Open Access

Iterative algorithm and estimation of solution for a fractional order differential equation

  • Jing Wu1,
  • Xinguang Zhang2, 4Email author,
  • Lishan Liu3, 4,
  • Yonghong Wu4 and
  • Benchawan Wiwatanapataphee4
Boundary Value Problems20162016:116

https://doi.org/10.1186/s13661-016-0608-5

Received: 18 February 2016

Accepted: 10 May 2016

Published: 17 June 2016

Abstract

In this paper, we establish an iterative algorithm and estimation of solutions for a fractional turbulent flow model in a porous medium under a suitable growth condition. Our main tool is the monotone iterative technique.

Keywords

iterated algorithmfractional order turbulent flow model p-Laplacian operatornonlocal boundary value problem

1 Introduction

Over the past decades, a large number of investigations have been carried out to study various natural and engineering systems and processes that involve fluid flow through porous media [15], such as petroleum extraction, where the flow accelerates toward the pumping well while crossing regions of variable porosity, and a turbulent regime eventually occurs and affects the overall pressure drop and well performance. Dybbs and Edwards [1] conducted a flow visualization study, and they showed that fluid flow in a porous medium exhibits turbulent characteristics when the pore-Reynolds number (based on the pore scale and velocity) becomes higher than a few hundred. Leibenson [5] introduced a p-Laplacian equation to describe turbulent flows in a porous medium. Inspired by the above work, many authors studied the existence and uniqueness of solution for differential equation involving p-Laplacian under various boundary conditions, here we refer the reader to the work of Li et al. [6], Chen et al. [7], Zhang et al. [810], Goodrich [11, 12], Ding et al. [13], and the references therein.

In this paper, we study a fractional order differential equation involving the p-Laplacian
$$ -\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\beta} \bigl( \varphi _{p} \bigl(-\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha}x \bigr) \bigr) (t)=f\bigl(x(t)\bigr), $$
(1.1)
subject to the following nonlocal boundary conditions:
$$\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha }x(0)=\boldsymbol { \mathscr{D}}_{\mathbf{t}}^{\alpha }x(1)=\boldsymbol{\mathscr {D}}_{\mathbf{t}}^{\gamma}x(0)=0,\qquad \boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\gamma}x(1)= \int ^{1}_{0}\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\gamma}x(s) \,dA(s), $$
where \(\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha }\), \(\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\beta}\), \(\boldsymbol {\mathscr{D}}_{\mathbf{t}}^{\gamma}\) are the standard Riemann-Liouville derivatives, \(\int^{1}_{0}x(s)\,dA(s)\) denotes a Riemann-Stieltjes integral and \(0<\gamma\le1<\alpha \), \(\beta\le2\), \(\alpha-\gamma>1\), A is a function of the bounded variation and dA can be a signed measure, \(\varphi_{p}\) is the p-Laplacian operator defined by \(\varphi_{p}(s) = \vert s\vert ^{p-2}s\), \(p > 1\). Obviously, \(\varphi_{p}(s)\) is invertible and its inverse operator is \(\varphi_{q}(s)\), where \(q > 1\) is a constant such that \(\frac{1}{p} + \frac{1}{q} =1\).

A fractional derivative gives a perfect aid to characterize the memory and hereditary properties of various processes and materials. Therefore differential equations of fractional order are being used in modeling of the electrical and mechanical properties of various real materials, the rheological properties of rock, as well as the hereditary properties of various biological processes [1420], and many extensive researches have been carried out to develop numerical schemes and analytical solutions [2132].

In this paper, we focus on the iterative algorithm and estimation of positive solutions for the problem (1.1) by introducing a new growth condition on the nonlinearity f:
  1. (F)
    \(f: [0,+\infty)\to[0,+\infty)\) is continuous and non-decreasing, and there exists a constant \(\epsilon>0\) such that
    $$ \sup_{s>0}\frac{f ( s ) }{s^{\epsilon}}< +\infty. $$
    (1.2)
     
The assumption (F) is a relatively weaker condition than the superlinear or sublinear or the mixed superlinear and sublinear condition, which includes many interesting cases [6, 3337]. Furthermore, (1.2) can be adopted to extend some results of partial differential equations [35, 36]. Some basic examples satisfying (F) are
  1. (1)

    \(f(s)=as^{\kappa}\), where \(a, \kappa>0\).

     
  2. (2)

    \(f(s)=s^{\mu}\arctan s\), \(\mu>0\).

     
  3. (3)

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

     
  4. (4)

    \(f(s)=\ln(a+s)\), \(a>0\).

     

Here we also comment that there are relatively few results on fractional order equations with nonlocal Riemann-Stieltjes integral boundary conditions, and no work has been done concerning the iterative algorithm of solutions of Equation (1.1). The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas which are used in the rest of the paper. In Section 3, we establish the existence of the maximal and minimal solutions, estimation of the lower and upper bounds of the extremal solutions and an iterative scheme converging to the exact extremal solutions.

