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Monotone iterative procedure and systems of a finite number of nonlinear fractional differential equations
Advances in Difference Equations volume 2015, Article number: 167 (2015)
Abstract
The aim of the paper is to present a nontrivial and natural extension of the comparison result and the monotone iterative procedure based on upper and lower solutions, which were recently established in (Wang et al. in Appl. Math. Lett. 25:1019-1024, 2012), to the case of any finite number of nonlinear fractional differential equations.
1 Introduction
Fractional derivatives and integrals are used for a better description of material properties. In the literature we can find many interesting papers concerning this theory; see e.g., [1–13]. The study of systems involving fractional differential/integral equations is also important as such systems occur in various problems of applied nature; for example, see [14–22]. Some basic theory of fractional differential equations involving the Riemann-Liouville differential operator can be found in [23–25].
In the paper we consider the following system of nonlinear fractional differential equations:
where \(D^{\alpha}\) is the standard Riemann-Liouville fractional derivative of order α, \(0\leq\alpha\leq1\), \(T>0\), \(f^{i} \in C([0,T]\times\mathbb{R}^{n},\mathbb{R})\), \(1\leq i \leq n\), and \(x^{1}_{0},\ldots,x^{n}_{0} \in\mathbb{R}\) satisfy
We investigate system (1.1) with respect to the existence of a solution via the method of upper and lower solutions. There is also presented the concept of an iterative procedure, where the appropriately constructed sequences are convergent to the extreme solution. The paper is a continuation of the investigations in [10] of Wang et al., where the authors examined system (1.1) in the case \(n=2\). After proving the main results we state, for convenience of the reader, the introduced techniques in the case of three nonlinear fractional differential equations and also present a concrete example.
2 Preliminaries
First, let us recall the needed notations and crucial results which will be needed in the next sections of the article.
Denote by \(C_{1-\alpha}([0,T])\) the family of all functions \(u \in C((0,T])\) such that \(t^{1-\alpha}u \in C([0,T])\). A basic theorem concerning the existence of the result and its uniqueness for the linear fractional equation is as follows.
Lemma 2.1
([23])
Let \(0<\alpha\leq1\), \(M \in\mathbb{R}\), and \(\sigma\in C_{1-\alpha }([0,T])\) be fixed. Then the linear initial value problem
has a unique solution, given by the following formula:
where \(E_{\alpha,\beta}\) is the Mittag-Leffler function, i.e. the function of the form
The comparison result for the initial value problem (2.1) due to Wang et al. is as follows.
Lemma 2.2
([10])
Let \(0<\alpha\leq1\) and \(M\in\mathbb{R}\) be given. Then, if \(w\in C_{1-\alpha}([0,T])\) satisfies
then \(w(t)\geq0\) for all \(t\in(0,T]\).
The same authors also proved the following result, which will be needed in the sequel.
Lemma 2.3
([10])
Let \(0<\alpha\leq1\), \(M\in\mathbb{R}\), and \(N\geq0\) be given. Assume that \(u,v \in C_{1-\alpha}([0,T])\) satisfy
Then \(u(t)\geq0\), \(v(t)\geq0\) for all \(t\in(0,T]\).
3 The results
In the sequel we will use the following notation:
\(C_{1-\alpha}([0,T])^{n}\) denotes \(C_{1-\alpha}([0,T])\times C_{1-\alpha }([0,T])\times\cdots\times C_{1-\alpha}([0,T])\) (n times).
Lemma 3.1
Let \(0<\alpha\leq1\) be fixed, \(M_{i} \in\mathbb{R}\), \(\sigma_{i} \in C_{1-\alpha}([0,T])\), \(i=1,2,\ldots, n\). Then the linear problem of n equations
has a unique solution in \(C_{1-\alpha}([0,T])^{n}\).
Proof
First observe that for any \(p_{1}, p_{2},\ldots, p_{n} \in C_{1-\alpha}([0,T])\) the system
has exactly one solution, which is a consequence of the fact that
Next, observe that system (3.1) can be transformed to system (3.2), where \(p_{1}, p_{2}, \ldots, p_{n}\) solve the following n problems:
Finally, observe that the solutions of the above equations are unique due to Lemma 2.1, which ends the proof. □
Now we can state and proof the comparison result for system (3.1).
