- Research
- Open Access
Stability for a class of semilinear fractional stochastic integral equations
- Allan Fiel^{1},
- Jorge A León^{1}Email author and
- David Márquez-Carreras^{2}
https://doi.org/10.1186/s13662-016-0895-2
© Fiel et al. 2016
Received: 7 December 2015
Accepted: 12 June 2016
Published: 23 June 2016
Abstract
In this paper we study some stability criteria for some semilinear integral equations with a function as initial condition and with additive noise, which is a Young integral that could be a functional of fractional Brownian motion. Namely, we consider stability in the mean, asymptotic stability, stability, global stability, and Mittag-Leffler stability. To do so, we use comparison results for fractional equations and an equation (in terms of Mittag-Leffler functions) whose family of solutions includes those of the underlying equation.
Keywords
- comparison results for fractional differential equations
- fractional Brownian motion
- Mittag-Leffler function
- stability criteria
- Young integral for Hölder continuous functions
MSC
- 34A08
- 60G22
- 26A33
- 93D99
1 Introduction
Currently fractional systems are of great interest because of the applications they have in several areas of science and technology, such as engineering, physics, chemistry, mechanics, etc. (see, e.g., [1–4] and the references therein). Particularly we can mention system identification [1], robotics [5], control [1, 4], electromagnetic theory [6], chaotic dynamics and synchronization [7–10], applications on viscoelasticity [11], analysis of electrode processes [12], Lorenz systems [7], systems with retards [13], quantic evolution of complex systems [14], numerical methods for fractional partial differential equations [15–17], among other. A nice survey of basic properties of deterministic fractional differential equations is in Lakshmikantham and Vatsala [18]. Also, many researchers have established stability criteria of mild solutions of stochastic fractional differential equations using different techniques.
For deterministic systems, the stability of fractional linear equations has been analyzed by Matignon [19] and Radwan et al. [20]. Besides, several authors have studied nonlinear cases using Lyapunov method (see, e.g., Li et al. [21] and references therein). In particular, nonlinear fractional systems with a function as initial condition using also the Lyapunov technique have been considered in the PhD thesis of Martínez-Martínez [22]. Moreover, in the work of Junsheng et al. [23] the form of the solution for a linear fractional equation with a constant initial condition in terms of the Mittag-Leffler function is given by means of the Adomian decomposition method. Wen et al. [24] have established stability results for fractional nonlinear equations via the Gronwall inequality. Equality (3) below can be seen as an extension of the results in [10] and the Gronwall inequality stated in [24]. In [24], the stability is used to obtain synchronization of fractional systems.
On the other hand, a process used frequently in the literature is fractional Brownian motion \(B^{H}=\{ B_{t}^{H}, t\ge0\}\) due to the wide range of properties it has, such as long range memory (when the Hurst parameter H is greater than one half) and intermittency (when \(H<1/2\)). Unfortunately, in general, it is not a semimartingale (the exception is \(H=1/2\)). Thus, we cannot use classical Itô calculus in order to integrate processes with respect to \(B^{H}\) when \(H\neq1/2\), but we may use other approaches, such as Young integration (see Gubinelli [25], Young [26], Zähle [27], Dudley and Norvaiša [28], Lyons [29]). The reader may also refer to Nualart [30], and Russo and Vallois [31] for other types of integrals. As a consequence, an important application is the analysis of stochastic integral equations driven by fractional Brownian motion that has been considered by several authors these days for different interpretations of stochastic integrals (see, e.g., Lyons [29], Quer-Sardanyons and Tindel [32], León and Tindel [33], Nualart [30], Friz and Hairer [34], Lin [35] and Nualart and Răşcanu [36]).
We observe that equation (1) can be a useful model for applications in several areas of science. For example, (1) provides a fractional version of the Verhlust-Pearl equation (see Scudo [46], p.5), where \(h(x)=\lambda x^{2}\), for some \(\lambda\in \mathbb{R}\). So, now the stability of a single insolated species can be analyzed by means of fractional systems. Also, as pointed out in [24], for some fractional systems in engineering h is a \(O(x)\) as \(x\to0\).