2 Preliminaries and lemmas

This work restricts attention to Riemann-Liouville fractional derivatives; for details, see the monographs [14, 15]. First of all, according to the definition and properties of the Riemann-Liouville fractional calculus, we make a change of variable, \(x(t)=I^{\gamma}v(t)\), \(v\in C[0,1]\), then we have
$$ \begin{aligned}&\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha}x(t)= \frac {d^{n}}{dt^{n}}I^{n-\alpha }x(t)=\frac{d^{n}}{dt^{n}}I^{n-\alpha}I^{\gamma}v(t) =\frac{d^{n}}{dt^{n}}I^{n-\alpha+\gamma}v(t)=\boldsymbol{\mathscr {D}}_{\mathbf{t}}^{\alpha -\gamma}v(t), \\ &\begin{aligned}\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha +1}x(t)&= \frac{d^{n}}{dt^{n}}I^{n-\alpha -1}x(t)=\frac{d^{n}}{dt^{n}}I^{n-\alpha-1}I^{\gamma}v(t) =\frac{d^{n}}{dt^{n}}I^{n-\alpha-1+\gamma}v(t) \\ & =\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha-\gamma+1}v(t), \end{aligned} \\ & \boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\gamma}x(t)= \boldsymbol { \mathscr{D}}_{\mathbf{t}}^{\gamma}I^{\gamma }v(t)=v(t).\end{aligned} $$
(2.1)
Thus, using (2.1), the BVP (1.1) reduces to the following modified fractional differential equation:
$$ -\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\beta} \varphi_{p} \bigl(-\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha -\gamma}v(t) \bigr)= f\bigl(I^{\gamma}v(t)\bigr), $$
(2.2)
with boundary condition
$$\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha-\gamma }v(0)=\boldsymbol{ \mathscr{D}}_{\mathbf{t}}^{\alpha -\gamma}v(1)=0,\quad v(0)=0, v(1)= \int^{1}_{0}v(s)\,dA(s). $$

As the inverse is also valid, the BVP (2.2) is indeed equivalent to the BVP (1.1).

Now denote
$$\begin{aligned} &G(\beta,t,s)=\frac{1}{\Gamma(\beta)} \textstyle\begin{cases} [t(1-s)]^{\beta-1},& 0\leq t\leq s\leq1 ,\\ [t(1-s)]^{\beta-1}-(t-s)^{\beta-1},& 0\leq s\leq t\leq 1, \end{cases}\displaystyle \end{aligned}$$
(2.3)
$$\begin{aligned} &\mathcal{A}= \int_{0}^{1}t^{\alpha-\gamma-1}\,dA(t),\qquad \mathcal{G}_{A}(s)= \int_{0}^{1}G(\alpha-\gamma,t, s)\,dA(t), \end{aligned}$$
(2.4)
and assume \(0\le\mathcal{A} < 1\) and \(\mathcal{G}_{A}(s)\ge0\) for \(s\in[0,1]\).

Suppose \(0<\gamma\le1<\alpha \), \(\beta\le2\), \(\alpha-\gamma>1\). The following results have been obtained in [8, 38].

Lemma 2.1

(see [38])

Given \(h\in L^{1}(0, 1)\), then the problem
$$\textstyle\begin{cases} -\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha -\gamma}v(t)=h(t) ,\quad 0< t< 1 ,\\ v(0)=v(1)=0, \end{cases} $$
has the unique solution \(v(t)= \int_{0}^{1}G(\alpha-\gamma,t,s)h(s)\,ds\).

Lemma 2.2

(see [8])

Let \(h\in L^{1}(0,1)\), then the fractional boundary value problem
$$\textstyle\begin{cases} -\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\beta }\varphi_{p} (-\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha -\gamma}v(t) )= h(t), \\ \boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha-\gamma }v(0)=\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\alpha -\gamma}v(1)=0,\quad v(0)=0, v(1)= \int^{1}_{0}v(s)\,dA(s), \end{cases} $$
has the unique solution
$$v(t)= \int_{0}^{1}H(t,s)\varphi_{q} \biggl( \int_{0}^{1}G(\beta,s,\tau)h(\tau)\,d\tau \biggr) \,ds, $$
where
$$ H(t,s)=\frac{t^{\alpha-\gamma-1}}{1-\mathcal{A}}\mathcal {G}_{A}(s)+G(\alpha- \gamma,t,s). $$
(2.5)

Lemma 2.3

(see [8])

The function \(G(\beta,t,s)\) and \(H(t,s)\) have the following properties:
  1. (1)

    \(G(\beta,t,s)> 0\), \(H(t,s)>0\), for \(t,s\in(0,1)\).

     
  2. (2)

    \(\frac{t^{\beta-1}(1-t)s(1-s)^{\beta -1}}{\Gamma(\beta)}\leq G(\beta,t,s) \leq\frac{\beta-1}{\Gamma(\beta)}t^{\beta-1}(1-t)\), for \(t,s\in [0,1]\), and \(\frac{t^{\beta-1}(1-t)s(1-s)^{\beta -1}}{\Gamma(\beta)}\leq G(\beta,t,s) \leq\frac{\beta-1}{\Gamma(\beta)}s(1-s)^{\beta-1}\), for \(t,s\in [0,1]\).

     
  3. (3)
    There exist two positive constants d, e such that
    $$ d{t^{\alpha-\gamma-1}}\mathcal{G}_{A}(s)\le H(t,s)\le e{t^{\alpha -\gamma -1}}, \quad t,s\in[0,1]. $$
    (2.6)
     
Let \(E=C[0,1]\) be the Banach space of all continuous functions equipped the norm \(\Vert v\Vert =\max\{v(t): t \in[0,1]\}\). Define a cone P of E by
$$\begin{aligned} P= \bigl\{ & v\in E: \mbox{there exist nonnegative numbers } l_{v}< L_{v} \mbox{ such that} \\ & l_{v}t^{\alpha-\gamma-1}\leq v(t)\leq L_{v}t^{\alpha-\gamma-1}, t\in[0, 1] \bigr\} \end{aligned} $$
and an operator T by
$$\begin{aligned}& (Tv) (t)= \int_{0}^{1}H(t,s)\varphi_{q} \biggl( \int_{0}^{1}G(\beta ,s,\tau )f \bigl(I^{\gamma}v( \tau) \bigr)\,d\tau \biggr)\,ds, \end{aligned} $$
then the fixed point of operator T in E is the solution of Equation (2.2), and \(x=I^{\gamma}v(\tau)\) is the solution of Equation (1.1).
Now we define the constant
$$M=\sup_{s>0}\frac{f ( s ) }{s^{\epsilon}}. $$
We have the following lemma.