Theorem 3.1
Let \(0<\alpha\leq1\), \(M_{1} \in\mathbb{R}\), \(M_{2}, \ldots, M_{n} \geq0\), and let \(u_{1},\ldots, u_{n} \in C_{1-\alpha }([0,T])\) satisfy
Then
Proof
Put \(r(t)=\sum_{s=1}^{n}u_{s}(t)\). Using (3.3) we obtain
Observe that
Hence, we obtain
Moreover, observe that
Applying (3.8) and (3.9) to Lemma 2.2 we get (3.4).
Now, consider any \(2\leq s\leq n\) and denote
By (3.3) we have
Again, using (3.7), we obtain
Moreover, observe that (3.3) implies
Finally, note that (3.10) and (3.11) applied to Lemma 2.3 give (3.5) and (3.6). □
Now, we are in a position to enunciate the main result.
Theorem 3.2
Suppose that there exist \(u^{1}_{0},u^{2}_{0},\ldots, u^{n}_{0} \in C_{1-\alpha }([0,T])\), \(u^{1}_{0}\leq\sum_{i=2}^{n}u_{0}^{i}\), satisfying
and there exist \(M_{1}\in\mathbb{R}\), \(M_{2},\ldots, M_{n} >0\) such that
-
(i)
$$ f_{1}(t,\alpha_{1},\ldots,\alpha_{n})-f_{1}(t, \beta_{1},\ldots,\beta_{n}) \geq- M_{1}( \alpha_{1}-\beta_{1}) - \sum_{i,j=2}^{n}M_{j} \delta_{ji}(\alpha_{i} - \beta_{i}), $$(3.13)
-
(ii)
$$\begin{aligned}& f_{s}(t,\alpha_{1},\ldots,\alpha_{n}) -f_{s}(t,\beta_{1},\ldots,\beta_{n}) \\& \quad \geq \Biggl(-M_{1}+\sum_{i=2}^{n}M_{i}-M_{s} \Biggr) (\alpha_{s}-\beta_{s}) -M_{s}\Biggl( \alpha_{1}-\beta_{1}+\alpha_{s}-\beta_{s} -\sum_{i=2}^{n}(\alpha_{i}- \beta_{i})\Biggr), \end{aligned}$$
where \(\alpha_{i}, \beta_{i} \in\mathbb{R}\), \(1\leq i \leq n\) satisfy for all \(t \in[0,T]\) and \(2\leq s \leq n\),
$$\begin{aligned}& u_{0}^{1}(t)-\Biggl(\sum_{i=2}^{n}u_{0}^{i}(t)-u_{0}^{s}(t) \Biggr) \leq\beta_{1} - \Biggl(\sum_{i=2}^{n} \beta_{i} - \beta_{s}\Biggr) \leq\alpha_{1} - \Biggl(\sum_{i=2}^{n}\alpha_{i} - \alpha_{s}\Biggr)\leq u_{0}^{s}(t), \\& u_{0}^{1}(t)-\Biggl(\sum_{i=2}^{n}u_{0}^{i}(t)-u_{0}^{s}(t) \Biggr)\leq\alpha_{s} \leq\beta_{s} \leq u^{s}_{0}(t), \end{aligned}$$ -
(iii)
$$\begin{aligned}& \sum_{s=2}^{n}f_{s} \bigl(t,u^{1}(t),u^{2}(t),\ldots, u^{n}(t)\bigr) -f_{1}\bigl(t,u^{1}(t),u^{2}(t),\ldots, u^{n}(t)\bigr) \\& \quad \geq\Biggl(-M_{1}+\sum_{s=2}^{n}M_{s} \Biggr) \Biggl(\sum_{s=2}^{n}u^{s}(t)-u^{1}(t) \Biggr), \end{aligned}$$(3.14)
where
$$u_{0}^{1}-\Biggl(\sum_{i=2}^{n}u_{0}^{i}-u_{0}^{s} \Biggr) \leq u^{1} - \Biggl(\sum_{i=2}^{n}u^{i} - u^{s}\Biggr) \leq u^{s} \leq u_{0}^{s}, \quad 2\leq s \leq n. $$
Then there exists a solution \((\bar{u}^{1},\bar{u}^{2},\ldots,\bar{u}^{n})\) of system (1.1) such that
Moreover, there exist iterative sequences \((u^{1}_{k}), (u^{2}_{k}),\ldots, (u^{n}_{k})\) such that \(u^{i}_{k}\to\bar{u}^{i}\), \(k \to\infty\), \(i=1,2,\ldots ,n\), uniformly on compact subsets of \((0,T]\).
