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Monotone iterative technique for periodic problem involving Riemann–Liouville fractional derivatives in Banach spaces
Boundary Value Problems volume 2018, Article number: 119 (2018)
Abstract
In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence and uniqueness of periodic solutions for a class of fractional differential equations in an ordered Banach space E. Under some monotonicity conditions and noncompactness measure conditions of nonlinearity, we obtain the existence of extremal solutions and a unique solution between lower and upper solutions.
1 Introduction
The theory of fractional derivatives equations is an important branch of differential equation theory, which has extensive background in physics, chemistry, control of dynamical systems and realistic mathematical model. It has been found that the differential equations involving fractional derivatives in time are more realistic to describe many phenomena in practical cases than those of integer order in time. Hence, the theory and application of fractional derivatives equations has been rapidly developed in recent years. In particular, the existence of solutions to such problems has been extensively studied by many authors. For details, see the monographs of Miller and Ross [1], Kiryakova [2], Podlubny [3], and Kilbas et al. [4] and the papers by Lakshmikantham and Vatsala [5], Agarwal et al. [6], Darwish et al. [7–10]. Some recent contributions to the theory of fractional differential equations can be seen in [11–23].
In [13], the authors studied periodic boundary value problems for fractional differential equations
where \(0< T<+\infty\), \(f\in C([0, T]\times R)\), and \(D^{\alpha}_{0}\) is the Riemann–Liouville fractional derivative of order \(0<\alpha\leq1\). Through discussing the properties of well-known Mittag-Leffler function, they established a comparison result for problem (1.1) and obtained the existence and uniqueness of solution for (1.1) by using the monotone iterative method.
However, all of the papers mentioned above are in scalar spaces \(\mathbb {R}\). To the best of our knowledge, the work on the periodic solution for fractional differential equations in abstract spaces is yet to be initiated. Motivated by the consideration and [11, 13], in this article, we discuss the periodic boundary value problems for fractional differential equations in an ordered Banach space E
where \(0< T<+\infty\), \(f\in C([0, T]\times E, E)\), and \(D^{\alpha}_{0}\) is the Riemann–Liouville fractional derivative of order \(0<\alpha\leq 1\). By combining the theory of measure of noncompactness and the method of lower and upper solutions coupled with the monotone iterative technique, we construct two monotone iterative sequences and prove that the sequences monotonically converge to the minimal and maximal periodic solutions of problem (1.2), respectively, under some monotone conditions and noncompactness measure conditions of f. Our results are more general than those in [11, 13]. Because we consider problem (1.2) in a more general Banach space, therefore, it has more extensive application background. Our main results will be given in Sect. 3. Some preliminaries to discuss problem (1.2) are presented in Sect. 2.
Remark 1.1
We call a function \(u(t)\) a classical solution of problem (1.2) if
-
(i)
\(u(t)\) is continuous on \((0, T]\), \(t^{1-\alpha}u(t)\) is continuous on \([0, T]\), and its fractional integral \(I^{1-\alpha}u(t)\) is continuously differentiable for \((0, T]\);
-
(ii)
\(u(t)\) satisfies problem (1.2).
2 Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this article.
Let E be an ordered Banach space with the norm \(\|\cdot\|\) and the partial order ≤, whose positive cone \(P=\{u\in E\mid u\geq\theta\} \) is normal with normal constant N. Generally, \(C([0,T],E)\) denotes the space of all continuous E-value functions on the interval \([0, T]\). Evidently, \(C([0,T],E)\) is an ordered Banach space with the norm \(\|u\|_{C}=\max_{t\in[0,T]}\|u(t)\|\) and the partial order ≤ deduced by the positive cone \(P_{C}=\{u\in C([0,T],E)\mid u(t)\geq\theta\}\). \(P_{C}\) is also normal with the same normal constant N. Let \(C_{r}([0,T],E)=\{ u\in C((0,T],E)\mid t^{r}u\in C([0,T],E)\}\), then \(C_{r}([0,T],E)\) is also a Banach space when endowed with the norm \(\|u\|_{r}=\max\{t^{r}\| u(t)\|:t\in[0,T]\}\). It is easy to verify that \(C_{r}([0,T],E)\subset L^{1}([0,T],E)\) if \(r<1\), where \(L^{1}([0,T],E)\) denotes the Banach space of all E-value Bochner integrable functions defined on \([0,T]\) with the norm \(\|u\|_{1}=\int_{0}^{1}\|u(t)\|\,dt\).
Definition 2.1
(see [4])
The Riemann–Liouville fractional integral of order \(\alpha>0\) of a function \(y:(0, +\infty)\rightarrow E\) is given by
provided the right-hand side is pointwise defined on \((0,+\infty)\).
