The a posteriori error estimate in fractional differential equations using generalized Jacobi functions

1.   Introduction

This article is devoted to the following nonlinear fractional differential equation with periodic boundary condition

where $\lambda\leq0$, $0 < \alpha\leq1$ and $^{c}D_{0^+}^{\alpha}$ is Caputo fractional derivative

Differential equations of fractional order occur more frequently on different research areas and engineering, such as physics, economics, chemistry, control theory, etc. In recent years, boundary value problems for fractional differential equation have become a hot research topic, see ^{[2,3,4,5,6,7,9,10,11,12,13,15,16,17,18,19,20,21,24,25]}. In ^[27], Zhang studied the boundary value problem for nonlinear fractional differential equation

where $1 < \alpha\leq2$, $f:[0, 1]\times[0, +\infty)\rightarrow [0, +\infty)$ is continuous and $^{c}D_{0^+}^{\alpha}$ is Caputo fractional derivative

The author obtained the existence of the positive solutions by using the properties of the Green function, Guo-Krasnosel'skill fixed point theorem and Leggett-Williams fixed point theorem.

Ahmad and Nieto ^[1] studied the anti-periodic boundary value problem of fractional differential equation

where $f:[0, T]\times\mathbb{R}\rightarrow \mathbb{R}$ is continuous. The authors obtained some existence and uniqueness results by applying fixed point principles. The anti-periodic boundary value condition in this article corresponds to the anti-periodic condition $u(0) = -u(T), u'(0) = -u'(T)$ in ordinary differential equation.

In ^[26], Zhang studied the following fractional differential equation

where $0 < \delta < 1$, $T > 0$, $u_0\in\mathbb{R}$ and $D_{0^+}^{\delta}$ is Riemann-Liouville fractional derivative

The author obtained the existence and uniqueness of the solutions by the method of upper and lower solutions and monotone iterative method.

In ^[7], Belmekki, Nieto and Rodriguez-Lopez studied the following equation

where $0 < \delta < 1$, $\lambda\in \mathbb{R}$, $f$ is continuous. The authors obtained the existence and uniqueness of the solutions by using the fixed point theorem. Cabada and Kisela ^[8] studied the following equation

where $0 < \delta < 1$, $\lambda\neq0 (\lambda\in \mathbb{R})$, $f$ is continuous. The authors studied the existence and uniqueness of periodic solutions by using Krasnosel'skii fixed point theorem and monotone iterative method. In ^[7,8], the boundary condition $\lim_{t\rightarrow 0^{+}}t^{1-\delta}u(t) = u(1)$ was called as periodic boundary value condition of Riemann-Liouville fractional differential equation, which is different from the periodic condition for ordinary differential equation. The boundary value condition $u(0) = u(1)$ is not suitable for Riemann-Liouville fractional differential equation.

For the ordinary differential equation, the periodic boundary value problem is closely related to the periodic solution. For the Caputo fractional differential equation, the periodic boundary value condition $u(0) = u(w)$ is meaningful. As far as we know, few work involves the periodic boundary value problem for Caputo fractional. The aim of this paper is to show the existence of positive solutions of (1.1) by using Krasnosel'skii fixed point theorem. Meanwhile, we also use the monotone iterative method to study the extremal solutions problem

The paper is organized as follows. In Section 2, we recall and derive some results on Mittag-Leffler functions. In Section 3, we use the Laplace transform to obtain the solution of a linear problem and discuss some properties of Green's function. In Section 4, the existence of positive solution is studied by using the Krasnosel'skii fixed point theorem. In Section 5, the existence of extremal solutions is proved by utilizing the monotone iterative technique. Section 6 is conclusion of the paper.

2.   Preliminaries

A key role in the theory of linear fractional differential equation is played by the well-known two-parameter Mittag-Leffler function

We recall and derive some of their properties and relationships summarized in the following.