Lemma 2.4

Assume that (F) holds. Then \(T: P \to P\) is a continuous, compact, and increasing operator.

Proof

Let \(v\in P\). Then there exist two nonnegative numbers \(L_{v}>l_{v}\ge0\) such that
$$ l_{v} t^{\alpha-\gamma-1}\leq v(t)\leq L_{v} t^{\alpha-\gamma-1}, \quad t\in[0, 1]. $$
(2.7)
Since
$$\frac{1}{\Gamma(\gamma)} \int_{0}^{t}(t-s)^{\gamma-1} s^{\alpha-\gamma-1} \,ds=\frac{\Gamma(\alpha-\gamma)}{\Gamma (\alpha)} t^{\alpha-1}, $$
we have
$$ \frac{\Gamma(\alpha-\gamma)l_{v}}{\Gamma(\alpha)} t^{\alpha-1}\le I^{\gamma}v(\tau)\le \frac{\Gamma(\alpha-\gamma)L_{v}}{\Gamma (\alpha)} t^{\alpha-1}. $$
(2.8)
If \(v\equiv0\), then it follows from (2.6) that
$$ \begin{aligned}[b] (Tv) (t)&\le e t^{\alpha-\gamma-1} \int_{0}^{1}\varphi_{q} \biggl( \int _{0}^{1}G(\beta,s,\tau)f (0 )\,d\tau \biggr) \,ds \\ &\le e \biggl[\frac{(\beta-1)f(0)}{\Gamma(\beta)} \biggr]^{q-1} t^{\alpha-\gamma-1} \end{aligned} $$
(2.9)
and
$$ \begin{aligned}[b] (Tv) (t)&\ge d t^{\alpha-\gamma-1} \int_{0}^{1}\mathcal {G}_{A}(s) \varphi_{q} \biggl( \int_{0}^{1}G(\beta,s,\tau)f (0 )\, d\tau \biggr) \,ds \\ &\ge d \int_{0}^{1}\mathcal{G}_{A}(s)s^{(\beta-1)(q-1)}(1-s)^{q-1} \,ds \biggl[\frac{f(0)}{\Gamma(\beta+2)} \biggr]^{q-1} t^{\alpha-\gamma-1}. \end{aligned} $$
(2.10)
Otherwise, by (F), T is increasing on x, and thus from (F) and (2.8), we have
$$ \begin{aligned}[b] (Tv) (t)&\le e t^{\alpha-\gamma-1} \int_{0}^{1}\varphi_{q} \biggl( \int _{0}^{1}G(\beta,s,\tau) f \bigl(I^{\gamma}v(\tau) \bigr)\,d\tau \biggr)\,ds \\ &\le e t^{\alpha -\gamma-1} \int_{0}^{1}\varphi_{q} \biggl( \int_{0}^{1}G(\beta,s,\tau)M \bigl(I^{\gamma}v(\tau ) \bigr)^{\epsilon}d\tau \biggr)\,ds \\ &\le e t^{\alpha-\gamma-1} \biggl[\frac{(\beta-1)M}{\Gamma(\beta)} \biggl(\frac{\Gamma(\alpha -\gamma )L_{v}}{\Gamma(\alpha)} \biggr)^{\epsilon}\biggr]^{q-1}.\end{aligned} $$
(2.11)
On the other hand,
$$ \begin{aligned}[b] (Tv) (t)&\ge d t^{\alpha-\gamma-1} \int_{0}^{1}\mathcal{G}_{A}(s) \varphi_{q} \biggl( \int _{0}^{1}G(\beta,s,\tau)f \bigl(I^{\gamma}v(\tau) \bigr)\,d\tau \biggr)\,ds \\ &\ge d \int_{0}^{1}\mathcal{G}_{A}(s)s^{(\beta-1)(q-1)}(1-s)^{q-1} \,ds \biggl[\frac{f(0)}{\Gamma(\beta+2)} \biggr]^{q-1} t^{\alpha-\gamma-1}. \end{aligned} $$
(2.12)
Take
$$\begin{aligned}&L^{*}_{v}=\max \biggl\{ e \biggl[\frac{(\beta-1)M}{\Gamma(\beta )} \biggl(\frac{\Gamma(\alpha-\gamma)L_{v}}{\Gamma(\alpha)} \biggr)^{\epsilon}\biggr]^{q-1}, e \biggl[ \frac{(\beta-1)f(0)}{\Gamma(\beta)} \biggr]^{q-1} \biggr\} , \\ & l^{*}_{v}=d \int_{0}^{1}\mathcal{G}_{A}(s)s^{(\beta-1)(q-1)}(1-s)^{q-1} \,ds \biggl[\frac{f(0)}{\Gamma(\beta+2)} \biggr]^{q-1},\end{aligned} $$
then by (2.9)-(2.12), we have \(l^{*}_{v}t^{\alpha-\gamma-1}\le(Tv)(t)\le L^{*}_{v}t^{\alpha-\gamma-1}\). Thus T is well defined and uniformly bounded and \(T(P) \subset P\).

