Abstract
In this paper, we focus on the convergence analysis and error estimation for the unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. By introducing a double iterative technique, in the case of the nonlinearity with singularity at time and space variables, the unique positive solution to the problem is established. Then, from the developed iterative technique, the sequences converging uniformly to the unique solution are formulated, and the estimates of the error and the convergence rate are derived.
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1 Introduction
This paper is motivated by the following singular nonlocal fractional differential equation:
where χ is a function of bounded variation satisfying \(\chi (x)=0\), \(x\in [ 0,\frac{1}{3} ) \), \(\chi(x)=\frac{1}{2}\), \(x\in [ \frac{1}{3},\frac{2}{3} ) \), \(\chi(x)= 1\), \(x\in [ \frac {2}{3},1 ] \), which exhibits a blow-up behaviour at \(x=0\) and \(z=0\). These types of singular behaviours [1–11] as well as impulsive phenomena [12–21] often exhibit some blow-up properties [22, 23] which occur in many complex physical processes, for example, in mechanics process [1], the stress near the crack tip in elastic fracture exhibits a singularity of \(r^{-0.5}\), where r is the distance measured from the crack tip.
Inspired by the above problem, this paper presents the convergence analysis and error estimation for the unique solution of the general fractional differential equation with singular decreasing nonlinearity and a p-Laplacian operator
where \(\pmb{\mathscr{D}_{x}}^{\alpha}\), \(\pmb{\mathscr{D}_{x}}^{\gamma}\) are the standard Riemann–Liouville derivatives with \(\gamma,\alpha \in(1,2]\), \(\int^{1}_{0}z(x)\,d\chi(x)\) is a Riemann–Stieltjes integral and χ is a function of bounded variation, \(\varphi_{p}(x) = \vert x \vert ^{p-2}x\), \(p > 1\) is the p-Laplacian operator with conjugate index \(q> 1\) satisfying \(\frac{1}{p} + \frac{1}{q} =1\).
Fractional calculus is a new research area of analytical mathematics which provides many useful tools for modelling various complex physical and biological processes with long memory [24–31]. For example, in fluid dynamics, laboratory data [24] and numerical experiment [25] show that solutes moving through a highly heterogeneous aquifer do not abide by Fick’s first law, and thus in order to improve the accuracy of the model, one can adopt fractional order advection–dispersion equation to describe the convection–diffusion process in a highly heterogeneous aquifer, see [24, 32–40]. In biomedicine, Arafa et al. [41] introduced a fractional-order HIV-1 infection of CD4+ T cells dynamics model and then used the generalised Euler method to find a numerical solution of the HIV-1 infection fractional order model. Subsequently, by analytical techniques, Wang et al. [42] and Zhang et al. [43] studied the existence of positive solution for some abstract fractional dynamic systems for bioprocess, respectively.
On the other hand, the p-Laplacian equation is a second order quasilinear differential operator with the ability of modelling various fundamental nonlinear phenomena in non-Newtonian fluids, nonlinear elasticity, torsional creep problem, radiation of heat, etc. [44–56]. Thus fractional order differential equations with p-Laplacian operator not only can describe the nonlinear phenomena in non-Newtonian fluids but also can model complex processes with long memory. For example, by using the monotone iterative technique, Wu et al. [57] investigated the existence of twin iterative solutions for a fractional differential turbulent flow model
where \(\pmb{\mathscr{D}_{x}}^{\gamma}\), \(\pmb{\mathscr{D}_{x}}^{\alpha}\) are the standard Riemann–Liouville derivatives such that \(1 <\alpha ,\gamma\le 2\), and \(h:[0,+\infty)\to[0,\infty)\) is a continuous and increasing function in the variable. The above work (also see [58–67]) shows that the monotone iterative technique is an effective analysis tool for obtaining iterative solutions and numerical solutions of the relative differential equations. However, to the best of our knowledge, in the application of iterative techniques, almost all works require the nonlinear term to be increasing in space variables and not to have singularity at space variables. So, even for the simplest case as Eq. (1.1), iterative solutions are difficult to construct by using classical iterative techniques. Thus in this paper, by introducing a double iterative technique, we study the convergence analysis and error estimation of the unique solution for the case where the nonlinearity in the equation is decreasing in space variables and is allowed to be singular at some time and space variables.
