Twin iterative solutions for a fractional differential turbulent flow model
- Jing Wu^{1},
- Xinguang Zhang^{2, 3}Email author,
- Lishan Liu^{3, 4} and
- Yonghong Wu^{3}
Received: 3 January 2016
Accepted: 6 May 2016
Published: 13 May 2016
Abstract
We investigate the existence of twin iterative solutions for a fractional p-Laplacian equation with nonlocal boundary conditions. Using the monotone iterative technique, we establish a new existence result on the maximal and minimal solutions under suitable nonlinear growth conditions. We also consider some interesting particular cases and give an example to illustrate our main results.
Keywords
extremal solutions monotone iterative technique p-Laplacian operator nonlocal boundary value problem fractional differential equation1 Introduction
It is well known that the p-Laplacian equation can describe a fundamental mechanics problem arising from turbulent flow in a porous medium; see [1]. Based on this background, some interesting results relative to the equation \((\varphi_{p}(x'(t)))'=f(t,x(t))\) subject to certain boundary value conditions have been obtained in [2–6] and references therein. On the other hand, fractional calculus has been greatly developed in recent years. In particular, fractional-order models have been proved to be more accurate than integer-order models for the description of many physical phenomena with long memory, such as viscoelasticity, electrochemistry control, porous media, electromagnetic, polymer rheology, and some hereditary properties of various materials and processes (for the reseach of fractional models and relative problems, we refer readers to [7–27]). Thus, fractional-order differential equations with p-Laplacian operator have attracted great interest from the mathematical research community.
Motivated by the mentioned works, in this paper, we consider the twin iterative solutions of fractional-order model for turbulent flow in a porous medium. Differently from the mentioned works, we not only obtain the minimal and maximal solutions of the nonlocal boundary value problem of the fractional p-Laplacian equation (1.1), but we also derive estimates of the lower and upper bounds of the extremal solutions and construct a convergent iterative scheme for finding these solutions. In addition, we consider some particular cases and give an example to illustrate our main results.
2 Preliminaries and lemmas
Our work is carried out based on various definitions and semigroup properties of the Riemann-Liouville fractional calculus. We give some preliminaries and lemmas for convenience of the reader.
Definition 2.1
Definition 2.2
Proposition 2.1
- (1)If \(x\in L^{1}(0, 1)\) and \(\alpha >\beta > 0\), then$$I^{\alpha }I^{\beta }x(t)=I^{\alpha +\beta }x(t), \qquad \mathscr{D}_{\mathbf{t}}^{\beta }I^{\alpha } x(t)=I^{\alpha -\beta } x(t),\qquad \mathscr{D}_{\mathbf{t}}^{\beta }I^{\beta } x(t)=x(t). $$
- (2)If \(\alpha >0\), \(\beta >0\), then$$\mathscr{D}_{\mathbf{t}}^{\alpha } t^{\beta -1}=\frac{\Gamma(\beta )}{\Gamma(\beta -\alpha )}t^{\beta -\alpha -1}. $$
Proposition 2.2
Lemma 2.1
(see [29])
Lemma 2.2
(see [33])
Lemma 2.3
(see [31])
- (1)
\(G_{\alpha}(t,s)\) and \(H(t,s)\) are nonnegative and continuous for \((t,s)\in[0,1]\times[0,1]\).
- (2)\(G_{\alpha}(t,s)\) satisfies$$ \frac{t^{\alpha -1}(1-t)s(1-s)^{\alpha-1}}{\Gamma(\alpha)}\leq G_{\alpha}(t,s) \leq \frac{\alpha-1}{\Gamma(\alpha)}s(1-s)^{\alpha -1} \quad \textit{for } t,s\in [0,1]. $$(2.7)
- (3)There exist two constants a, b such that$$ a{t^{\alpha-1}}\mathcal{G}_{A}(s)\le H(t,s)\le b t^{\alpha-1},\quad s,t\in[0,1]. $$(2.8)
Lemma 2.4
Proof
- (H0)
A is a function of bounded variation satisfying \(\mathcal{G}_{A}(s)\ge0\) for \(s\in[0, 1]\) and \(0\le\mathcal{A}<1\).