Definition 2.2
(see [4])
The Riemann–Liouville fractional derivative of order \(\alpha>0\) of a function \(y:(0, +\infty)\rightarrow E\) is given by
where \(n=[\alpha]+1\), provided that the right-hand side is pointwise defined on \((0,+\infty)\).
For problem (1.2), we have the following definitions of upper and lower solutions.
Definition 2.3
A function \(v\in C_{1-\alpha}([0,T],E)\) is called a lower solution of problem (1.2) if it satisfies
If all the inequalities of (2.1) are inverse, we call it an upper solution of problem (1.2).
Remark 2.1
In what follows, if v and w are lower solution and upper solution of problem (1.2), respectively, we assume that
Let \(\mu(\cdot)\) denote the Kuratowski measure of noncompactness of the bounded set. For details of the definition and properties of the measure of noncompactness, see [24]. The following result is important to proving our main results.
Lemma 2.1
Let \(H\subset C_{1-\alpha}([0,T],E)\) be bounded and equicontinuous. Then
where \(H(t)=\{u(t)\mid u\in H\}\subset E\), \(t\in[0, T]\).
Proof
By hypotheses, for every \(u\in H\) and any \(\varepsilon >0\), there exists \(\delta>0\) such that when \(| t_{1}-t_{2}|<\delta \), for any \(t_{1}, t_{2}\in[0, T]\), we have
Let
be a division of \([0, T]\) such that \(\|\Delta\|<\delta\), where \(\|\Delta \|=\max\{t_{i}-t_{i-1}; i=1,2,\ldots, n\}\). Let \(B=\bigcup_{i=1}^{n}t_{i}^{1-\alpha}H(t_{i})\). There is a division \(B=\bigcup_{j=1}^{m}B_{j}\) such that
where \(d(B_{j})\) denotes the diameter of \(B_{j}\). Let G be the set of all mappings from \(\{1,2,\ldots, n\}\) into \(\{1,2,\ldots, m\}\). It is clear that G is a finite set. For any \(\beta\in G\), let
It is clear that \(H=\bigcup_{\beta\in G}H_{\beta}\). For any \(u, v\in H_{\beta}\), and \(t\in[0, T]\), we have \(t\in[t_{i-1}, t_{i}]\) for some \(i\in\{1,2,\ldots, n\}\), and so, (2.3) and (2.4) imply that
From this and the definition of norm for \(C_{r}([0,T],E)\) it follows that
consequently,
which implies \(\mu(H)\leq\mu(B)+3\varepsilon\). Since ε is arbitrary, we get
On the other hand, for any \(\varepsilon>0\), there is a division \(H=\bigcup_{l=1}^{k}H_{l}\) such that
Hence, for \(\forall t\in[0, T]\), \(\forall x_{1}, x_{2}\in H_{l}\), \(l=1,2,\ldots,k\), we have
Since \(\{t^{1-\alpha}H(t)\}=\bigcup_{l=1}^{k}t^{1-\alpha}H_{l}(t)\), together with (2.5) and (2.6) we get
that is,
Because ε is arbitrary, we obtain
Consequently,
To sum up, the proof of Lemma 2.1 is complete. □
Lemma 2.2
(see [25])
Let \(B=\{u_{n}\}\subset C_{1-\alpha }([0,T],E)\) be bounded and countable set. Then \(\mu(B(t))\) is Lebesgue integral on \([0, T]\), and
Let M be the positive constant. For \(h\in C([0, T],E)\), we consider the linear periodic boundary value problems
By an argument similar to that in [11, Theorem 3.2] or [13, Lemma 1.1], we can obtain the following result.
Lemma 2.3
The linear periodic boundary value problem (2.7) has a unique solution u given by
where \(E_{\alpha,\alpha}(x)=\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma ((k+1)\alpha)}\) is the Mittag-Leffler function.
Remark 2.2
The well-known two-parameter Mittag-Leffler function
converges uniformly in \(\mathbb{R}\).
Remark 2.3
As showed in [16, Lemma 2.1 and Lemma 2.2], for \(0<\alpha\leq1\), we have
Hence, \([1-\Gamma(\alpha)E_{\alpha,\alpha}(-MT^{\alpha})]>0\). If \(h\geq \theta\), the solution of (2.7) \(u\geq\theta\). This comparison result will play a very important role in this paper.
3 Main results
For \(v, w\in C_{1-\alpha}([0,T],E)\), we denote
Our main results are as follows.
Theorem 3.1
Let E be an ordered Banach space, whose positive cone P is normal, \(f:[0, T]\times E\rightarrow E\) be continuous. Assume that \(v_{0}, w_{0}\in C_{1-\alpha}([0,T],E)\) are lower and upper solutions of (1.2) such that (2.2) holds. If the following conditions are satisfied:
-
(H1)
There exists a constant \(M > 0\) such that
$$f(t, x_{2})-f(t, x_{1})\geq-M(x_{2}-x_{1}) $$for \(\forall t\in[0, T]\) and \(v_{0}\leq x_{1}\leq x_{2}\leq w_{0}\).