Proposition 2.1. Let $\alpha\in(0, 1], \beta > 0, \lambda\in\mathbb{R}$ and $\xi > 0$. Then it holds

$(C_{1})$ $\lim_{t\rightarrow0^{+}}E_{\alpha, \beta}(\lambda t^{\alpha}) = \frac{1}{\Gamma(\beta)}, \lim_{t\rightarrow0^{+}}E_{\alpha, 1}(\lambda t^{\alpha}) = 1.$

$(C_{2})$ $E_{\alpha, \alpha+1}(\lambda t^{\alpha}) = \lambda^{-1}t^{-\alpha}(E_{\alpha, 1}(\lambda t^{\alpha})-1).$

$(C_{3})$ $E_{\alpha, \alpha}(\lambda t^{\alpha}) > 0$, $E_{\alpha, 1}(\lambda t^{\alpha}) > 0$ for all $t\geq 0.$

$(C_{4})$ $E_{\alpha, \alpha}(\lambda t^{\alpha})$ is decreasing in $t$ for $\lambda < 0$and increasing for $\lambda > 0$ for all $t > 0.$

$(C_{5})$ $E_{\alpha, 1}(\lambda t^{\alpha})$ is decreasing in $t$ for $\lambda < 0$ and increasing for$\lambda > 0$ for all $t > 0.$

$(C_{6})$ $\int_{0}^{\xi}t^{\beta-1}E_{\alpha, \beta}(\lambda t^{\alpha})dt = \xi^{\beta}E_{\alpha, \beta+1}(\lambda \xi^{\alpha}).$

Proof. $(C_{1})$ It is obtained by an immediate calculation from (2.1).

$(C_{2})$ By (2.1), we get

Hence,

$(C_{3})$ It follows from [23,Lemma 2.2].

$(C_{4})$ It follows from [8,Proposition 1].

$(C_{5})$ By a direct calculation, we get

since $\alpha\in (0, 1]$, $t > 0$ and $E_{\alpha, \alpha}(\lambda t^{\alpha})$ is positive by Proposition $(C_{3})$, the assertion is proved.

$(C_{6})$ It follows from (1.99) of ^[20].

3.   Linear problem

In this section, we deal with the linear case that $f(t, x) = f(t)$ is a continuous function by mean of the Laplace transform for caputo fractional derivative

where $L$ denotes the Laplace transform operator, $X(s)$ denotes the Laplace transform of $x(t)$.

From Lemma 3.2 of ^[14], we get

and

We do Laplace transform to the equation

By (3.1), we obtain

where $F$ denotes the Laplace transform of $f$. By (3.2) and (3.3), we obtain that

Hence,

which implies that

if $E_{\alpha, 1}(\lambda\omega^{\alpha})\neq 1$. Therefore, if $E_{\alpha, 1}(\lambda\omega^{\alpha})\neq 1$, the solution of the problem (3.4) is

Theorem 3.1. Let $E_{\alpha, 1}(\lambda\omega^{\alpha})\neq1$, the periodic boundary value problem (3.4) has a unique solutiongiven by

where

Remark 3.2. The unique solution $x$ of (3.4) is continuous on $[0, \omega]$.

Lemma 3.3. Let $0 < \alpha\leq1, \lambda\neq0$ and sign$(\eta)$ denotes the signumfunction. Then

$(F_{1})$ $\lim_{t\rightarrow 0^{+}}G_{\alpha, \lambda}(t, s) = \frac{E_{\alpha, \alpha}(\lambda(\omega-s)^{\alpha})}{(1-E_{\alpha, 1}(\lambda\omega^{\alpha}))(\omega-s)^{1-\alpha}}$ for anyfixed $s\in [0, \omega),$

$(F_{2})\; \lim_{s\rightarrow \omega^{-}}G_{\alpha, \lambda}(t, s) = sign(-\lambda)\cdot\infty$ for any fixed $t\in[0, \omega],$

$(F_{3})\; \lim_{t\rightarrow s^{+}}G_{\alpha, \lambda}(t, s) = \infty$ for any fixed$s\in[0, \omega),$

$(F_{4})\; G_{\alpha, \lambda}(t, s) > 0$ for $\lambda < 0$ and for all $t\in[0, \omega]$ and $s\in[0, \omega),$

$(F_{5})\; G_{\alpha, \lambda}(t, s)$ changes its sign for $\lambda > 0$ for $t\in[0, \omega]$ and $s\in[0, \omega).$

Proof. $(F_{1})$ When $0\leq t\leq s < \omega$, by Proposition 2.1 $(C_{1})$ we can get $(F_{1})$.

