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Twin iterative solutions for a fractional differential turbulent flow model
Boundary Value Problems volumeÂ 2016, ArticleÂ number:Â 98 (2016)
Abstract
We investigate the existence of twin iterative solutions for a fractional pLaplacian 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.
1 Introduction
In this paper, we are concerned with the existence of twin iterative solutions for the following nonlocal fractional differential equation with a pLaplacian operator:
where \(\mathscr{D}_{\mathbf{t}}^{\alpha}\), \(\mathscr{D}_{\mathbf{t}}^{\beta}\) are the standard RiemannLiouville derivatives with \(1 <\alpha ,\beta \le 2\), \(\int^{1}_{0}x(s)\,dA(s)\) is the RiemannStieltjes integral with respect to a function A of bounded variation, and \(\varphi_{p}\) is the pLaplacian operator defined by \(\varphi_{p}(s) = s^{p2}s\), \(p > 1\). Obviously, \(\varphi_{p}(s)\) is invertible, and its inverse operator is \(\varphi_{q}(s)\), where \(q > 1\) is the constant such that \(\frac{1}{p} + \frac{1}{q} =1\).
It is well known that the pLaplacian 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 [2â€“6] and references therein. On the other hand, fractional calculus has been greatly developed in recent years. In particular, fractionalorder models have been proved to be more accurate than integerorder 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 [7â€“27]). Thus, fractionalorder differential equations with pLaplacian operator have attracted great interest from the mathematical research community.
Recently, Wang et al. [22] investigated the existence of multiple positive solutions for nonlocal fractional pLaplacian equation
where \(0 < \alpha\le2\), \(0 < \beta\le1\), \(0 \le a \le1\), \(0 <\xi< 1\). By using Krasnoselâ€™skiiâ€™s fixed point theorem and the LeggettWilliams theorem, some results on the existence of positive solutions are obtained. And then, by means of the upper and lower solutions method, Wang et al. [28] 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. [29] considered the following fractionalorder model for turbulent flow in a porous medium:
where \(\mathscr{D}_{\mathbf{t}}^{\alpha }\), \(\mathscr{D}_{\mathbf{t}}^{\beta }\), \(\mathscr{D}_{\mathbf{t}}^{\gamma}\) are the standard RiemannLiouville derivatives, \(\int^{1}_{0}x(s)\,dA(s)\) is the RiemannStieltjes integral, \(0<\gamma\le1<\alpha \le2<\beta <3\), \(\alpha\gamma>1\), A is a function of bounded variation, and dA can be a signed measure. In the case where the nonlinearity \(f(u,v)\) may be singular at both \(u=0\) and \(v=0\), the uniqueness of a positive solution for a fractional model of turbulent flow in a porous medium was established via the fixed point theorem of the mixed monotone operator.
Motivated by the mentioned works, in this paper, we consider the twin iterative solutions of fractionalorder 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 pLaplacian 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 RiemannLiouville fractional calculus. We give some preliminaries and lemmas for convenience of the reader.
Definition 2.1
The RiemannLiouville fractional integral of order \(\alpha>0\) of a function \(x:(0,+\infty)\rightarrow\mathbb{R}\) is given by
provided that the righthand side is pointwise defined on \((0,+\infty)\).
Definition 2.2
The RiemannLiouville fractional derivative of order \(\alpha>0\) of a function \(x:(0,+\infty)\rightarrow\mathbb{R}\) is given by
where \(n=[\alpha]+1\) with \([\alpha]\) denoting the integer part of a number Î±, provided that the righthand side is pointwise defined on \((0,+\infty)\).
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
For \(\alpha> 0\), if \(f(x)\) is integrable, then
where \(c_{i}\in\mathbb{R}\) (\(i=1,2,\ldots,n\)), and \(n=[\alpha]\).
Now consider the following linear fractional differential equation with nonlocal boundary condition:
Let
Lemma 2.1
(see [29])
Given \(r\in L^{1}(0, 1)\), the boundary value problem
has the unique solution
On the other hand, it follows from PropositionÂ 2.1 that the unique solution of the problem
is \(t^{\alpha1}\). Let
According to the strategy of [31] and [32], we have the following lemmas.
Lemma 2.2
(see [33])
If \(1<\alpha \le2\) and \(r\in L^{1}(0,1)\), then the boundary value problem (2.1) has the unique solution
where
Lemma 2.3
(see [31])
Let \(0\le\mathcal {A} < 1\) and \(\mathcal{G}_{A}(s)\ge0\) for \(s\in[0, 1]\). Then \(G_{\alpha}(t,s)\) and \(H(t,s)\) have the following properties:

(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
$$ \frac{t^{\alpha 1}(1t)s(1s)^{\alpha1}}{\Gamma(\alpha)}\leq G_{\alpha}(t,s) \leq \frac{\alpha1}{\Gamma(\alpha)}s(1s)^{\alpha 1} \quad \textit{for } t,s\in [0,1]. $$(2.7) 
(3)
There exist two constants a, b such that
$$ a{t^{\alpha1}}\mathcal{G}_{A}(s)\le H(t,s)\le b t^{\alpha1},\quad s,t\in[0,1]. $$(2.8)
Let \(\frac{1}{q}+\frac{1}{p}= 1\), where p is given by (1.1). We consider the associated linear boundary value problem
for \(r\in L^{1}(0,1)\) and \(r\ge0\).