This work is organized as follows. In Section 2 we introduce a fractional integral equation, whose family of solutions includes those of (1). Also, in Section 2, we state a comparison result for fractional systems that becomes the main tool for our results. In Section 3, we study some stability criteria for equation (1) in the case that \(\mathcal{Z}\equiv0\). These results can be seen as extensions of the results given in [45] and [24]. Finally, the stability of equation (1) in the case that θ is either a Hölder continuous process or a functional of fractional Brownian motion is considered in Section 4.
2 Preliminaries
In this section we introduce the framework and the definitions that we use to prove our results. Part of the main tool that we need is the stability of some fractional linear systems as presented by Fiel et al. [45] and a comparison result (see Lemma 1 below).
2.1 The Young integral
We observe that this integral has been extended by Zähle [27], Gubinelli [25], Lyons [29], among others. For a detailed exposition on the Young integral the reader is referred to the paper of Dudley and Norvais̆a [28] (see also Gubinelli [25], and León and Tindel [33]).
2.2 Semilinear Volterra integral equations with additive noise
In this work we use comparison methods in order to obtain the stability of some fractional systems. We can find comparison theorems in the literature for fractional evolution equations (see, e.g., Theorem 4.2 in [18]), but, unfortunately these results are not suitable for our purpose. Thus, we give the following lemma, which is a version of Theorem 2.2.5 in Pachpatte [47] and allows us to prove stability for the semilinear equations that we study. Hence, this result is a fundamental tool in the development of this paper.
Lemma 1
- (i)
\(k(\cdot, x)\) is measurable on \([0,T]\) for each \(x\in\mathbb{R}\);
- (ii)
There is a constant \(M>0\) such that \(|k(s,x)-k(s,y)|\le M|x-y|\), for any \(s\in[0,T]\) and \(x,y\in\mathbb{R}\);
- (iii)
k is bounded on bounded sets of \([0,T]\times\mathbb{R}\);
- (iv)
\(k(s,\cdot)\) is non-decreasing for any \(s\in[0,T]\).
Remark
The assumptions of k entail that equation (5) has a unique continuous solution.
Proof
3 A class of nonlinear fractional-order systems
In this section we establish two sufficient conditions for the stability of a deterministic semilinear Volterra integral equation. Thus, we improve the results in [45] for this kind of systems when the noise is null (i.e., \(\mathcal{Z}\) in (1) is equal to zero).
3.1 A constant as initial condition
- (H1)
There is a constant \(C>0\) such that \(A+C<0\) and \(|h(x)|\le C|x|\), for all \(x\in\mathbb{R}\).
- (H2)
There are \(\delta_{0}>0\) and \(C>0\) such that \(A+C<0\) and \(|h(x)|\le C|x|\), for \(|x|<\delta_{0}\).
Now, we consider several definitions of stability.
Definition 1
- (i)
globally stable in the large if \(X(t)\) goes to zero as t tends to infinity, for all \(x_{0}\in\mathbb{R}\);
- (ii)Mittag-Leffler stable if there is \(\delta>0\) such that \(|x_{0}|<\delta\) implieswhere \(\beta\in(0,1)\), \(B<0\), \(b>0\), and m is a positive and locally Lipschitz function with \(m(0)=0\);$$\bigl\vert X(t)\bigr\vert \le \bigl[ m(x_{0})E_{\beta,1} \bigl(Bt^{\beta}\bigr) \bigr]^{b}, \quad t\ge0, $$
- (iii)
stable if for \(\varepsilon>0\), there is \(\delta>0\) such that \(|x_{0}|<\delta\) implies \(|X(t)|<\varepsilon \), for all \(t\ge0\);
- (iv)
stable in the large if there is \(\delta>0\) such that \(|x_{0}|<\delta\) implies \(\lim_{t\to\infty}X(t)=0\);
- (v)
asymptotically stable if it is stable and stable in the large.
Remark 1
Our first stability result for any continuous solution of equation (6) is the following.
Proposition 1
Assume either (H2) or (H1) is satisfied. Then any continuous solution X to equation (6) is stable.
Proof
Now we establish the main result of this subsection.