$(F_{2})$ When $0\leq t\leq s < \omega$, it follows by Proposition 2.1 $(C_{5})$ that $1-E_{\alpha, 1}(\lambda\omega^{\alpha})$ is positive for $\lambda < 0$ and negative for $\lambda > 0.$ The unboundedness is implied by continuity of Mittag-Leffler function, Proposition 2.1 $(C_{1})$ and the relation $\lim_{t\rightarrow 0^{+}}t^{-r} = \infty$ for $r > 0$.

$(F_{3})$ When $0\leq s < t\leq\omega$, the first term of $(3.6)$ is finite due to the continuity of the involved functions. And by a similar argument as in the previous point of this proof we have the second term tends to infinity.

$(F_{4})$ It is obtained by the positivity of all involved functions (Proposition 2.1 $(C_{3})$) and the inequation $1-E_{\alpha, 1}(\lambda\omega^{\alpha}) > 0$ for $\lambda < 0$.

$(F_{5})$ When $0\leq s < t\leq\omega$, the second term of $(3.6)$ is positive due to

and by the positivity of all involved functions (Proposition 2.1 ($C_{3})$) we get the proof. When $0\leq t\leq s < \omega$, it is obtained by $1-E_{\alpha, 1}(\lambda\omega^{\alpha}) < 0$ for $\lambda > 0$ and the positivity of all involved functions (Proposition 2.1 ($C_{3})$).

Proposition 3.4. Let $\alpha\in(0, 1]$ and $\lambda < 0$. Then the Green's function (3.6) satisfies

$(K_{1})$ $G_{\alpha, \lambda}(t, s)\geq m = :\frac{E_{\alpha, 1}(\lambda\omega^{\alpha})E_{\alpha, \alpha}(\lambda\omega^{\alpha})}{|\lambda|\omega E_{\alpha, \alpha+1}(\lambda\omega^{\alpha})} > 0,$

$(K_{2})\; \int_{0}^{\omega}G_{\alpha, \lambda}(t, s)ds = M = :\frac{1}{|\lambda|}$ for all $t\in[0, \omega]$.

Proof. $(K_{1})$ For $0\leq t\leq s < \omega$, we deduce from Proposition 2.1 $(C_{4}), (C_{5})$ that $G_{\alpha, \lambda}$ has the minimum on the line $t = s$. Hence,

For $0\leq s < t\leq\omega$, we have

$(K_{2})$ Employing Proposition 2.1, we get

which completes the proof.

4.   Existence of positive solution

Let $C[0, \omega]$ be the space continuous function on $[0, \omega]$ with the norm $\|x\| = \sup\{|x(t)|:t\in [0, \omega]\}$.

In this section, we always assume that $\lambda < 0$. Clearly, $x$ is a solution of (1.1) if and only if

where $G_{\alpha, \lambda}$ is Green's function defined in Theorem 3.1.

The following famous Krasnosel'skii fixed point theorem, which is main tool of this section.

Theorem 4.1. ^[22] Let $B$ be a Banach space, and let $P\subset B$ be a cone. Assume$\Omega_{1}, \Omega_{2}$ two open and bounded subsets of $B$ with$0\in\Omega_{1}, \Omega_{1}\subset\Omega_{2}$ and let$A:P\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\rightarrow P$ bea completely continuous operator such that one of the followingconditions is satisfied:

$(L_{1})$ $\|Ax\|\leq\|x\|$, if $x\in P\cap\partial\Omega_{1}$, and$\|Ax\|\geq\|x\|$, if $x\in P\cap\partial\Omega_{2}$,

$(L_{2})$ $\|Ax\|\geq\|x\|$, if $x\in P\cap\partial\Omega_{1}$, and$\|Ax\|\leq\|x\|$, if $x\in P\cap\partial\Omega_{2}$.