This paper is organised as follows. In Sect. 2, we firstly recall the definitions and properties of the Riemann–Liouville fractional derivative and integral, and then give some lemmas which will be used in the rest of this paper. In Sect. 3, we introduce a double iterative technique and establish the condition for which Eq. (1.2) has a unique positive solution, then from the developed iterative technique, the sequences converging uniformly to the unique positive solution are formulated, and the estimates of the approximation error and the convergence rate are derived.
2 Preliminaries and lemmas
In this section, we firstly recall the definitions and properties of the Riemann–Liouville fractional derivative and integral, and then give some useful lemmas.
Definition 2.1
([68])
The Riemann–Liouville fractional integral of order \(\gamma>0\) of a function \(z:(0,+\infty)\rightarrow\mathbb{R}\) is given by
provided that the right-hand side is pointwise defined on \((0,+\infty)\).
Definition 2.2
([68])
The Riemann–Liouville fractional derivative of order \(\gamma>0\) of a function \(z:(0,+\infty)\rightarrow\mathbb{R}\) is given by
where \(n=[\gamma]+1\), \([\gamma]\) denotes the integer part of number γ, provided that the right-hand side is pointwise defined on \((0,+\infty)\).
Property 2.1
([68])
-
(1)
If \(z\in L^{1}(0, 1)\), \(\gamma>\alpha> 0\), then
$$I^{\gamma}I^{\alpha}z(x)=I^{\gamma+\alpha}z(x), \quad\quad \pmb{\mathscr {D}_{x}}^{\alpha}I^{\gamma} z(x)=I^{\gamma-\alpha} z(x),\quad\quad \pmb{\mathscr{D}_{x}}^{\alpha}I^{\alpha } z(x)=z(x). $$ -
(2)
If \(\gamma>0\), \(\alpha>0\), then
$$\pmb{\mathscr{D}_{x}}^{\gamma} x^{\alpha-1}= \frac{\Gamma(\alpha )}{\Gamma(\alpha-\gamma)}x^{\alpha-\gamma-1}. $$ -
(3)
Let \(\gamma> 0\), and \(z(x)\) is integrable, then
$$I^{\gamma}\pmb{\mathscr{D}_{x}}^{\gamma}z(x)=z(x)+c_{1}x^{\gamma -1}+c_{2}x^{\gamma-2}+ \cdots+c_{n}x^{\gamma-n}, $$where \(c_{i}\in\mathbb{R}\) (\(i=1,2,\ldots,n\)), n is the smallest integer greater than or equal to γ.
According to the definitions and properties of the Riemann–Liouville fractional derivative and integral and discussion in [34], we have the following lemma.
Lemma 2.1
Given \(h\in L^{1}(0, 1)\), the following boundary value problem
has the unique solution
where
with an index α.
On the other hand, by using Property 2.1(3), we get that the unique solution of the equation
is \(x^{\gamma-1}\). Thus let
and according to the strategy of [45], we have the following lemma.
Lemma 2.2
Suppose \(1<\gamma\le2\) and \(h\in L^{1}(0,1)\), then the following nonlocal boundary value problem
has the unique solution
where
Lemma 2.3
([69])
Let \(0\le\mathcal{L} < 1\) and \(\mathcal{K}_{\chi}(y)\ge0\) for \(y\in[0, 1]\), then \(K_{\alpha}(x,y)\) and \(H(x,y)\) have the following properties:
-
(1)
\(K_{\alpha}(x,y)\) and \(H(x,y)\) are nonnegative and continuous for \((x,y)\in[0,1]\times[0,1]\).
-
(2)
\(K_{\alpha}(x,y)\) satisfies
$$ \frac{x^{\alpha -1}(1-x)y(1-y)^{\alpha-1}}{\Gamma(\alpha)}\leq K_{\alpha}(x,y) \leq\frac{\alpha-1}{\Gamma(\alpha)}y(1-y)^{\alpha -1},\quad \textit{for } x,y\in [0,1]. $$(2.7) -
(3)
There exist two constants a, b such that
$$ a{x^{\gamma-1}}\mathcal{K}_{\chi}(y)\le H(x,y)\le b x^{\gamma-1}, \quad y,x\in[0,1]. $$(2.8)
Let q be the conjugate index of p, and consider the following associated linear nonlocal boundary value problem:
for \(h\in L^{1}(0,1)\) and \(h\ge0\). We have the following result.