-
(H2)
There exists a constant \(K > 0\) with
$$\frac{2KT^{\alpha}\Gamma(\alpha)}{\Gamma(2\alpha)} \biggl(\frac {1}{[1-\Gamma(\alpha)E_{\alpha,\alpha}(-MT^{\alpha})]}+1 \biggr)< 1 $$such that
$$\mu \bigl(\bigl\{ f\bigl(t, u_{n}(t)\bigr)+Mu_{n}(t)\bigr\} \bigr)\leq K\mu \bigl(\bigl\{ u_{n}(t)\bigr\} \bigr) $$for \(\forall t\in[0, T]\), and a monotonous sequence \(\{u_{n}\}\subset [v_{0}, w_{0}]\).
Then problem (1.2) has minimal and maximal solutions between \(v_{0}\) and \(w_{0}\), which can be obtained by a monotone iterative procedure starting from \(v_{0}\) and \(w_{0}\), respectively.
Proof
For any \(h\in[v_{0}, w_{0}]\), consider the linear periodic boundary value problem
By Lemma 2.3, we obtain that problem (3.1) has unique solution u, which can be expressed as follows:
Firstly, we need to show that the operator \(A:[v_{0}, w_{0}]\rightarrow C_{1-\alpha}([0,T],E)\) is well defined, i.e., for \(h\in[v_{0}, w_{0}]\), \(Ah\in C_{1-\alpha}([0,T],E)\). By (H1), for \(h\in[v_{0}, w_{0}]\), we have
We denote
By the normality of the cone P, there exists \(L>0\) such that\(\|F(h)\| _{1-\alpha}\leq L\), that is,
By (2.8) and (3.3), we have that
That is to say, the integral in (3.2) exists and belongs to \(C_{1-\alpha }([0,T],E)\).
By Lemma (2.3), the solution of problem (1.2) is equivalent to the fixed point of the operator A. Now, we complete the proof by four steps.
Step 1. We show that the operator \(A: [v_{0}, w_{0}]\rightarrow C_{1-\alpha}([0,T],E)\) is equicontinuous. For any \(u\in[v_{0}, w_{0}]\) and \(0\leq t_{1}\leq t_{2}\leq T\), we have
For the first term of the above formula, by (2.8) and (3.3), we have
The function \(E_{\alpha,\alpha}(-Mt^{\alpha})\) is continuous, so the previous expression has limit zero as \(| t_{2}-t_{1}| \rightarrow0\).
For the rest, by the properties of \(E_{\alpha,\alpha}(x)\) discussed in [16, Proposition 1], we have
Obviously, the previous expression also tends to zero for \(| t_{2}-t_{1}|\rightarrow0\). That is to say, \(A:[v_{0}, w_{0}]\rightarrow C_{1-\alpha}([0,T],E)\) is equicontinuous.
Step 2. We show that \(v_{0}\leq Av_{0}, Aw_{0}\leq w_{o}\), and \(Au_{1}\leq Au_{2}\) for any \(u_{1}, u_{2}\in[v_{0}, w_{0}]\) with \(u_{1}\leq u_{2}\).
Let
then, by Definition 2.3, we have \(\sigma(t)\leq F(v_{0})(t)\). Hence,
namely \(v_{0}\leq Av_{0}\). Similarly, we can show that \(Aw_{0}\leq w_{0}\). For any \(u_{1}, u_{2}\in[v_{0}, w_{0}]\) with \(u_{1}\leq u_{2}\), by assumption (H1),
which implies that \(Au_{1}\leq Au_{2}\).
Step 3. From Step 2 we know that A maps \([v_{0}, w_{0}]\) into itself, and \(A:[v_{0}, w_{0}]\rightarrow[v_{0}, w_{0}]\) is a continuously increasing operator. We can now define the sequences
Then from the monotonicity of A it follows that
Obviously, \(\{v_{n}\}, \{w_{n}\}\subset[v_{0}, w_{0}]\) are equicontinuous. Next, we show that \(\{v_{n}\}\) and \(\{w_{n}\}\) are convergent in \(C_{1-\alpha}([0,T],E)\).
From (H2), Lemma 2.1, and Lemma 2.2, for any \(t\in[0, T]\), we have that
Since \(\{v_{n}\}\) is equicontinuous, using Lemma (2.1), we have
While
hence \(\mu (\{v_{n}\} )=0\). So, \(\{v_{n}\}\) are relatively compact in \(C_{1-\alpha}([0,T],E)\). Hence, \(\{v_{n}\}\) has a convergent subsequence in \(C_{1-\alpha}([0,T],E)\). Combining this with the monotonicity (3.5), we easily prove that \(\{v_{n}\}\) itself is convergent in \(C_{1-\alpha }([0,T],E)\).