Proof
Let us first consider the linear system of the form
where \(u^{1}, u^{2},\ldots,u^{n} \in C_{1-\alpha}([0,T])\). Due to Lemma 3.1 there exists a system of solutions \((u^{1}_{1}, u^{2}_{1},\ldots, u^{n}_{1}) \in C([0,T])^{n}\) for system (3.15). Using induction we obtain the sequence \((u^{1}_{k}, u^{2}_{k},\ldots, u^{n}_{k}) \in C([0,T])^{n}\), \(k \in\mathbb{N}\), satisfying
Now, put \(p^{1}_{1}=u^{1}_{1}-u^{1}_{0}\), \(p^{s}_{1}=u^{s}_{0}-u^{s}_{1}\), \(2\leq s \leq n\). From (3.12) and (3.15), for all \(t\in(0,T]\), we obtain
Hence, using Theorem 3.1, we have
and
Consider now \(q_{1}=\sum_{i=2}^{n}u^{i}_{1}-u^{1}_{1}\). Using (3.14) and (3.15) we have
Moreover, (1.2) implies
Now, from Lemma 2.2 we conclude
Combining (3.17) and (3.18) with (3.19) we obtain for all \(2\leq s \leq n\) the inequalities
Let \(2\leq s\leq n\) be fixed and suppose now that for some \(k \in \mathbb{N}\) the following inequalities hold:
Denote \(p^{1}_{k+1}=u^{1}_{k+1}-u^{1}_{k}\), \(p^{s}_{k+1}=u^{s}_{k}-u^{s}_{k+1}\), \(2\leq s \leq n\). From (3.13), (3.16), and (3.20) we obtain
Also observe that \(t^{1-\alpha}p^{1}_{k+1}(t)|_{t=0}=t^{1-\alpha }p^{s}_{k+1}(t)|_{t=0}=0\), which, together with the above, due to Theorem 3.1, gives
Consider now \(q_{k}=\sum_{i=2}^{n}u^{i}_{k}-u^{1}_{k}\). Using the same arguments as with \(q_{1}\) we obtain
and
which, due to Lemma 2.2, gives
Summarizing, by (3.21)-(3.23) and induction, we obtain the following inequalities describing the sequences \((u_{k}^{s})_{k \in\mathbb {N}\cup\{0\}}\):
where \(2\leq s \leq n\). The inequalities (3.24) imply
Observe that
In order to show that the sequence \((u_{k}^{1})\) is convergent observe first that from (3.24) there exists a function \(x^{*}\) such that
Hence, putting \(\bar{u}^{1}=x^{*}+\sum_{s=2}^{n-1}\bar{u}^{s}\), we have
In order to show the uniform convergence of sequences \((u^{2}_{k}), (u^{3}_{k}),\ldots, (u^{n}_{k})\), observe that from (3.24) and from the fact that \(u^{s}_{k} \to\bar{u}^{s}\), \(s=2,3,\ldots,n\), we have
Then, the uniform convergence of sequences \((u_{k}^{s})\), \(s=2,3,\ldots, n\), on a compact subset of \((0,T]\) is a straightforward consequence of Dini’s theorem, which states that if a monotone sequence of continuous functions is convergent on a compact set, then it converges uniformly.