On the other hand, according to the Arezela-Ascoli theorem and the Lebesgue dominated convergence theorem, it is easy to see that \(T:P \to P\) is completely continuous. □

3 Main results

Denote
$$r_{q}= \biggl[e\varphi_{q} \biggl(\frac{(\beta-1)M}{\Gamma(\beta)\Gamma ^{\epsilon}(\gamma+1)} \biggr) \biggr]^{\frac{1}{1-\epsilon(q-1)}},\qquad \rho _{q}= \biggl[ \frac{1}{M} \biggl(\frac{\Gamma(\gamma+1)}{e}\varphi_{q} \biggl( \frac {\Gamma (\beta)}{\beta-1} \biggr) \biggr)^{\epsilon}\biggr]^{\frac{1}{\epsilon (q-1)-1}}, $$
then we have the following main results.

Theorem 3.1

Assume (F) holds, and the following conditions are satisfied:
$$ \epsilon(q-1)>1, \qquad f(0)\le\rho_{q}. $$
(3.1)
Then Equation (1.1) has a minimal solution \(x^{*}\) and a maximal solution \(y^{*}\), and there exist some nonnegative numbers \(m_{i}< n_{i}\), \(i=1,2\), such that
$$ m_{1}t^{\alpha-1}\le x^{*}(t) \le n_{1}t^{\alpha -1}, \qquad m_{2}t^{\alpha-1}\le y^{*}(t) \le n_{2}t^{\alpha-1}, \quad t\in[0,1]. $$
(3.2)
Moreover, for initial values \(v^{(0)}=0\), \(w^{(0)}=r\), let \(\{{v^{(n)}}\}\) and \(\{{w^{(n)}}\}\) be the iterative sequences generated by
$$ \begin{aligned}&v^{(n)}(t) = \bigl(T{v^{(n-1)}} \bigr) (t) = \bigl(T^{n}v^{(0)}\bigr) (t), \\ & {w^{(n)}}(t) = \bigl(T{w^{(n-1)}}\bigr) (t) = \bigl(T^{n}w^{(0)}\bigr) (t). \end{aligned} $$
(3.3)
Then
$$\lim_{n\to+\infty}{v^{(n)}}= \boldsymbol{ \mathscr{D}}_{\mathbf {t}}^{\gamma}x^{*}, \qquad \lim_{n\to+\infty}{w^{(n)}}= \boldsymbol{\mathscr{D}}_{\mathbf {t}}^{\gamma}y^{*}, $$
uniformly for \(t\in[0,1]\).

Proof

We construct a closed convex set of P by \(P[0, r_{q}] = \{v \in P : \Vert v\Vert \le r_{q}\}\), and we prove \(T(P[0, r_{q}]) \subset P[0, r_{q}]\).