Lemma 2.4
The associated linear nonlocal boundary value problem (2.9) has a unique positive solution with the form
Proof
Let \(w =-\pmb{\mathscr{D}_{x}}^{\gamma}z\), \(v = \varphi _{p}(w)=\varphi_{p}(-\pmb{\mathscr{D}_{x}}^{\gamma}z)\), then we have
Now consider the fractional Dirichlet boundary value problem
It follows from Lemma 2.1 that
Thus by (2.9)–(2.11), one gets that the solution of (2.9) satisfies
Hence, according to Lemma 2.2, the solution of the boundary value problem (2.9) can be written by
As \(h(y)\ge0\), \(y\in[0,1]\), the solution of Eq. (2.9) is also positive. □
3 Main results
In this section, we firstly list some assumptions and then give the proof of our main results.
- (\(K_{0}\)):
-
χ is a function of bounded variation satisfying \(\mathcal{K}_{\chi}(y)\ge0\) for \(y\in[0, 1]\) and \(0\le\mathcal{L}<1\).
- (\(F_{1}\)):
-
\(f\in C((0, 1)\times(0, +\infty), [0, +\infty))\), and \(f(x, z)\) is decreasing in z and for any \(r\in(0, 1)\), there exists a constant \(0<\mu<\frac{1}{p-1}\) such that, for any \((x, z)\in(0, 1)\times(0, +\infty)\),
$$ f(x, rz)\leq r^{-\mu}f(x, z). $$(3.1)
Remark 3.1
Obviously, if \(p=\frac{3}{2}\), then \(f(x,z)=x^{-1}z^{-\frac{1}{2}}\) satisfies the assumption (\(F_{1}\)) which implies that f can be allowed to have singularity at \(x=0\) and \(z=0\).
Remark 3.2
If (\(F_{1}\)) holds, from (3.1), for any \(r\geq1\), one has the following equivalent statement:
In this paper, our work space is a Banach space \(E=C[0, 1]\) with the norm \(\Vert {z} \Vert = \max_{x\in[0, 1]} \mid{z(x)}\mid\) for any \(z\in E\). Let \(P={\{z\in C[0, 1]: z(x)\geq0, x\in[0, 1]\}}\), then P is a normal cone of E with normality constant 1. Now define a subset of P and a nonlinear integral operator T: \(E \rightarrow E\) by
and
It follows from Lemma 2.4 that \(z\in C[0, 1]\) is a solution of the p-Laplacian fractional differential Eq. (1.2) if and only if \(z\in C[0, 1]\) is a fixed point of the nonlinear operator T.
Theorem 3.1
Suppose (\(K_{0}\)) and (\(F_{1}\)) hold. If
then
-
(i)
the p-Laplacian fractional differential Eq. (1.2) has a unique positive solution \(z^{*}\in C[0,1]\);
-
(ii)
for any initial value \(z_{0}\in Q\), the sequence of functions \(\{z_{n}\}_{n\ge1}\) defined by
$$ \begin{aligned}&z_{n}= \int_{0}^{1}H(x,y) \biggl( \int _{0}^{1}K_{\alpha}(y,\tau)f \bigl( \tau,z_{n-1}(\tau) \bigr)\,d\tau \biggr) ^{q-1}\,dy, \quad n=1, 2, 3,\ldots, \end{aligned} $$(3.5)converge uniformly to the unique positive solution \(z^{*}\) of Eq. (1.2) on [0,1];
-
(iii)
the error between the iterative value \(z_{n}\) and the exact solution \(z^{*}\) can be estimated by
$$\begin{aligned} \bigl\Vert z_{n}-z^{*} \bigr\Vert \leq{ \bigl( 1- \epsilon^{[\mu (q-1)]^{2n}} \bigr) }\epsilon^{{-\frac{1}{2}}}, \end{aligned}$$with an exact convergence rate
$$\begin{aligned} \bigl\Vert {z_{n}- z^{*}} \bigr\Vert =o{ \bigl( 1- \epsilon^{[\mu(q-1)]^{2n}} \bigr) }, \end{aligned}$$where \(0<\epsilon<1\) is a positive constant.
-
(iv)
there exists a constant \(0< l<1\) such that the exact solution \(z^{*}\) of Eq. (1.2) intervenes between two known curves \(lx^{\gamma-1}\) and \(l^{-1} x^{\gamma-1}\), i.e.,
$$lx^{\gamma-1}\le z^{*}(x)\le l^{-1} x^{\gamma-1}, \quad x \in[0,1]. $$
Proof
Step 1. We show that T: \(Q\rightarrow Q\) is a compact operator.
In fact, for any \(z\in Q\), it follows from the definition of the set Q that there exists a constant \(0< l_{z}<1\) such that
Notice that \(f(x, z)\) is decreasing in z, by Lemma 2.3, (3.1), (3.4) and (3.6), one has
So T is well defined and uniformly bounded.
