u^{(1)}=Tu^{(0)}\le T\tilde{x}=\tilde{x}\le T(w^{(0)})=w^{(1)}\). By induction, we have \(u^{(n)}\le\tilde{x}\le w^{(n)}\), \(n=1,2,3,\ldots\) . Taking limit, we have \(x^{*}\le\tilde{x}\le y^{*}\). This implies that \(x^{*}\) and \(y^{*}\) are the maximal and minimal solutions of system (1.1), respectively. The proof is completed. □
Remark
In particular, if \(\alpha_{i}=\beta_{i}=p_{i}=2\), then the nonlocal fractional system (1.1) will reduce to a fourth order classical beam system of ordinary differential equation,
we have the following good result.
Corollary 3.1
Suppose conditions (H0) and (H1) hold, and
then system (3.14) has the minimal and maximal solutions, \(x^{*}=(x^{*}_{1},x^{*}_{2})\) and \(y^{*}=(y^{*}_{1},y^{*}_{2})\), which are positive; and there exist some nonnegative numbers \(m_{i}< n_{i}\), \(i=1,2,3,4\), such that
Moreover, for initial values \(u^{(0)}(t)=(0,0)\), \(w^{(0)}(t)=(r^{*},r^{*})\), let \(\{{u^{(n)}}\}\) and \(\{{w^{(n)}}\}\) be the iterative sequences generated by
then
uniformly for \(t\in[0,1]\).
In the end, we know that fractional order integral and derivative operators can describe an important characteristics exhibiting longmemory in time in many complex processes and systems. With this advantage, in many ecoeconomical systems and diffusive processes with long time memory behavior [2, 3, 15, 39], fractional calculus provides an excellent tool to describe the hereditary properties of them. Here we give a specific example arising from the above complex processes.
Example
Consider the following nonlocal boundary value problem of the fractional pLaplacian equation:
where
Then system (3.18) has the positive minimal and maximal solutions, \(x^{*}=(x^{*}_{1},x^{*}_{2})\) and \(y^{*}=(y^{*}_{1},y^{*}_{2})\); and there exist some nonnegative numbers \(m_{i}\le n_{i}\), \(i=1,2,3,4\), such that
By simple computation, problem (3.18) is equivalent to the following multipoint boundary value problem:
Let
and
Firstly, we have
and by simple computation, we have \(\mathcal{G}_{A_{i}}(s)\ge0\), \(i=1,2\), and so (H0) holds.
Obviously, \(f_{1},f_{2}: [0,+\infty)\times[0,+\infty)\to[0,+\infty)\) are continuous and nondecreasing in the first variable and second variable and \(f_{i}(0,0)\neq0\), \(i=1,2\). Noticing that
and then (2.11) is satisfied with \(\epsilon_{1}=2>\frac{1}{q_{1}1}=\frac{3}{2}\), \(\epsilon_{2}=1>\frac{1}{q_{2}1}=\frac{1}{2}\), \(M=3\). Thus, by Theorem 3.1, system (3.18) has maximal and minimal solutions which satisfy (3.19).
Conclusion
In this work, we have established an existence result on the maximum and minimum solutions for a class of fractional order differential systems involving a pLaplacian operator and nonlocal boundary conditions. This type of differential systems actually arise from some complex natural processes such as ecological economy phenomena and diffusive interaction, moreover fractional differential operator can more accurately depict the coevolution process of economic, social and ecological subsystems and the transport of solute in highly heterogeneous porous media. The main contribution is that we introduced some new type growth conditions for nonlinearity and adopted the monotone iterative technique to establish the existence and estimation of the lower and upper bounds of the maximum and minimum solutions. Furthermore, the iterative schemes converging to the exact maximum and minimum solutions, which start from some explicit initial values, are also constructed.
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Acknowledgements
We are thankful to the editor and the anonymous reviewers for many valuable suggestions to improve this paper.
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Funding
The first author and second author are partially supported by the Youth Project of MOE (Ministry of Education in China) Humanity and Social Science Foundation (16YJC630101), the Youth Project of Hunan Provincial Philosophy and Social Science Foundation (15YBA407), the Foundation of the Excellent Youth Scholars of Hunan Provincial Department of Education (14B188) and the Project of Hunan Provincial Situation Decisionmaking Consultation (16JCC009). The third author and fourth author are partially supported financially by the National Natural Science Foundation of China (11571296,11371221) and the Natural Science Foundation of Shandong Province of China (ZR2014AM009).
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The authors declare that they have no competing interests.
Authors’ contributions
The study was carried out in collaboration among all authors. TR, SL and XZ completed the main part of this article and gave one example; LL corrected the main theorems and polished the manuscript. All authors read and approved the final manuscript.
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Keywords
 maximum and minimum solutions
 fractional differential system
 pLaplacian operator
 nonlocal boundary value problem