Maximum and minimum solutions for a nonlocal p-Laplacian fractional differential system from eco-economical processes
- Teng Ren^{1}Email author,
- Sidi Li^{2},
- Xinguang Zhang^{3, 4} and
- Lishan Liu^{4, 5}
Received: 4 February 2017
Accepted: 3 August 2017
Published: 18 August 2017
Abstract
This paper focuses on the maximum and minimum solutions for a fractional order differential system, involving a p-Laplacian 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.
Keywords
1 Introduction
The interest in using fractional differential equations in modeling ecological economy and diffusive processes has wide literature. Especially when one wants to model long-range ecological economy phenomena and diffusive interaction, fractional differential operator has higher accuracy than integer order differential model in depicting the co-evolution 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 psycho-socio-economical 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].
However, to the best of our knowledge, there are relatively few results on a fractional order differential system involving the p-Laplacian operator and nonlocal Riemann-Stieltjes 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 p-Laplacian system subject to a nonlocal Riemann-Stieltjes 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.
2 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 Riemann-Liouville fractional derivatives; for details, see [15, 39, 40]. Here we only recall a famous semigroup property for Riemann-Liouville fractional calculus.
Proposition 2.1
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]
Lemma 2.2
see [16]
- (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]
- (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$$ \frac{t^{\alpha_{i} -1}(1-t)s(1-s)^{\alpha_{i} -1}}{\Gamma(\alpha_{i} )}\leq G_{\alpha_{i}}(t,s) \leq \frac{\alpha_{i}-1}{\Gamma(\alpha_{i} )}s(1-s)^{\alpha_{i} -1}. $$(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)
Lemma 2.4
Proof
- (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
- (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
- 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.
Lemma 2.5
Assume that (H0)-(H1) hold. Then \(T: P \to P\) is a continuous, compact operator.
Proof
On the other hand, according to the Arzela-Ascoli theorem and the Lebesgue dominated convergence theorem, we know that \(T:P \to P\) is completely continuous. □
3 Main results
Lemma 3.1
Proof
Theorem 3.1
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 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
Corollary 3.1
In the end, we know that fractional order integral and derivative operators can describe an important characteristics exhibiting long-memory in time in many complex processes and systems. With this advantage, in many eco-economical 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
4 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 p-Laplacian 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 co-evolution 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.
Declarations
Acknowledgements
We are thankful to the editor and the anonymous reviewers for many valuable suggestions to improve this paper.
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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