On the other hand, since \(H(x,y)\) is uniformly continuous on \([0, 1]\times[0, 1]\), let \(0\le x_{1} < x_{2} \le1\), for all \(z\in Q\), one has
which implies that \(T(Q)\) is equicontinuous, and then T is a compact operator in Q.
In the following, we shall show that \(T(Q)\subset Q\). In fact, by (2.7), (2.8), (3.6) and (3.1), for any \(z\in Q\), we have
and
where \(\widetilde{l}_{T_{z}}\) satisfies
Hence we have \(T(Q)\subset Q\).
Step 2. In this step, we prove that Eq. (1.2) has a unique positive solution \(z^{*}\in C[0,1]\).
In fact, let \(\eta(x)=x^{\gamma-1}\), then \(\eta\in Q\). By Step 1, we have \(T\eta\in Q\). Thus there exists a constant \(l_{T_{\eta}}\) such that \(0 < l_{T_{\eta}} < 1\) and
where \(l_{T_{\eta}}\) can be chosen as in (3.9). Notice that \(0<\mu (q-1)<1\), for some \(\kappa\in(0,1)\), we can choose a sufficiently large positive constant σ such that
Now fix the initial value \(z_{0}=\kappa^{\sigma}\eta(x)\) and let
We firstly show
In fact, since T is a decreasing operator in z, it follows from (3.10)–(3.12) that
and then
On the other hand, it follows from (3.1) and (3.10) that
and then by (3.2), (3.10), (3.16) and the monotonicity of T, one gets
Equation (3.14), (3.15) and (3.17) yield
Consequently, by applying induction for (3.18), we obtain (3.13).
Now, for any \(c\in(0, 1)\), from (3.1) and (3.3) we have
Noticing that \(T^{2}\) is a nondecreasing operator with respect to z, by using (3.19) repeatedly, we obtain
that is,
Consequently, for all natural numbers n and p, one has
and
It follows from (3.22), (3.23) and the fact that P is a normal cone with normality constant 1 that
Since \(\{z_{n}\}\in Q\) and \(T(Q)\subset Q\) is compact, \(\{z_{n}\}\) is a Cauchy sequence of compact set, and then \(\{z_{n}\} \) converges to some \(z^{*}\in Q\) with
So
Let \(n \longrightarrow\infty\) in (3.25), we get \(z^{*}(x) = Tz^{*}(x)\), which implies that \(z^{*}\) is a positive solution of Eq. (1.2).
Now we prove \(z^{*}\in Q\) is unique. Let z̃ be another positive solution of Eq. (1.2). Take \(r_{1}=\sup\{r>0 \mid\tilde{z}\geq r{z^{*}}\}\). Obviously, \(0< r_{1}< +\infty\). We assert \(r_{1}\geq1\). If not, we have \(0< r_{1}<1\), which leads to
Since \(r_{1}^{\mu^{2}(q-1)^{2}}>r_{1}\), this contradicts with the definition of \(r_{1}\). Hence \(r_{1}\geq1\) and \(\tilde{z} \geq z^{*}\). Similarly, we also have \(\tilde{z}\leq z^{*}\). Therefore \(\tilde{z}= z^{*}\), which implies that the positive solution of Eq. (1.2) is unique.
Step 3. At the end, we give the convergence analysis and error estimation for the unique solution of Eq. (1.2).
For any initial value \({\omega_{0}}\in Q\), there exists a constant \(l_{\omega_{0}}\in(0,1)\) such that
Since \(T(Q)\subset Q\), there still exists a constant \(l_{\omega _{1}}\in(0,1)\) such that
Choose sufficiently large \(\widetilde{\sigma}>2\sigma\) such that
where \(\kappa\in(0,1)\) and \(\sigma>0\) are defined by (3.11). Thus
which implies that \(\omega_{1}=T\omega_{0}\le Tz_{0}=z_{1}\), and then
Let
it follows from (3.26) and (3.27) that
Letting \(n \rightarrow\infty\) in (3.28) and using (3.25), we get that \(\omega_{n}\) uniformly converges to the unique positive solution \(z^{*}\) of Eq. (1.2).
Moreover, by (3.23) and (3.28), we have the following estimate of error:
with an exact rate of convergence
where \(0<\epsilon=\kappa^{2\sigma}<1\) is a positive constant which is determined by \(z_{0}=\kappa^{\sigma}\eta(x)\), that is, it is independent of the initial value \({\omega_{0}}\).