Proposition 2
Let h be a function satisfying (H2) (resp. (H1)). Then any continuous solution of equation (6) is Mittag-Leffler stable and therefore is also asymptotically stable (resp. globally stable in the large).
Proof
Let (H2) (resp. (H1)) be satisfied and \(0< x_{0}<\delta_{0}\) (resp. \(x_{0}>0\)). Then \(0< X(t)<\delta_{0}\) (resp. \(X(t)>0\)) by [22] (see (8)).
Remark
Let X be a solution to equation (6). Wen et al. [24] (Theorem 1) have proved that the solution to equation (6) is stable if \(\lim_{|x|\to0}\frac{|h(x)|}{|x|}\to0\). Also, Zhang and Li [51] have used an equality similar to (3) to prove that X is asymptotically stable for the case that \(\lim_{x\to0}\frac {|h(x)|}{|x|}=0\), \(\beta\in(1,2)\) and \(\beta+\frac{1}{|A|}<2\). Proposition 2 establishes that X is asymptotically stable under a weaker condition. Namely (H2). This is possible because we use a comparison type result and the fact that this solution does not change sign.
3.2 A function as initial condition
Here we treat the case that the initial condition is a function satisfying some suitable conditions.
On the other hand, in this paper we analyze several stability criteria for different classes \(\mathcal{E}\) of initial conditions. Sometimes \(\mathcal{E}\) is a subset of a normed linear space \(\mathcal{X}\) of continuous functions endowed with the norm \(\|\cdot\|_{\mathcal{X}}\). In other words we consider normed linear spaces \((\mathcal{X},\|\cdot\|_{\mathcal {X}})\). Mainly, in the remaining of this paper, we deal with the following classes of initial conditions.
Definition 2
- 1.
If the initial condition ξ is continuous on \([0,\infty)\) and we have \(\xi_{\infty}\in\mathbb{R}\) such that, given \(\varepsilon>0\), there exists \(t_{0}>0\) such that \(|\xi _{s}-\xi_{\infty}|\le\varepsilon\) for any \(s\ge t_{0}\), we say that ξ belongs to the family \(\mathcal{E}^{1}\).
- 2.\(\mathcal{E}^{2}\) is the set of all functions ξ of class \({C}^{1}(\mathbb{R}_{+})\) (i.e., ξ has a continuous derivative on \(\mathbb{R}_{+}\)) such that$$ \lim_{t \rightarrow\infty} \vert \xi_{t}\vert /t^{\beta}=0\quad \mbox{and} \quad \bigl\vert \xi_{t}' \bigr\vert \le\frac{\tilde{C}}{t^{1-\upsilon}},\quad \text{for some } \upsilon\in(0,\beta) \text{ and } \tilde{C}\in\mathbb{R}. $$
- 3.\(\mathcal{E}^{3}\) is the space of continuous functions of the formwith \(g \in L^{1}([0,\infty)) \cap L^{p}([0,\infty))\), \(\eta\in(0,\beta +1)\), and \(p>\frac{1}{\eta}\vee1\).$$ \xi_{t}=\frac{1}{\varGamma (\eta)} \int_{0}^{t} (t-s)^{\eta-1} g(s) \,ds, $$(10)
The stability concepts that we develop in this section are the following.
Definition 3
- (i)
globally stable in the large for the class \(\mathcal{E}\) (or globally \(\mathcal{E}\)-stable in the large) if \(X(t)\) tends to zero as \(t\to\infty\), for every \(\xi\in\mathcal{E}\);
- (ii)
\(\mathcal{E}\)-stable if for \(\varepsilon>0\), there is \(\delta>0\) such that \(\|X\|_{\infty,[0,\infty)} <\varepsilon\) for every \(\xi\in\mathcal{E}\) satisfying \(\|\xi \|_{\mathcal{X}}<\delta\);
- (iii)
asymptotically \(\mathcal{E}\)-stable if it is \(\mathcal {E}\)-stable and there is \(\delta>0\) such that \(\lim_{t\to\infty}X(t)=0\) for any \(\xi\in\mathcal{E}\) such that \(\| \xi \|_{\mathcal{X}}<\delta\).