In fact, for any \(v\in P[0, r_{q}]\), if \(v\equiv0\), it follows from Lemma 2.3 and (3.1) that
$$ \begin{aligned}[b] \bigl\Vert (Tv)\bigr\Vert &=\max _{t\in[0,1]} \biggl\{ \int _{0}^{1}H(t,s)\varphi_{q} \biggl( \int_{0}^{1}G(\beta,s,\tau)f \bigl(I^{\gamma}v( \tau) \bigr)\, d\tau \biggr)\,ds \biggr\} \\ &\le e \int_{0}^{1} \varphi_{q} \biggl( \int_{0}^{1} \frac{\beta-1}{\Gamma(\beta)}\tau(1- \tau)^{\beta-1}f(0)\,d\tau \biggr)\,ds\le e \varphi_{q} \biggl( \frac{\beta-1}{\Gamma(\beta)}f(0) \biggr) \\ &\le \varphi _{q} \biggl(\frac{\beta-1}{\Gamma(\beta)} \biggr) \biggl[ \frac{1}{M} \biggl(\frac {\Gamma(\gamma+1)}{e}\varphi_{q} \biggl( \frac{\Gamma(\beta)}{\beta -1} \biggr) \biggr)^{\epsilon}\biggr]^{\frac{q-1}{\epsilon(q-1)-1}} \\ &=e \varphi_{q} \biggl(\frac{\beta-1}{\Gamma(\beta)} \biggr) \biggl[M^{q-1} \biggl(\frac{e}{\Gamma(\gamma+1)}\varphi_{q} \biggl( \frac {\beta-1}{ \Gamma(\beta)} \biggr) \biggr)^{\epsilon(q-1)} \biggr]^{\frac {1}{1- \epsilon(q-1)}} \\ &= \biggl[ \biggl(e \varphi_{q} \biggl(\frac{\beta-1}{\Gamma(\beta )} \biggr) \biggr)^{1-\epsilon(q-1)}M^{q-1} \biggl(\frac{e}{\Gamma(\gamma +1)}\varphi _{q} \biggl(\frac{\beta-1}{ \Gamma(\beta)} \biggr) \biggr)^{\epsilon(q-1)} \biggr]^{\frac {1}{1-\epsilon (q-1)}} \\ &= r_{q}.\end{aligned} $$
(3.4)
Otherwise, we have
$$ 0< v(t) \le\max_{t\in[0,1]}v(t)\le r_{q}, \quad t\in(0,1), $$
(3.5)
and
$$ 0< I^{\gamma}v(t)=\frac{1}{\Gamma(\gamma)} \int _{0}^{t}(t-s)^{\gamma-1}v(s)\,ds \le \frac{r_{q}}{ \Gamma(\gamma+1)}, \quad t\in(0,1). $$
(3.6)
So by (F), (2) and (3) of Lemma 2.3, and (3.6), we get
$$ \begin{aligned}[b] \bigl\Vert (Tv)\bigr\Vert &=\max _{t\in[0,1]} \biggl\{ \int _{0}^{1}H(t,s)\varphi_{q} \biggl( \int_{0}^{1}G(\beta,s,\tau) f \bigl(I^{\gamma}v( \tau) \bigr)\,d\tau \biggr)\,ds \biggr\} \\ &\le et^{\alpha -\gamma-1} \int_{0}^{1}\varphi_{q} \biggl( \int_{0}^{1}G(\beta,s,\tau) \frac{f (I^{\gamma}v )}{(I^{\gamma}v)^{\epsilon}} \bigl(I^{\gamma }v\bigr)^{\epsilon}\,d\tau \biggr)\,ds \\ &\le e \int_{0}^{1} \varphi_{q} \biggl( \int_{0}^{1} \frac{\beta-1}{\Gamma(\beta)}\tau(1- \tau)^{\beta-1}M \frac {r_{q}^{\epsilon}}{\Gamma^{\epsilon}(\gamma+1)}\,d\tau \biggr)\,ds \\ &\le e \varphi_{q} \biggl(\frac{(\beta-1)M r_{q}^{\epsilon}}{ \Gamma(\beta)\Gamma^{\epsilon}(\gamma+1)} \biggr)=r_{q}. \end{aligned} $$
(3.7)
Equations (3.4) and (3.7) imply that \(T(P[0, r_{q}]) \subset P[0, r_{q}]\).
Let \(v^{(0)}(t) =0\) and \(v^{(1)}(t)= (Tv^{(0)})(t)=(T0)(t)\), \(t \in[0, 1]\), then it follows from \(0\in P([0,r_{q}])\) that \(v^{(1)}(t) \in T(P[0, r_{q}])\). Denote
$$v^{(n+1)} = Tv^{(n)} = T^{n+1} v^{(0)},\quad n = 1, 2, \ldots. $$
It follows from \(T(P[0, r_{q}]) \subset P[0, r_{q}]\) that \(v_{n} \in P[0, r_{q}]\) for \(n\ge1\). Noticing that T is compact, we see that \(\{v^{(n)}\}\) is a sequentially compact set.
On the other hand, since \(v^{(1)}\ge0=v^{(0)}\), we have
$$v^{(2)}(t) = \bigl(Tv^{(1)}\bigr) (t) \ge\bigl(Tv^{(0)} \bigr) (t)= v^{(1)}(t),\quad t \in[0, 1]. $$
By induction, we get
$$v^{(n+1)} \ge v^{(n)},\quad n = 1, 2, \ldots. $$
Consequently, there exists \(v^{*} \in P[0, r_{q}]\) such that \(v^{(n)} \to v^{*}\). Letting \(n \to+\infty\), from the continuity of T and \(Tv^{(n)} =v^{(n-1)}\), we obtain \(Tv^{*} = v^{*}\), which implies that \(v^{*}\) is a nonnegative solution of the BVP (2.2), and thus \(x^{*}=I^{\gamma}v^{*}(t)\) is a nonnegative solution of the BVP (1.1). By (2.8), there exist constants \(0\le m_{1}< n_{1}\) such that
$$m_{1}t^{\alpha-1}\le x^{*}(t)\le n_{1} t^{\alpha-1}, \quad t \in(0, 1). $$
Now let \(w^{(0)}(t)=r_{q}\) and
$$w^{(1)}(t)=\bigl(Tw^{(0)}\bigr) (t),\quad t \in[0, 1]. $$
Since \(w^{(0)}(t)=r_{q} \in P[0, r_{q}]\), \(w^{(1)}\in P[0, r_{q}]\). Let
$$w^{(n+1)} = Tw^{(n)} = T^{(n+1)}w^{(0)},\quad n = 1, 2, \ldots. $$
It follows from \(T(P[0, r_{q}]) \subset P[0, r_{q}]\) that
$$w^{(n)} \in P[0, r_{q}], \quad n = 0, 1, 2, \ldots. $$
From Lemma 2.4, T is compact, and consequently \(\{w^{(n)}\}\) is a sequentially compact set.
Now, since \(w^{(1)} \in P[0, r_{q}]\), we get
$$0 \le w^{(1)}(t) \le\bigl\Vert w^{(1)}\bigr\Vert \le r_{q} = w^{(0)}(t). $$
It follows from (F) that \(w^{(2)} = Tw^{(1)} \le Tw^{(0)}=w^{(1)}\). By induction, we obtain
$$w^{(n+1)} \le w^{(n)},\quad n = 0, 1, 2, \ldots. $$
Consequently, there exists \(w^{*}\in P[0, r_{q}]\) such that \(w^{(n)} \to w^{*}\). Letting \(n \to+\infty\), from the continuity of T and \(Tw^{(n)} = w^{(n-1)}\), we have \(Tw^{*} = w^{*}\), which implies that \(y^{*}=I^{\gamma}w^{*}(t)\) is another nonnegative solution of the boundary value problem (1.1) and \(y^{*}\) also satisfies (3.2) since \(w^{*}\in P\).