- (H1)\(f:[0,+\infty) \to[0,+\infty)\) is continuous and nondecreasing, and there exists a constant \(\epsilon> 0\) such that, for any \(x\in[0,+\infty)\),$$ f(\mu x) \ge\mu^{\epsilon} f(x), \quad \forall 0 < \mu\le1. $$(2.11)
- (H2)\(h\ge0\) satisfies$$ 0< \int_{0}^{1}s(1-s)^{\beta-1}h(s)\,ds< +\infty. $$(2.12)
Remark 2.1
Remark 2.2
- (i)
\(f(s)=\sum_{i=1}^{m}a_{i}s^{\gamma _{i}}\), where \(a_{i}, \gamma_{i}>0\), \(i=1,2,\ldots,m\).
- (ii)If \(0<\gamma_{i}, d_{i}<+\infty\) (\(i=1,2,\ldots, m\)) and \(\delta,c>0\), then$$f(s)= \Biggl[c+\sum_{i=1}^{m}d_{i}(t)x^{\gamma_{i}} \Biggr]^{\delta}. $$
- (iii)
\(f(s)=\frac{s^{\gamma}}{1+s^{\delta}}+s^{l}\), \(\gamma,\delta,l>0\), \(\gamma>\delta\).
- (iv)
\(f(s)=\frac{(a+s^{\gamma})s^{l}}{b+s^{\delta}}\), \(a,b,\gamma,\delta,l>0\), \(l>\delta\).
Proof
Lemma 2.5
Assume that (H0)-(H2) hold. Then \(T: P \to P\) is continuous, compact, and nondecreasing.
Proof
On the other hand, according to the Arezelà-Ascoli theorem and the Lebesgue dominated convergence theorem, it is easy to get that \(T:P \to P\) is completely continuous. It follows from (H1) that the operator T is nondecreasing. □
3 Main results
Theorem 3.1
Proof
Let \(P[0, c] = \{x \in P : 0 \le\|x\| \le c+1\}\). We first prove that \(T(P[0, c]) \subset P[0, c]\).
Now we prove that \(x^{*}\) and \(y^{*}\) are extremal solutions for a fractional differential equation (1.1). Let x̃ be any positive solution of the boundary value problem (1.1). Then \(x_{0}=0\le\tilde{x}\le c+1=y_{0}\), and \(x_{1}=Tx_{0}\le T\tilde{x}=\tilde{x}\le T(c+1)=w_{1}\). By induction we have \(x_{n}\le\tilde{x}\le y_{n}\), \(n=1,2,3,\ldots\) . Taking the limit, we have \(x^{*}\le\tilde{x}\le y^{*}\). This implies that \(x^{*}\) and \(y^{*}\) are the maximal and minimal solutions of the BVP (1.1), respectively. The proof is completed. □
Corollary 3.1
Proof
Remark 3.1
Corollary 3.1 is an interesting case of the boundary value problem (1.1). Because of the independence of ϵ and q, condition (3.6) is easy to be satisfied. For example, for \(q=4\) and \(\epsilon=\frac{1}{4}\), the BVP (1.1) has the minimal and maximal solutions if (H0)-(H2) are satisfied.
In addition, note that, when \(p=2\), the nonlinear operator \(\mathscr{D}_{\mathbf{t}}^{\beta} (\varphi_{p} (\mathscr{D}_{\mathbf{t}}^{\alpha} ) )\) reduces to the linear operator \(\mathscr{D}_{\mathbf{t}}^{\beta} (\mathscr{D}_{\mathbf{t}}^{\alpha } )\), and if \(0<\epsilon<\frac{1}{2}\), then (3.6) naturally holds, and so we have the following corollary.
Corollary 3.2
- (h1)\(f:[0,+\infty) \to(0,+\infty)\) is continuous and nondecreasing, and there exists a constant \(0<\epsilon<\frac{1}{2}\) such that, for any \(x\in[0,+\infty)\),$$f(cx) \ge c^{\epsilon} f(x), \quad \forall 0 < c \le1. $$
4 Example
Thus, by Corollary 3.1 the BVP (4.1) has maximal and minimal solutions that satisfy (4.2) and (4.3).
Declarations
Acknowledgements
The authors were supported financially by the National Natural Science Foundation of China (11571296, 11371221), the Natural Science Foundation of Shandong Province of China (ZR2014AM009) and the Project of Shandong Province Higher Educational Science and Technology Program (J14li07).
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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