In the following auxiliary result, \(\mathcal{E}^{4}\) is the family of functions ξ having the form (10) with \(\eta=\beta\) and g is a continuous function such that \(\lim_{t\to\infty} g(t)=0\). In this case, the involved norm is \(\|\xi\|_{\mathcal {X}}=\|g\|_{\infty,[0,\infty)}\).
Lemma 2
Proof
Now we give a general result.
Theorem 1
Proof
Remark
In the following result we see that the family \(\mathcal{E}:=\{ \xi\in C([0,\infty)): \xi=\sum_{i=1}^{3}\xi^{(i)},\xi^{(i)}\in \mathcal{E}^{i}\}\) is an example of a family of functions for which the assumptions of Theorem 1 is satisfied. Here, \(\|\cdot\|_{\mathcal{X}^{1}}= \|\cdot\|_{\infty,[0,\infty)}\), \(\|\xi^{(2)}\|_{\mathcal{X}^{2}}=\|\xi _{\cdot}^{(2)}E_{\beta,1}(A\cdot^{\beta})\|_{\infty,[0,\infty)} +\|\cdot^{1-\upsilon}{\xi_{\cdot}^{(2)}}'\|_{\infty,[0,\infty)}\) and \(\|\xi^{(3)}\|_{\mathcal{X}^{3}}=\|g\|_{L^{1}([0,\infty))}+ \|g\|_{L^{p}([0,\infty))}\), where \(\cdot^{1-\upsilon}{\xi_{\cdot }^{(2)}}'\) denotes \(s\mapsto s^{1-\upsilon}{\xi^{(2)}_{s}}'\) and \(\xi^{(3)}\) is given by the right-hand side of (10). Thus, in this case \(\|\xi\|_{\mathcal{X}}=\sum_{i=1}^{3} \|\xi^{(i)}\|_{\mathcal{X}^{i}}\).
Proposition 3
Let \(A<0\) and \(\beta\in(0,1)\). Then any solution to (11) is \(\mathcal{E}\)-stable and \(\mathcal{E}\)-stable in the large.
Proof
By the previous remark we only need that equation (11) is \(\mathcal{E}^{i}\)-stable and \(\mathcal{E}^{i}\)-stable in the large, for \(i=1,2,3\). To prove this, let Y be the solution to equation (11). The global \(\mathcal {E}^{i}\)-stability in the large has already been considered in [45] (Theorem 3.3). Now we divide the proof in three steps.
Remark
Observe that \(\mathcal{E}^{1}\) contains the bounded variation functions on compact sets of \(\mathbb{R}_{+}\) of the form \(\xi=\xi^{(1)}-\xi ^{(2)}\), where \(\xi^{(1)}\) and \(\xi^{(2)}\) are two non-decreasing and bounded functions on \(\mathbb{R}_{+}\).
The following result is an immediate consequence of Theorem 1 and Proposition 3.
4 Semilinear integral equations with additive noise
Definition 4
- (i)(\(\mathcal{E},p\))-stable if for \(\varepsilon>0\), there is \(\delta>0\) such that \(\|X\|_{\infty,[0,\infty)}<\varepsilon\) for any \((\xi,f,\theta)\) such that$$ \|\xi\|_{\mathcal{X}}+\|f\theta\|_{L^{1}([0,\infty))}+\|f\theta \|_{L^{p}([0,\infty))}+ \|\dot{f}\theta\|_{L^{1}([0,\infty))}< \delta; $$(13)
- (ii)
asymptotically (\(\mathcal{E},p\))-stable if it is (\(\mathcal{E},p\))-stable and there is \(\delta>0\) such that \(\lim_{t\to\infty} X(t)=0\) for any (\(\xi,f,\theta \)) satisfying (13).
An extension of Theorem 1 is the following.
Theorem 3
Proof
Observe \(X(0)=\xi_{0}\). Consequently the proof is similar to that of Theorem 1. □
Now we state a consequence of Theorem 3.