At the end, we prove that \(x^{*}\) and \(y^{*}\) are extremal solutions for Equation (1.1). Let ũ be any positive solution of Equation (2.2), then \(v^{(0)}=0\le\widetilde{u}\le r_{q}=w^{(0)}\), and \(v^{(1)}=Tv^{(0)}\le T\widetilde{u}=\widetilde{u}\le T(w^{(0)})=w^{(1)}\). By induction, we have \(v^{(n)}\le\widetilde{u}\le w^{(n)}\), \(n=1,2,3,\ldots\) . Taking the limit, we have \(v^{*}\le\widetilde{u}\le w^{*}\), which implies that \(x^{*}\) and \(y^{*}\) are the maximal and minimal solutions of Equation (1.1), respectively. The proof is completed. □

Corollary 3.1

Assume (F) holds, and
$$ \epsilon(q-1)>1, \quad f(0)=0. $$
(3.8)
Then Equation (1.1) has a minimal solution which is trivial and a positive maximal solution \(y^{*}\), and there exist positive numbers \(0< m_{2}< n_{2}\), such that
$$ m_{2}t^{\alpha-1}\le y^{*}(t) \le n_{2}t^{\alpha-1}, \quad t\in[0,1]. $$
(3.9)
Moreover, for the initial value \(w^{(0)}=r\), let \(\{{w^{(n)}}\}\) be the iterative sequence generated by
$$ {w^{(n)}}(t) = \bigl(T{w^{(n-1)}}\bigr) (t) = \bigl(T^{n}w^{(0)}\bigr) (t). $$
(3.10)
Then
$$\lim_{n\to+\infty}{w^{(n)}}=\boldsymbol{\mathscr{D}}_{\mathbf {t}}^{\gamma}y^{*} $$
uniformly for \(t\in[0,1]\).

Corollary 3.2

If \(p=2\) (\(\varphi_{p}\) reduces to the linear operator), assume that (F) holds and satisfies the following conditions:
$$ \epsilon>1, \quad f(0)\le\rho_{2}. $$
(3.11)
Then Equation (1.1) has a minimal solution \(x^{*}\) and a maximal solution \(y^{*}\), and there exist some nonnegative numbers \(m_{i}< n_{i}\), \(i=1,2\), such that
$$ m_{1}t^{\alpha-1}\le x^{*}(t) \le n_{1}t^{\alpha -1}, \qquad m_{2}t^{\alpha-1}\le y^{*}(t) \le n_{2}t^{\alpha-1}, \quad t\in[0,1]. $$
(3.12)
Moreover, for initial values \(v^{(0)}=0\), \(w^{(0)}=r_{2}\), let \(\{{v^{(n)}}\}\) and \(\{{w^{(n)}}\}\) be the iterative sequences generated by
$$ v^{(n)} = T{v^{(n-1)}} = T^{n}v^{(0)}, \qquad {w^{(n)}} = T{w^{(n-1)}} = T^{n}w^{(0)}. $$
(3.13)
Then \(\lim_{n\to+\infty}{v^{(n)}}= \boldsymbol{\mathscr {D}}_{\mathbf{t}}^{\gamma}x^{*}\), \(\lim_{n\to+\infty}{w^{(n)}}=\boldsymbol{\mathscr{D}}_{\mathbf {t}}^{\gamma}y^{*} \) uniformly for \(t\in[0,1]\).

Corollary 3.3

Suppose (3.1) holds and one of the following assumptions is satisfied:
  1. (f1)

    \(f: [0,+\infty)\to[0,+\infty)\) is continuous and non-decreasing, there exists a constant \(\epsilon>0\) such that \(\frac{f(x)}{x^{\epsilon}}\) is increasing with respect to x and \(\lim_{x\to+\infty}{\frac{f(x)}{x^{\epsilon}}}=M>0\).

     
  2. (f2)

    \(f: [0,+\infty)\to[0,+\infty)\) is continuous and non-decreasing, there exists a constant \(\epsilon>0\) such that \(\frac{f(x)}{x^{\epsilon}}\) is nonincreasing with respect to x and \(\lim_{x\to+0}{\frac{f(x)}{x^{\epsilon}}}=M>0\).

     
  3. (f3)

    \(f: [0,+\infty)\to[0,+\infty)\) is continuous and non-decreasing, and there exists a constant \(\epsilon>0\) such that \(\lim_{x\to+0}{\frac{f(x)}{x^{\epsilon}}}=a>0\), \(\lim_{x\to+\infty}{\frac{f(x)}{x^{\epsilon}}}=b>0\), \(M=\max\{a,b\}\).

     
Then Equation (1.1) has a minimal solution \(x^{*}\) and a maximal solution \(y^{*}\), and there exist some nonnegative numbers \(m_{i}< n_{i}\), \(i=1,2\), such that
$$ m_{1}t^{\alpha-1}\le x^{*}(t) \le n_{1}t^{\alpha -1}, \qquad m_{2}t^{\alpha-1}\le y^{*}(t) \le n_{2}t^{\alpha-1}, \quad t\in[0,1]. $$
(3.14)
Moreover, for initial values \(v^{(0)}=0\), \(w^{(0)}=r\), let \(\{{v^{(n)}}\}\) and \(\{{w^{(n)}}\}\) be the iterative sequences generated by
$$ v^{(n)} = T{v^{(n-1)}} = T^{n}v^{(0)}, \qquad w^{(n)} = T{w^{(n-1)}} = T^{n}w^{(0)}. $$
(3.15)
Then
$$\lim_{n\to+\infty}{v^{(n)}} = \boldsymbol{ \mathscr{D}}_{\mathbf{t}}^{\gamma}x^{*}, \qquad \lim_{n\to+\infty}{w^{(n)}} = \boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\gamma}y^{*}, $$
uniformly for \(t\in[0,1]\).