At the end, it follows from \(z^{*}\in Q\) that there exists a constant \(0< l_{1}<1\) such that
The proof is completed. □
4 Example
Now we recall the singular nonlocal fractional differential Eq. (1.1). By simple computation, we get that Eq. (1.1) is equivalent to the following 4-point boundary value problem:
In the following, we shall verify that Eq. (1.1) satisfies all conditions of Theorem 3.1. Let \(\alpha=\frac{3}{2}\), \(\gamma=\frac {4}{3}\), \(p=\frac{3}{2}\) and
then \(f\in C((0, 1)\times[0,\infty), [0,+\infty))\), and for any fixed \(x \in(0, 1)\), \(f(x,z)\) is nondecreasing in z.
Take \(\mu=\frac{2}{3}\), then \(0<\mu<\frac{1}{p-1}=2\). For any \(r\in (0, 1)\) and \((x, z)\in(0, 1)\times(0, +\infty)\), we have
Thus condition (\(F_{1}\)) is satisfied.
Next we verify condition (\(K_{0}\)). In fact, since
we have
and
So condition (\(K_{0}\)) is also satisfied.
Now we check condition (3.1). In fact, substituting \(f(x,z)=x^{-1}z^{-\frac{1}{2}}\) into (3.1), we get
which implies that (3.1) holds. Thus, according to Theorem 3.1, we have the following conclusions:
-
(i)
the p-Laplacian fractional differential Eq. (1.1) has a unique positive solution \(z^{*}\in C[0,1]\);
-
(ii)
for any initial value \(z_{0}\in Q\), the sequence of functions \(\{z_{n}\}_{n\ge1}\) defined by
$$\begin{aligned}z_{n}&= \int_{0}^{1} \biggl[ \frac{x^{\frac{1}{3}}}{0.2635} \biggl( \frac{1}{2}K_{\frac{4}{3}} \biggl( \frac{1}{3},y \biggr) + \frac {1}{2}K_{\frac{4}{3}} \biggl( \frac{2}{3},y \biggr) \biggr) +K_{\frac {4}{3}}(x,y) \biggr] \\ &\quad{}\times \biggl( \int_{0}^{1}K_{\frac{3}{2}}(y,\tau) \tau^{-1}z_{n}^{-\frac {1}{2}}(\tau)\,d\tau \biggr) ^{2}\,dy, \\ & \quad n=1, 2, 3,\ldots, \end{aligned} $$converges uniformly to the unique positive solution \(z^{*}\) of Eq. (1.1) on [0,1];
-
(iii)
the error between the iterative value \(z_{n}\) and the exact solution \(z^{*}\) can be estimated by
$$\begin{aligned} \bigl\Vert z_{n}-z^{*} \bigr\Vert \leq{ \bigl( 1- \epsilon^{[\frac {4}{3}]^{2n}} \bigr) }\epsilon^{{-\frac{1}{2}}}, \end{aligned}$$and the convergence rate can be formulated by
$$\begin{aligned} \bigl\Vert {z_{n}- z^{*}} \bigr\Vert =o{ \bigl( 1- \epsilon^{[\frac{4}{3}]^{2n}} \bigr) }, \end{aligned}$$where \(0<\epsilon<1\) is a positive constant which is determined by the fixed function \(\kappa^{\sigma}x^{\frac{4}{3}}\);
-
(iv)
there exists a constant \(0< l<1\) such that the exact solution \(z^{*}\) of Eq. (1.1) intervenes between two known curves \(lx^{\frac{2}{3}}\) and \(l^{-1} x^{\frac{2}{3}}\), i.e.,
$$lx^{\frac{2}{3}}\le z^{*}(x)\le l^{-1} x^{\frac{2}{3}}, \quad x \in[0,1]. $$
5 Conclusion
In this paper, by introducing a double iterative technique, we established the convergence analysis and error estimation for the unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. The equation we studied in the present paper exhibits a blow-up behaviour at time and space variables, which occurs in many complex physical processes, such as mechanics processes, the convection-diffusion process and the bioprocess with long memory. The developed double iterative technique can be applied for solving the case where the nonlinear term is decreasing and has singularity at time and space variables.
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Wu, J., Zhang, X., Liu, L. et al. The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. Bound Value Probl 2018, 82 (2018). https://doi.org/10.1186/s13661-018-1003-1
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DOI: https://doi.org/10.1186/s13661-018-1003-1