Theorem 4
Assume (H2) (resp. (H1)) holds. Let ξ be as in Proposition 3, \(f\in C^{1}((0,\infty))\) such that \(\dot{f}\theta\in L^{1}([0,\infty))\) and \(f\theta\in L^{1}([0,\infty))\cap L^{p}([0,\infty))\) for some \(p>\frac {1}{\alpha-1}\), and \(\beta+1>\alpha\). Then any continuous solution to (12) is asymptotically \((\mathcal{E},p)\)-stable (resp. globally \(\mathcal{E}\)-stable in the large).
Proof
4.1 Stochastic integral equations with additive noise
In the remaining of this paper we suppose that all the introduced random variables are defined on a complete probability space \((\varOmega ,\mathcal{F},P)\).
Remark 2
The last remark motivates the following.
Definition 5
A continuous solution X to equation (12) is said to be globally \(\mathcal{E}\)-stable in the mean if \(\mathbf{E} |X(t)|\to0\) as \(t\to\infty\) for any process \(\xi\in\mathcal{E}\).
An immediate consequence of the proof of Theorem 1, we can state the following extension of Theorem 3.
Theorem 5
Let h satisfy (H1), \(A<0\), \(\mathcal {E}\) a family of continuous processes and \(f,\theta\) as in Remark 2 such that the solution to equation (14) is stable in the mean. Then any continuous solution to equation (12) is also \(\mathcal{E}\)-stable in the mean.
Remark
In [45] (Theorem 4.3) we can find examples of families of processes for which the solution of (14) is \(\mathcal{E}\)-stable in the mean.
Another definition motivated by Remark 2 is the following.
Definition 6
Let \(\mathcal{E}\subset\mathcal{X}\) be a family of continuous functions. We say that a continuous process ξ belongs to \(\mathcal{E}\) in the mean (\(\xi\in \mathcal{E}_{m}\) for short) if \(\mathbf{E}(|\xi|)\in\mathcal{E}\).
The following definition is also inspired by Remark 2.
Definition 7
Remark
In this definition, if \(\xi=\sum_{i=1}^{n}\xi ^{(i)}\), with \(\xi^{(i)}\in\mathcal{E}_{m}\), then we set \(\|\xi\|_{\mathcal{X}}=\sum_{i=1}^{n} \|\xi^{(i)}\|_{\mathcal{X}}\).
Theorem 6
Let (H2) be true, ξ as in Proposition 3, \(p>\frac {1}{\beta}\), and \(f \in C^{1} ((0,\infty))\) a positive function with negative derivative such that \((r\mapsto r^{\gamma}|\dot{f}(r)|)\in L^{1} ([0,\infty))\) and \(( r\mapsto r^{\gamma}{f}(r) )\in L^{1} ([0,\infty))\cap L^{p}([0,\infty))\). Moreover, let h be a non-decreasing and locally Lipschitz function, which is concave on \(\mathbb{R}_{+}\) and convex on \(\mathbb{R}_{-} \cup\{ 0\}\). Then the solution to equation (17) is \((\tilde {\mathcal{E}},p)\)-stable in the mean, where \(\xi\in\tilde{\mathcal{E}}\) if and only if \(\xi=\xi^{(1)}-\xi^{(2)}\) with \(\xi^{1}\), \(\xi^{2}\) two non-negative, non-decreasing, and continuous processes in \(\mathcal{E}_{m}\).
Proof
Example 1
Example 2
5 Conclusion
In this work we show that a useful tool to study several definitions of stability for some fractional equations is comparison results for fractional systems (see Lemma 1) and an equation in terms of the Mittag-Leffler functions (see representation (3)). Hence we can apply the properties of the Mittag-Leffler function to consider fractional systems with a function as initial condition and an additive noise, which is a Young integral that could be a functional of fractional Brownian motion.
Declarations
Acknowledgements
The authors thank the referees for their suggestions for improvements. Also we are thankful to Cinvestav-IPN and Universitat de Barcelona for their hospitality and economical support. The first author was partially supported by the CONACyT fellowship 259100. The second author was partially supported by the CONACyT grant 220303. The third author was partially supported by the MTM2012-31192 ‘Dinámicas Aleatorias’ del Ministerio de Economía y competitividad.
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
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