An example

Consider the following nonlocal boundary value problem of the fractional p-Laplacian equation:
$$ \textstyle\begin{cases} -\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{4}{3}} (\varphi _{\frac{5}{2}} (-\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{3}{2}}x ) )(t) =x^{2}(t), \quad t\in(0,1), \\ \boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{3}{2}}x(0)=\boldsymbol {\mathscr{D}}_{\mathbf{t}}^{\frac{3}{2}}x(1)=\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{1}{4}}x(0)=0, \qquad \boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{1}{4}}(1)=\int ^{1}_{0}\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{1}{4}}x(s)\,dA(s), \end{cases} $$
(3.16)
where A is a bounded variation function satisfying \(A(t)=0\) for \(t\in[0,\frac{1}{2})\), \(A(t)=2\) for \(t\in [\frac{1}{2},\frac{3}{4})\), \(A(t)=1\) for \(t\in[\frac{3}{4},1]\). Then Equation (3.16) has a positive maximal solution \(y^{*}\); and there exist positive numbers \(0< m< n\), such that
$$ mt^{\frac{1}{2}}\le y^{*}(t) \le nt^{\frac{1}{2}}, \quad t \in[0,1]. $$
(3.17)

Proof

By a simple computation, the problem (3.16) reduces to the following multi-point boundary value problem:
$$\textstyle\begin{cases} -\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{4}{3}} (\varphi _{\frac{5}{2}} (-\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{3}{2}}x ) )(t) =x^{2}(t),\quad t\in(0,1), \\ \boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{3}{2}}x(0)=\boldsymbol {\mathscr{D}}_{\mathbf{t}}^{\frac{3}{2}}x(1)=\boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{1}{4}}x(0)=0, \qquad \boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{1}{4}}x(1)=2 \boldsymbol {\mathscr{D}}_{\mathbf{t}}^{\frac{1}{4}}x (\frac{1}{2} ) - \boldsymbol{\mathscr{D}}_{\mathbf{t}}^{\frac{1}{4}}x (\frac{3}{4} ). \end{cases} $$
Let
$$\alpha=\frac{3}{2},\qquad \beta=\frac{4}{3},\qquad p= \frac{5}{2},\qquad f(x)=x^{2}. $$
First, we have
$$\mathcal{A}= \int_{0}^{1}t^{\alpha-1}\,dA(t)=2\times \biggl( \frac{1}{2} \biggr)^{\frac{1}{2}}- \biggl(\frac{3}{4} \biggr)^{\frac{1}{2}}=0.5482< 1, $$
and by a simple computation, we have \(\mathcal{G}_{A}(s)\ge0\).
Obviously, \(f(0)=0\) and
$$\sup_{{s>0}}\frac{f ( s ) }{s^{2}}=1,\qquad \epsilon(q-1)=2 \biggl( \frac{5}{3}-1 \biggr)=\frac{4}{3}>1. $$
By Corollary 3.2, Equation (3.16) has a positive maximal solution which satisfies (3.17). □

Declarations

Acknowledgements

The authors were supported financially by the National Natural Science Foundation of China (11571296,11371221) and the Natural Science Foundation of Shandong Province of China (ZR2014AM009).

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

(1)
School of Economic Mathematics, Southwestern University of Finance and Economics
(2)
School of Mathematical and Informational Sciences, Yantai University
(3)
School of Mathematical Sciences, Qufu Normal University
(4)
Department of Mathematics and Statistics, Curtin University

