Stability for a class of semilinear fractional stochastic integral equations

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.


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. [4], [15], [18], [36] and the references therein). Particularly we can mention system identification [4], robotics [26], control [4,36], electromagnetic theory [14], chaotic dynamics and synchronization [12,13,42,44], applications on viscoelasticity [2], analysis of electrode processes [16], Lorenz systems [12], systems with retards [6], quantic evolution of complex systems [19], numerical methods for fractional partial differential equations [5,30,31], among other. A nice survey of basic properties of deterministic fractional differential equations is in Lakshmikantham and Vatsala [20]. 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 [27] and Radwan et al. [38]. Besides, several authors have studied non-linear cases using Lyapunov method (see, e.g. Li et al. [23] and its references). In particular, non-linear fractional systems with a function as initial condition using also the Lyapunov technique have been considered in the Ph.D.Thesis of Martínez-Martínez [28]. Moreover, in the work of Junsheng et al. [17] the form of the solution for a linear fractional equation with a constant initial condition in terms of Mittag-Leffler function is given by means of the Adomian decomposition method. Wen et al. [41] have established stability results for fractional non-linear equations via the Gronwall inequality. Lemma 2.2 below can be seen as an extension of the results in [44] and the Gronwall inequality stated in [41]. In [41], the stability is used to obtain synchronization of fractional systems.
On the other hand, a process used frequently in literature is fractional Brownian motion B H = {B H t , t ≥ 0} 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 intermitency (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 = 1/2, but we may use another approaches such as Young integration (see Gubinelli [11], Young [45], Zähle [49], Dudley and Norvaisa [8], Lyons [25]). The reader can also see Nualart [33], and Russo and Vallois [39] 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 [25], Quer-Sardanyons and Tindel [37], León and Tindel [21], Nualart [33], Friz and Hairer [9], Lin [24] and Nualart and Rȃşcanu [34]).
Stability of stochastic systems driven by Brownian motion has been also studied. Some authors use fundamental solution of this equations in order to investigate the stability of random systems. An example of this is the paper of Applebay and Freeman [1], who give the solution in terms of the principal matrix of integrodifferential equations with an Itô integral noise and find the equivalence between almost sure exponential convergence and the p-th mean exponential convergence to zero for these systems. Bao [3] uses Gronwall inequality to state the mean square stability for Volterra-Itô equations with a function as initial condition and bounded kernels. Several researchers have studied stability of stochastic systems via Lyapunov function techniques. An example of this is the paper of Li et al. [22], who proves stability in probability for Itô-Volterra integral equation, also Zhang and Li [47] have stated a stochastic type stability criteria for stochastic integrodifferential equations with infinite retard, and, Zhang and Zhang [48] have dealed with conditional stability of Skorohod Volterra type equations with anticipative kernel. Nguyen [32] present the solution via the fundamental solution for linear stochastic differential equations with time-varying delays to obtain the exponential stability of these systems. The noise is an additive one and has form · 0 σ(s)dW H s . Here W is a Brownian motion and σ is a deterministic function such that ∞ 0 σ 2 (s)e 2λs ds < ∞, for some λ > 0. Also, Zeng et al. [50] utilize the Lyapunov function techniques to prove stability in probability and moment exponential stability for stochastic differential equation driven by fractional Brownian motion with parameter H > 1/2 . Yan and Zhang [43] proved sufficient conditions for the asymptotical stability in p-th moment for the closed form of the solution to a fractional impulsive partial neutral stochastic integro-differential equation with state dependent retard in Hilbert space. In the linear case, Fiel et al. [10] have used the Adomian decomposition method to find the mild solution of an stochastic fractional integral equation with a function as initial condition driven by a Hölder continuous process in terms of Young or Skorohod integrals. This closed form is given in terms of Mittag-Leffler functions. The stability in the large and stability in the mean sense of these random systems is also analized. As an application, the stability of equations driven by a functional of fractional Brownian motion is derived. In this paper we extend the results given in [10] and [41], that is, we study the stability of the solution to the equation there are C > 0 and δ > 0 such that |h(x)| ≤ C|x| for |x| < δ), β ∈ (0, 1), A < 0 and Z is a Young integral of the form Here, θ = {θ s , s ≥ 0} is a γ-Hölder continuous function that may represent the pahts of a functional of fractional Brownian motion, where γ ∈ (0, 1), α ∈ (1, 2) and α + γ > 2. Unlike other papers where the involved kernels are bounded functions we consider the case that kernels are not bounded, and use comparison results as a main tool. It is worth mentioning that it is considered the stability of the solution to (1.1) in [41] with Z ≡ 0 and ξ a constant.
This work is organized as follows. In Section 2 we introduce a fractional integral equation, whose family of solutions include those of (1.1). 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 (1.1) in the case that Z ≡ 0. These results can be seen as extensions of the results given in [10] and [41]. Finally, the stability of equation (1.1) in the case that θ is either a Hölder continuous process, or a functional of fractional Brownian motion is considered in Section 4.

Preliminaries
In this section we introduce the framework and the definitions that we use to prove our results. Although some results are well-known, we give them here for the convenience of the reader. Part of the main tool that we need is the stability of some fractional linear systems as it was presented by Fiel et al. [10] and a comparison result (see Lemma 2.3 below).