Lemma 2.4
The associated linear BVP (2.9) has the unique positive solution
Proof
Let \(w =\mathscr{D}_{\mathbf{t}}^{\alpha}x\) and \(v = \varphi_{p}(w)\). Consider the boundary value problem
By LemmaÂ 2.1 we have
Noting that \(\mathscr{D}_{\mathbf{t}}^{\alpha}x=w\), \(w = \varphi^{1}_{p}(v)\), we get from (2.9) and (2.10) that the solution of (2.9) satisfies
By LemmaÂ 2.2 the solution of the BVP (2.9) can be written as
Since \(r(s)\ge0\), \(s\in[0,1]\), the solution of equation (2.9) is
â€ƒâ–¡
For convenience of presentation, we list some assumptions to be used throughout the rest of the paper.

(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
$$ 0< \int_{0}^{1}s(1s)^{\beta1}h(s)\,ds< +\infty. $$(2.12)
Remark 2.1
By (H1), for any \(\nu>1\), it is easy to get
Remark 2.2
There are a large number of functions that satisfy (H1). In particular, (H1) can cover mixed cases of the superlinear and sublinear cases. Some basic examples of f that satisfy (H1) are:

(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
(i) and (ii) are obvious. For (iii) and (iv), obviously, \(f:[0,+\infty) \to [0,+\infty)\) is continuous and nondecreasing, and for any \(0<\mu\le1\), noticing that \(\gamma,\delta,l>0\), we have
and
â€ƒâ–¡
Let \(\mathbb{N}\) be the set of all positive integers, \(\mathbb{R}\) be the set of all real numbers, and \(\mathbb{R}_{+}\) be the set of all nonnegative real numbers. Let \(C([0,1], \mathbb{R})\) be the Banach space of all continuous functions from \([0,1]\) into \(\mathbb{R}\) with the norm
Define the cone P in \(C([0,1], \mathbb{R}_{+})\) by
and the operator T by
Then each fixed point of the operator T on P is a positive solution of the BVP (1.1).
Lemma 2.5
Assume that (H0)(H2) hold. Then \(T: P \to P\) is continuous, compact, and nondecreasing.
Proof
For any \(x\in P\), we can find two positive numbers \(L_{x}>l_{x}\ge0\) such that
It follows from (H1) that T is increasing with respect to x, and thus by (2.8) we have
and
where
which implies that T is well defined and \(T(P) \subset P\). Moreover, T is also uniformly bounded for any bounded set of P. In fact, let \(D\subset P\) be any bounded set. Then there exists a constant \(L>0\) such that \(\ x\\leq L\) for any \(x\in D\). Moreover, for any \(x \in D\), \(s \in[0,1]\), from \(\x\\leq L< L+1\) and (2.13) we have
Consequently,
Therefore, \(T(D)\) is uniformly bounded.
On the other hand, according to the ArezelÃ 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.â€ƒâ–¡
3 Main results
Define the constant
Theorem 3.1
Suppose that conditions (H0)(H2) hold. If there exists a constant \(c> 0\) such that
where A is defined by (3.1), then the BVP (1.1) has the minimal and maximal solutions \(x^{*}\) and \(y^{*}\), which are positive, and there exist some nonnegative constants \(m_{i}\le n_{i}\), \(i=1,2\), such that
Moreover, for initial values \(x_{0} =0\), \(y_{0} =c+1\), let \(\{{x_{n}}\}\) and \(\{{y_{n}}\}\) be the iterative sequences generated by
Then
uniformly for \(t\in[0,1]\).
Proof
Let \(P[0, c] = \{x \in P : 0 \le\x\ \le c+1\}\). We first prove that \(T(P[0, c]) \subset P[0, c]\).
In fact, for any \(x \in P[0, c]\), since
from (H1) and (3.5) we have
which implies that \(T(P[0, c]) \subset P[0, c]\).
Let \(x_{0}(t) =0\) and \(x_{1}(t) = (Tx_{0})(t)\), \(t \in[0, 1]\). Since \(x_{0}(t) \in P[0, c]\), we have \(x_{1} \in P[0, c]\). Denote
It follows from \(T(P[0, c]) \subset P[0, c]\) that \(x_{n} \in P[0, c]\). Noticing that T is compact, we get that \(\{x_{n}\}\) is a sequentially compact set.