Then, $A$ has at least one fixed point in$P\cap(\overline{\Omega}_{2}\setminus\Omega_{1})$.

Proposition 4.2. Assume that there exist $0 < r < R, 0 < c_1 < c_2$ such that

Let $P\subset C[0, \omega]$ be the cone

Then the operator $A:\overline{P}_R\setminus P_r\rightarrow P$ givenby

is completely continuous, where $P_l = \{u\in P:\|u\| < l\}$.

Proof. Let $x\in \overline{P}_R\setminus P_r$, then

We first show that $A$ is well-defined, i.e. that $A:\overline{P}_R\setminus P_r\rightarrow P$. Note that

where $k = \frac{1}{1-E_{\alpha, 1}(\lambda\omega^{\alpha})}$ and

Clearly, for $t\in (0, \omega]$

which implies that $q$ is continuous at $t = 0$. On the other hand,

where

Since

we obtain that $H_{1} \in C[0, \omega]$ and

Noting that $u_{k}\in C(0, \omega]$

we have $H_{2}\in C(0, \omega].$ Hence, $q\in C[0, \omega]$.

Moreover,

which means that $A:\overline{P}_R\setminus P_r\rightarrow P.$

Next, we show that $A$ is continuous on $\overline{P}_R\setminus P_r$. Let $x_n, x\in \overline{P}_R\setminus P_r$ and $\|x_n-x\|\rightarrow 0$. From (4.2), we have $\|f(t, x_n(t))-f(t, x(t))\|\rightarrow 0$,

which implies that $A$ is continuous. From (4.6), we get that $Ax(t)$ is uniformly bounded. Finally, we show that $\{Ax|x\in\overline{P}_R/P_r\}$ is an equicontinuity in $C[0, \omega]$. By (4.6), we have

Since $E_{\alpha, 1}(\lambda t^{\alpha})\in C[0, \omega]$, $q(t)\in C[0, \omega]$ are uniformly continuous, $|E_{\alpha, 1}(\lambda t_{1}^{\alpha})-E_{\alpha, 1}(\lambda t_{2}^{\alpha})|$ and $|q(t_{1})-q(t_{2})|$ tend to zero as $|t_{1}-t_{2}|\rightarrow 0$. Hence, $\{Ax(t)|x\in \overline{P}_R\setminus P_r\}$ is equicontinuous in $C[0, \omega]$.

Finally, by Arzela-Ascoli theorem, we can obtain that $A$ is compact. Hence, it is completely continuous.

Theorem 4.3. Assume that there exist $0 < r < R, 0 < c_1 < c_2$ such that (4.2)and (4.3) hold. Further suppose one of the followingconditions is satisfied

$(i)\; f(t, u)\geq\frac{Mc_{2}}{m^{2}\omega^{2}c_{1}}u, \ \ \forall(t, u)\in [0, \omega]\times\left[\frac{mc_{1}\omega}{Mc_{2}}r, r\right],$

$f(t, u)\leq |\lambda|u, \ \ \forall(t, u)\in[0, \omega]\times\left[\frac{mc_{1}\omega}{Mc_{2}}R, R\right],$

$(ii)\; f(t, u)\leq |\lambda|u, \ \ \forall(t, u)\in [0, \omega]\times\left[\frac{mc_{1}\omega}{Mc_{2}}r, r\right],$

$f(t, u)\geq\frac{Mc_{2}}{m^{2}\omega^{2}c_{1}}u, \ \ \forall(t, u)\in [0, \omega]\times\left[\frac{mc_{1}\omega}{Mc_{2}}R, R\right].$

Then (1.1) has at least a positive solution $x$ with $r\leq\|x\| \leq R$.