Using a similar argument to that for \(\{w_{n}\}\), we can prove that \(\{ w_{n}\}\) is also convergent in \(C_{1-\alpha}([0,T],E)\). Then there are \(\underline{u}, \bar{u} \in C_{1-\alpha}([0,T],E)\) such that
Letting \(n\rightarrow\infty\) in (3.4), we see that
Therefore, \(\underline{u}, \bar{u} \in C_{1-\alpha}([0,T],E)\) are fixed points of A.
Step 4. We prove the minimal and maximal property of \(\underline{u}\), ū. Assume that ũ is a fixed point of A in \([v_{0}, w_{0}]\), then we have
By the monotonicity of A, it is easy to see that
Furthermore, we have
Letting \(n\rightarrow\infty\) in (3.6), we obtain \(\underline{u}\leq \tilde{u}\leq\bar{u}\). So \(\underline{u}\), ū are the minimal and maximal fixed points of A in \([v_{0}, w_{0}]\), and therefore, they are the minimal and maximal solutions of problem (1.2) in \([v_{0}, w_{0}]\), respectively.
This completes the proof of Theorem 3.1. □
Remark 3.1
When \(E=\mathbb{R}\), we do not need condition (H3), \(\{v_{n}\}\) and \(\{w_{n}\}\) defined in (3.4) are convergent in \(C_{1-\alpha}([0,T],\mathbb{R})\) automatically. Therefore, Theorem 3.1 improves the main results in [13].
Next, we discuss the uniqueness of the solution to problem (1.2) in \([v_{o}, w_{o}]\).
Theorem 3.2
Let E be an ordered Banach space, whose positive cone P is normal, \(f:[0, T]\times E\rightarrow E\) be continuous. Assume that \(v_{0}, w_{0}\in C_{1-\alpha}([0,T],E)\) are lower and upper solutions of (1.2) such that (2.2) holds. If conditions (H1), (H2) and the following condition are satisfied:
-
(H3)
There exists a constant \(C > 0\) with
$$\frac{N(M+C)T^{\alpha}\Gamma(\alpha)}{\Gamma(2\alpha)} \biggl(\frac {1}{[1-\Gamma(\alpha)E_{\alpha,\alpha}(-MT^{\alpha})]}+1 \biggr)< 1 $$such that
$$f(t, x_{2})-f(t, x_{1})\leq C(x_{2}-x_{1}), $$for \(\forall t\in[0, T]\), and \(v_{0}\leq x_{1}\leq x_{2}\leq w_{0}\), where N is a normal constant.
Then problem (1.2) has a unique solution between \(v_{0}\) and \(w_{0}\), which can be obtained by a monotone iterative procedure starting from \(v_{0}\) or \(w_{0}\).
Proof
From the proof of Theorem 3.1, we know that the iterative sequences \(\{v_{n}\}\) and \(\{w_{n}\}\) defined by (3.4) satisfy (3.5). Now, we show that there exists a unique \(u^{\star} \in C_{1-\alpha }([0,T],E)\) such that \(u^{\star}=Au^{\star} \). For \(t\in[0, T]\), by (H3), we have
From the normality of the cone P, it follows that
Therefore,
Again using the above inequality, we get
which implies that for \(n\rightarrow\infty\) we have \(\|w_{n}-v_{n}\| _{1-\alpha}\rightarrow0\). Then there exists a unique \(u^{\star} \in C_{1-\alpha}([0,T],E)\) such that
So let \(n\rightarrow\infty\) in (3.4), we have \(u^{\star}=Au^{\star}\), which means that \(u^{\star}\) is a unique solution of problem (1.2).
This completes the proof of Theorem 3.2. □
4 Conclusion
By using the method of lower and upper solutions coupled with the monotone iterative technique, combining the theory of measure of noncompactness, we present some monotone conditions and noncompactness measure conditions of f such that problem (1.2) has minimal and maximal periodic solutions. In addition, we investigate the uniqueness of the solution for this problem. Our results are more general than those in [11, 13], because we consider problem (1.2) in a more general Banach space, it has more extensive application background. Our main results improve the main results in [11, 13].
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Research is supported by the National Natural Science Foundation of China (11661071, 11261053, 11361055) and the Youth Science Foundation of Tianshui Normal University (No. TSA1510).
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Ding, Y., Li, Y. Monotone iterative technique for periodic problem involving Riemann–Liouville fractional derivatives in Banach spaces. Bound Value Probl 2018, 119 (2018). https://doi.org/10.1186/s13661-018-1037-4
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DOI: https://doi.org/10.1186/s13661-018-1037-4