Since \(x_{1} = Tx_{0} = T0\in P[0, c]\), we have
By induction we get
Consequently, there exists \(x^{*} \in P[0, c]\) such that \(x_{n} \to x^{*}\). Letting \(n \to+\infty\), from the continuity of T and \(Tx_{n} =x_{n1}\) we obtain \(Tx^{*} = x^{*}\), which implies that \(x^{*}\) is a nonnegative solution of the nonlinear integral equation (1.1). Since \(x^{*}\in P\), there exist constants \(0\le m_{1}< n_{1}\) such that
and consequently \(x^{*}\) is a positive solution of the boundary value problem (1.1), and (3.3) holds.
On the other hand, let \(y_{0}(t) =c+1\), \(y_{1} = Ty_{0}\), \(t \in[0, 1]\). Then \(y_{0}(t) \in P[0, c]\) and \(y_{1} \in P[0, c]\). Let
It follows from \(T(P[0, c]) \subset P[0, c]\) that
By LemmaÂ 2.3, T is compact, and consequently \(\{y_{n}\}\) is a sequentially compact set.
Now, since \(y_{1} \in P[0, c]\), we get
It follows from LemmaÂ 2.3 that \(y_{2} = Ty_{1} \le Ty_{0} = y_{1}\). By induction we obtain
Consequently, there exists \(y^{*}\in P[0, c]\) such that \(y_{n} \to y^{*}\). Letting \(n \to+\infty\), from the continuity of T and \(Ty_{n} = y_{n1}\) we have \(Ty^{*} = y^{*}\), which implies that \(y^{*}\) is another nonnegative solution of the boundary value problem (1.1) and \(y^{*}\) also satisfies (3.3) since \(y^{*}\in P\).
Now we prove that \(x^{*}\) and \(y^{*}\) are extremal solutions for a fractional differential equation (1.1). Let xÌƒ 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 \(x_{n}\le\tilde{x}\le y_{n}\), \(n=1,2,3,\ldots\)â€‰. 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.â€ƒâ–¡
Corollary 3.1
Suppose that conditions (H0)(H2) hold. If
then the BVP (1.1) has the minimal and maximal solutions \(x^{*}\) and \(y^{*}\), which are positive, and there exist some constants \(0 \le m_{i}\le n_{i}\), \(i=1,2\), such that
Moreover, there exists a positive constant c such that for initial values \(x_{0} =0\), \(y_{0} =c+1\), the iterative sequences generated by
converge uniformly to \(x^{*}\) and \(y^{*}\) for \(t\in[0,1]\), namely,
Proof
From (3.6) we have
which implies that we can find a constant \(c>0\) large enough such that
By TheoremÂ 3.1 the conclusion of CorollaryÂ 3.1 holds.â€ƒâ–¡
Remark 3.1
CorollaryÂ 3.1 is an interesting case of the boundary value problem (1.1). Because of the independence of Ïµ 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
Suppose that \(p=2\) and (H0), (H2) hold. Moreover, suppose that f satisfies

(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 \(x\in[0,+\infty)\),
$$f(cx) \ge c^{\epsilon} f(x), \quad \forall 0 < c \le1. $$
Then the BVP (1.1) has the minimal and maximal solutions \(x^{*}\) and \(y^{*}\), which are positive, and there exist constants \(0\le m_{i}\le n_{i}\), \(i=1,2\), such that
Moreover, there exists a positive constant c such that for initial values \(x_{0} =0\), \(y_{0} =c+1\), the iterative sequences generated by
converge uniformly to \(x^{*}\) and \(y^{*}\) for \(t\in[0,1]\), namely,
4 Example
Consider the following nonlocal boundary value problem of the fractional pLaplacian equation:
where Î³, Î´, l are positive constants satisfying \(l>\delta\), \(\gamma+l<2\), A is a boundedvariation function satisfying
Then the BVP (4.1) has the minimal and maximal solutions \(x^{*}\) and \(y^{*}\), which are positive, and there exist constants \(0\le m_{i}< n_{i}\), \(i=1,2\), such that
Moreover, there exists a positive constant c such that for initial values \(x_{0} =0\), \(y_{0} =c+1\), the iterative sequences generated by
converge uniformly to \(x^{*}\) and \(y^{*}\) for \(t\in[0,1]\), namely,
Let
Then by simple computation problem (4.1) is equivalent to the following multipoint boundary value problem:
First, we have
and by simple computation we have \(\mathcal{G}_{A}(s)\ge0\), and so (H0) holds.
Next, from RemarkÂ 2.2, for any \(0<\mu\le1\), we have
and thus (H1) holds. Now we compute
so (H2) is satisfied.
Thus, by CorollaryÂ 3.1 the BVP (4.1) has maximal and minimal solutions that satisfy (4.2) and (4.3).
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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).
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Wu, J., Zhang, X., Liu, L. et al. Twin iterative solutions for a fractional differential turbulent flow model. Bound Value Probl 2016, 98 (2016). https://doi.org/10.1186/s1366101606049
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DOI: https://doi.org/10.1186/s1366101606049