Proof. Here we only consider the case (i). By Proposition 4.2, $A: \overline{P}_R\setminus P_r\rightarrow P$ is completely continuous. For $x \in \partial P_r$, we have

and

Similarly, if $x\in \partial P_{R}$,

By Theorem 4.1, there exists $x\in \overline{P}_R\setminus P_r$ such that $Ax = x$ and $x$ is a solution of (1.1). Moreover,

Corollary 4.4. Let $c_{1} < c_{2}$ be positive reals and $f(t, x)$ satisfy the conditions

$(i)$ $c_{1}\leq f(t, x)\leq c_{2}$ for all $x\geq 0$,

$(ii)$ $f:[0, w]\times(0, +\infty)\rightarrow \mathbb{R}$ is a continuous function.

Then problem (1.1) has a positive solution.

Proof. Let $0 < r < \frac{c_{1}^{2}m^{2}\omega^{2}}{Mc_2},$ $R > \frac{c_{2}^{2}M}{|\lambda|m_{1}c_{1}\omega}$, then (4.2) and (4.3) are satisfied. Clearly, for $(t, u)\in [0, \omega]\times\left[\frac{mc_{1}\omega}{Mc_{2}}r, r\right]$,

and for $(t, u)\in[0, \omega]\times\left[\frac{mc_{1}\omega}{Mc_{2}}R, R\right]$,

Hence, by Theorem 4.3 (1.1) has at least a positive solution.

Example 4.5. Consider the equation

where $0 < \alpha\leq1, \beta > 1$ and

Choosing $c_1 = 1, c_2 = 2, r = \frac{1}{10}\min\{1, \frac{m^{2}\omega^{2}}{2M}\}, R = 1.$ It is easy to check that (4.2) and (4.3) hold. For $\lambda\in\Lambda$,

Hence, (4.10) has at least one positive solution for $\lambda\in\Lambda$.

5.   Existence of solutions via monotone iterative techniques

In this section, by using the monotone iterative method, we discuss the existence of solutions when $\lambda = 0$ in (1.1). Firstly, we give the definition of the upper and lower solutions and get monotone iterative sequences with the help of the corresponding linear equation. Finally, we prove the limits of the monotone iterative sequences are solutions of (1.7).

Definition 5.1. Let $h, k\in C^1[0, \omega]$. $h$ and $k$ are called lower solution and upper solution of problem (1.7), respectively if $h$ and $k$ satisfy

Clearly, if $g$ the lower solution or upper solution of (1.7), then $^{c}D_{0^+}^{\alpha}g$ is continuous on $[0, \omega]$.

Lemma 5.2. Let $\delta\in C[0, \omega]$ with $\delta\geq0$ and $p\in\mathbb{R}$ with $p\leq0$. Then

has a unique solution $z(t)\geq0$ for $t\in [0, \omega]$, where $0 < \alpha\leq1, \lambda < 0$.

Proof. Let $z_1$, $z_2$ are two solutions of (5.3) and $v = z_1-z_2$, then

Using Theorem 3.1, (5.4) has trivial solution $v = 0$.

By (3.5), we can verify that problem (5.3) has a unique solution

As consequence, by Proposition 2.1 ($C_3$), ($C_5$) and Lemma 3.3 ($F_4$), we conclude that $z(t)\geq0$. This completes the proof.

Theorem 5.3. Assume that $h, k$ are the lower and upper solutions ofproblem (1.7) and $h\leq k$. Moreover, supposethat $f$ satisfies the following properties:

(M) there is $\lambda < 0$ such that for all fixed $t\in [0, \omega]$, $f(t, x)-\lambda x$ is nondecreasing in $h(t)\leq x\leq k(t)$,

(J) $f:[0, \omega]\times[h(t), k(t)]\rightarrow \mathbb{R}$ is a continuous function.

Then there are two monotone sequences $\{h_n\}$ and $\{k_n\}$ are nonincreasingand nondecreasing, respectively with $h_0 = h$ and $k_0 = k$such that $\lim_{n\rightarrow \infty}h_n = \overline{h}(t)$, $\lim_{n\rightarrow \infty}k_n = \overline{k}(t)$uniformly on $[0, \omega]$, and $\overline{h}, \overline{k}$ are the minimal and the maximalsolutions of (1.7) respectively, such that

on $[0, \omega]$, where $x$ is any solution of (1.7) such that $h(t)\leq x(t)\leq k(t)$on $[0, \omega]$.