References

  1. Dybbs, A, Edwards, V: A new look at porous media fluid mechanics - Darcy to turbulent. In: Bear, J, Corapcioglu, MY (eds.) Fundamentals of Transport Phenomena in Porous Media, pp. 199-254. Springer, Netherlands (1984) View ArticleGoogle Scholar
  2. Macdonald, I, El-Sayed, M, Mow, K, Dullien, F: Flow through porous media - Ergun equation revisited. Ind. Eng. Chem. Fundam. 18, 199-208 (1979) View ArticleGoogle Scholar
  3. Kirkhara, C: Turbulent flow in porous media - an analytical and experimental study. Dept. of Civil Engng., Univ. of Melbourne, Australia, Feb (1967) Google Scholar
  4. Mickeley, H, Smith, A, Korchak, I: Fluid flow in packed beds. Chem. Eng. Sci. 23, 237-246 (1965) View ArticleGoogle Scholar
  5. Leibenson, L: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk Kirgiz. SSSR 9, 7-10 (1983) (in Russian) MathSciNetGoogle Scholar
  6. Li, S, Zhang, X, Wu, Y, Caccetta, L: Extremal solutions for p-Laplacian differential systems via iterative computation. Appl. Math. Lett. 26, 1151-1158 (2013) MathSciNetView ArticleMATHGoogle Scholar
  7. Chen, T, Liu, W: An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator. Appl. Math. Lett. 25, 1671-1675 (2012) MathSciNetView ArticleMATHGoogle Scholar
  8. Zhang, X, Liu, L, Wu, Y: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26-33 (2014) MathSciNetView ArticleMATHGoogle Scholar
  9. Zhang, X, Liu, L: Positive solutions of fourth-order four-point boundary value problems with p-Laplacian operator. J. Math. Anal. Appl. 336, 1414-1423 (2007) MathSciNetView ArticleMATHGoogle Scholar
  10. Zhang, X, Liu, L, Wiwatanapataphee, B, Wu, Y: The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition. Appl. Math. Comput. 235, 412-422 (2014) MathSciNetMATHGoogle Scholar
  11. Goodrich, C: The existence of a positive solution to a second-order delta-nabla p-Laplacian BVP on a time scale. Appl. Math. Lett. 25, 157-162 (2012) MathSciNetView ArticleMATHGoogle Scholar
  12. Goodrich, C: Existence of a positive solution to a first-order p-Laplacian BVP on a time scale. Nonlinear Anal. 74, 1926-1936 (2011) MathSciNetView ArticleMATHGoogle Scholar
  13. Ding, Y, Wei, Z, Xu, J, O’Regan, D: Extremal solutions for nonlinear fractional boundary value problems with p-Laplacian. J. Comput. Appl. Math. 288, 151-158 (2015) MathSciNetView ArticleMATHGoogle Scholar
  14. Podlubny, I: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, New York (1999) MATHGoogle Scholar
  15. Miller, K, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  16. El-Saka, H: The fractional-order SIS epidemic model with variable population size. J. Egypt. Math. Soc. 22, 50-54 (2014) MathSciNetView ArticleMATHGoogle Scholar
  17. Yin, D, Duan, X, Zhou, X: Fractional time-dependent deformation component models for characterizing viscoelastic Poisson’s ratio. Eur. J. Mech. A, Solids 42, 422-429 (2013) View ArticleGoogle Scholar
  18. Paola, M, Zingales, M: The multiscale stochastic model of Fractional Hereditary Materials (FHM). Procedia IUTAM 6, 50-59 (2013) View ArticleGoogle Scholar
  19. Mongiovi, M, Zingales, M: A non-local model of thermal energy transport: the fractional temperature equation. Int. J. Heat Mass Transf. 67, 593-601 (2013) View ArticleGoogle Scholar
  20. Tarasov, V: Lattice model of fractional gradient and integral elasticity: long-range interaction of Grünwald-Letnikov-Riesz type. Mech. Mater. 70, 106-114 (2014) View ArticleGoogle Scholar
  21. Zhang, X, Liu, L, Wu, Y, Lu, Y: The iterative solutions of nonlinear fractional differential equations. Appl. Math. Comput. 219, 4680-4691 (2013) MathSciNetMATHGoogle Scholar
  22. Zhang, X, Han, Y: Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. Appl. Math. Lett. 25, 555-560 (2012) MathSciNetView ArticleMATHGoogle Scholar
  23. Graef, J, Kong, L, Yang, B: Positive solutions for a fractional boundary value problem. Appl. Math. Lett. 56, 49-55 (2016) MathSciNetView ArticleMATHGoogle Scholar
  24. Zhang, X, Liu, L: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55, 1263-1274 (2012) MathSciNetView ArticleMATHGoogle Scholar
  25. Cui, Y: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48-54 (2016) MathSciNetView ArticleMATHGoogle Scholar
  26. Zhang, X, Liu, L, Wu, Y: The eigenvalue problem for a singular higher fractional differential equation involving fractional derivatives. Appl. Math. Comput. 218, 8526-8536 (2012) MathSciNetMATHGoogle Scholar
  27. Goodrich, C: On a fractional boundary value problem with fractional boundary conditions. Appl. Math. Lett. 25, 1101-1105 (2012) MathSciNetView ArticleMATHGoogle Scholar
  28. Zhang, X, Liu, L, Wu, Y: Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives. Appl. Math. Comput. 219, 1420-1433 (2012) MathSciNetMATHGoogle Scholar
  29. Goodrich, C: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 61, 191-202 (2011) MathSciNetView ArticleMATHGoogle Scholar
  30. Goodrich, C: Positive solutions to boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 75, 417-432 (2012) MathSciNetView ArticleMATHGoogle Scholar
  31. Zhang, X, Wu, Y, Caccetta, L: Nonlocal fractional order differential equations with changing-sign singular perturbation. Appl. Math. Model. 39, 6543-6552 (2015) MathSciNetView ArticleGoogle Scholar
  32. Zhang, X, Liu, L, Wu, Y, Wiwatanapataphee, B: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252-263 (2015) MathSciNetGoogle Scholar
  33. Zhang, X, Liu, L, Wu, Y: The uniqueness of positive solution for a singular fractional differential system involving derivatives. Commun. Nonlinear Sci. Numer. Simul. 18, 1400-1409 (2013) MathSciNetView ArticleMATHGoogle Scholar
  34. Agarwal, R, Henderson, J: Superlinear and sublinear focal boundary value problems. Appl. Anal. 60, 189-200 (1996) MathSciNetView ArticleMATHGoogle Scholar
  35. Axelsson, O, Kaporin, I: On the sublinear and superlinear rate of convergence of conjugate gradient methods. Numer. Algorithms 25, 1-22 (2000) MathSciNetView ArticleMATHGoogle Scholar
  36. Papageorgiou, N, Rocha, E: Pairs of positive solutions for p-Laplacian equations with sublinear and superlinear nonlinearities which do not satisfy the AR-condition. Nonlinear Anal. 70, 3854-3863 (2009) MathSciNetView ArticleMATHGoogle Scholar
  37. Ma, R, Thompson, B: Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities. J. Math. Anal. Appl. 303, 726-735 (2005) MathSciNetView ArticleMATHGoogle Scholar
  38. Bai, Z, Lv, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005) MathSciNetView ArticleGoogle Scholar

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