Showing a uniform convergence of \((u_{k}^{1})\) requires some observations. Take any \(2\leq s \leq n\) and denote
From (3.24) and the convergence of \((u_{k}^{1}), \ldots, (u_{k}^{n})\) we have
Applying again Dini’s result we get the uniform convergence of \((h_{k})\) on every compact subset of \((0,T]\). Finally note that
and thus \((u^{1}_{k})\) is uniformly convergent on a compact subset of \((0,T]\) to \(\bar{u}^{1}\) as a linear combination of sequences uniformly convergent.
Moreover, observe that the limit functions satisfy the properties
Taking k to ∞ in (3.16) we see that \((\bar{u}^{1},\bar {u}^{2},\ldots,\bar{u}^{n})\) is a system of solutions of system (1.1). Also observe that from (3.24) we have the following relations between the limit functions:
which ends the proof. □
Remark 3.1
Observe that using the same methods as in the proof of Theorem 3.2 we can see that \((\bar{u}^{1},\bar{u}^{2},\ldots,\bar{u}^{n})\) is an extremal solution of system (1.1) in the sense that if \((u^{1},\ldots,u^{n})\) were any other solution such that
for any \(2\leq s \leq n\), then we would have
4 The system of three fractional differential equations
In order to see the nature of the iterative procedure introduced in the proof of Theorem 3.2, we consider the case \(n=3\).
Corollary 4.1
If there exist \(u_{0},v_{0},w_{0} \in C_{1-\alpha}([0,T])\), \(u_{0}\leq v_{0} + w_{0}\) such that
and there exist \(M\in\mathbb{R}\), \(N,S \geq0\) satisfying
where \(\alpha_{i}, \beta_{i} \in\mathbb{R}\), \(1\leq i \leq3\) satisfy, for all \(t \in[0,T]\),
and
where
Then there exists a solution
of (4.1) and the sequences \((u_{n}) \subseteq [2u_{0}-v_{0}-w_{0},v_{0}+w_{0}]\), \((v_{n}) \subseteq[u_{0}-w_{0},v_{0}] \), \((w_{n}) \subseteq [u_{0}-v_{0},w_{0}]\) such that \(u_{n} \to u^{*}\), \(v_{n} \to v^{*}\), \(w_{n} \to w^{*}\) uniformly on compact subsets of \((0,T]\). Moreover, the following inequalities hold:
4.1 Example
Consider the nonlinear problem of the form
where \(t\in[0,1]\). Taking
and
we obtain, for all \(t\in[0,1]\),
Next, for all \(\alpha_{i}, \beta_{i} \in\mathbb{R}\), \(1\leq i \leq3\) such that
one can calculate that
Therefore it is sufficient to take in Corollary 4.1 \(M=0\), \(N=0\), \(S=\Gamma(1.5)^{-1}\). Finally observe that condition (4.2) also holds. Thus, the system of fractional differential equations (4.3) has a solution \((u^{*},v^{*},w^{*}) \in[-2t,2t]\times[-t,t] \times[-t,t]\).
Now, using the proof of Theorem 3.2 and Lemma 3.1, we can derive the iterative procedure \((u_{k},v_{k},w_{k})\) convergent to the solution \((u^{*},v^{*},w^{*})\). First observe that the sequences \((u_{k})\), \((v_{k})\), \((w_{k})\) satisfy the following system of linear equations:
which can be equivalently transformed to the system
where \(p_{k}\), \(q_{k}\), \(r_{k}\) are the solutions of the following systems:
The solutions of the above systems, due to Lemma 2.1, are given by the formulas
In consequence, the iterative sequences are of the form
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Acknowledgements
The author is very grateful to the reviewers for the remarks, which improved the final version of the manuscript. This article was financially supported by University of Łódź as a part of donation for the research activities aimed at the development of young scientists, grant no. 545/1117.
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The author formulated and proved all the results in the article, produced the illustrative example, wrote the manuscript, and read and approved it.
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Wardowski, D. Monotone iterative procedure and systems of a finite number of nonlinear fractional differential equations. Adv Differ Equ 2015, 167 (2015). https://doi.org/10.1186/s13662-015-0504-9
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DOI: https://doi.org/10.1186/s13662-015-0504-9