Proof. Let $[h, k] = \{u\in C[0, \omega]: h(t)\leq u(t)\leq k(t), t\in[0, \omega]\}$. For any $\eta\in[h, k]$, we consider the equation

Theorem 3.1 implies the above problem has a unique solution

Define an operator $B$ by $x = B\eta$, we shall show that

(a) $k\geq Bk, Bh\geq h$,

(b) $B$ is nondecreasing on $[h, k]$.

To prove $(a)$. Denote $\theta = k-Bk$, we have

and $\theta(w)-\theta(0)\leq 0$. Since $k\in C^1[0, \omega]$,

By Lemma 5.2, $\theta\geq0$, i.e. $k\leq Bk$. In an analogous way, we can show that $Bh\geq h$.

To prove $(b)$. We show that $B\eta_1\leq B\eta_2$ if $h\leq\eta_1\leq\eta_2\leq k$. Let $z_1 = B\eta_1$, $z_2 = B\eta_2$ and $z = z_2-z_1$, then by $(M)$, we have

and $v(\omega) = v(0)$. By Lemma 5.2, $z(t)\geq0$, which implies $B\eta_1\leq B\eta_2$.

Define the sequence $\{h_{n}\}$, $\{k_{n}\}$ with $h_{0} = h, k_{0} = k$ such that $h_{n+1} = Bh_{n}, k_{n+1} = Bk_{n}$ for $n = 0, 1, 2, ...$. From $(a)$ and $(b)$, we have

on $t\in[0, \omega]$, and

Therefore, there exist $\overline{h}, \overline{k}$ such that $\lim_{n\rightarrow \infty}h_n = \overline{h}$, $\lim_{n\rightarrow \infty}k_n = \overline{k}$.

Similar to the proof of Proposition 4.2, we can show that $B:[h, k]\rightarrow [h, k]$ is a completely continuous operator. Therefore, $\overline{h}, \overline{k}$ are solutions of (1.7).

Finally, we prove that if $x\in[h_0, k_0]$ is one solution of (1.7), then $\overline{h}(t)\leq x(t)\leq \overline{k}(t)$ on $[0, \omega]$. To this end, we assume, without loss of generality, that $h_n(t)\leq x(t)\leq k_n(t)$ for some $n$. From property (b), we can get that $h_{n+1}(t)\leq x(t)\leq k_{n+1}(t), t\in [0, \omega].$ Since $h_0(t)\leq x(t)\leq k_0(t)$, we can conclude that

Passing the limit as $n\rightarrow \infty$, we obtain $\overline{h}(t)\leq x(t)\leq \overline{k}(t), \ t\in [0, \omega].$ This completes the proof.

Example 5.4. Consider the equation

It easy to check that $h = 1, k = 2$ are the low solution and upper solution of (5.6), respectively. Let $\lambda = -10$. For all $t\in [0, \omega]$,

is nondecreasing on $u\in {[1,2]}$ and

is continuous on $[0, \omega]\times [1,2]$.

Hence, there exist two monotone sequences $\{h_n\}$ and $\{k_n\}$, nonincreasing and nondecreasing respectively, that converge uniformly to the extremal solutions of (5.6) on $[h, k].$

6.   Conclusions

This paper focuses on the existence of solutions for the Caputo fractional differential equation with periodic boundary value condition. We use Green's function to transform the problem into the existence of the fixed points of some operator, and we prove the existence of positive solutions by using the Krasnosel'skii fixed point theorem. On the other hand, the existence of the extremal solutions for the special case of the problem is obtained from monotone iterative technique and lower and upper solutions method. Since the fractional differential equation is nonlocal equation, the process of verifying the compactness of operator is very tedious, and we will search for some better conditions to prove the compactness of the operator $A$ in the follow-up research. Meanwhile, since the existence result for $0 < \alpha\leq 1$ is obtained in present paper, we will discuss the existence of solutions for the Caputo fractional differential equation when $n-1 < \alpha\leq n$ in follow-up research.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.