The Mittag-Leffler function
The Mittag-Leffler function is an important tool of fractional calculus due to its properties and applications. As we can see in Lemma 2.2 below, solutions of semilinear fractional integral equations depend on it. In order to see a more detailed exposition on this function, the reader is refered to the book of Podlubny [36]. For z ∈ R, this function is defined as where Γ is the Gamma function. A function f defined on (0, ∞) is said to be completely monotonic if it possesses derivatives f (n) for all n ∈ N ∪ {0}, and if for all t > 0. In particular, we have that each completely monotonic function on (0, ∞) is positive, decreasing and convex, with concave first derivative (see, e.g. [29] and [40]). It is well-known that for a, b ≥ 0, z → E a,b (−z) is completely monotonic if and only if a ∈ (0, 1] and b ≥ a (see Schneider [40]). Moreover, for a < 2, there is a positive constant C a,b such that (see [36], Theorem 1.6). For a, b > 0 and λ ∈ R, this function satisfies the following (see (1.83) in [36]):

The Young integral
Here we introduce the Young integral, which is an integral with respect to Hölder continuous functions. This was initially defined for functions with p-variation in Young [45].
Using the following result, we can understand easily the basic properties of Young integral for Hölder continuous functions. The proof of this theorem can be found in [11] (see also [21]). Sometimes we write J st (f dg) instead of t s f u dg u .
We observe that this integral has been extended by Zähle [49], Gubinelli [11], Lyons [25], among others. For a more detailed exposition on the Young integral the reader is refered to the paper of Dudley and Norvaisa [8] (see also Gubunelli [11], and León and Tindel [21]).

Semilinear Volterra integral equations with additive noise
Here we consider the Volterra integral equation where the initial condition ξ = {ξ t , t ≥ 0} is bounded on compact sets and measurable, β ∈ (0, 1), A ∈ R, α ∈ (1, 2), θ = {θ s , s ≥ 0} is a γ-Hölder continuous function with γ ∈ (0, 1) and Γ is the Gamma function. The second integral in Now we give two lemmas that we need in the remaining of the paper. The following result provides a closed form for the solution to equation (2.4) and its proof is in [10].

Lemma 2.2
Let α + γ > 2 and A ∈ R. Then, the solution to (2.4) has the form where η > 0 and g ∈ L 1 ([0, M )) for each M > 0, then we have 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 [20]), but, unfortunately this results are not suitable for our purpose. Thus, we give the following lemma, that is a version of Theorem 2.2.5 in Pachpatte [35] and allows us to prove stability for the semi-linear equations that we study. Hence, this result is a fundamental tool in the development of this paper.
Also, let B ∈ R, β ∈ (0, 1), and x and y two continuous functions on [0, T ] such that (2.6) where u is the solution to the equation Remark. The assumptions of k yield that equation (2.7) has a unique continuous solution.

Proof. Denote by C([0, T ]) the family of continuous functions on
It is not difficult to see that Hyphoteses i) and iii) imply that G is well-defined. It means, G(z) is a continuous function for . Then, from the continuity of E β,β and hypothesis ii), there is a constantM > 0 such that, for every z,z ∈ C([0, T ]), we have Similarly, forT ≤ T, we are able to see that , due to Hypothesis iv), (2.6), and E β,β being a completely monotonic function. Thus the result is true if we writeT instead of T. Now, suppose the lemma holds for the interval [0, nT ], n ∈ N. Then, by (2.6) we can write and using the fact that equation (2.7) has a unique solution due to Hypothesis ii), we can proceed as in the first part of this proof to see that x ≤ u on [0, (n + 1)T ]. Thus, the result follows using induction on n.

A class of a 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 [10] for this kind of systems when the noise is null (i.e., Z in (1.1) is equal to zero).

A constant as initial condition
This part is devoted to refine Theorem 1 of [41] in the one-dimensional case. Toward this end, in this section, we suppose that the initial condition is a constant. That is, we first consider the fractional equation with x 0 ∈ R, β ∈ (0, 1), A < 0 and h : R → R a measurable function.
In the remaining of this paper we deal with the following hypotheses.
(H1) There is a constant C > 0 such that A + C < 0 and |h(x)| ≤ C|x|, for all x ∈ R.

Definition 3.1 Any solution X to equation (3.1) is said to be:
i) globally stable in the large if X(t) goes to zero as t tends to infinity, for all where β ∈ (0, 1), B < 0, b > 0 and m is a positive and locally Lipschitz function with Similarly, for a continuous function h satisfiying (H2), we introduce the function Then, using [20] (Theorems 3.1 and 4.2) again, the equation has at least one solution defined on [0, ∞) due to |Ax+ h(ϕ(x))| ≤ |Ax|+ C|ϕ(x)| ≤ (|A|+ C)|x|. Hence equation (3.1) has at least one continuous solution on [0, ∞) if (3.2) is stable and x 0 is small enough because, in this case, the solution of (3.2) is also a solution of equation (3.1) and h • ϕ is bounded. So, without loss of generality we can assume that (3.1) has at least one continuous solution because one of the main purposes of the paper is to deal with the stability of (1.1).
We need the following lemma to prove some of our results. The main idea of its proof is in the paper of Martínez-Martínez et al. [28]. Here we give an sketch of the proof for the convenience of the reader. (H2) (resp. (H1)). Then, for 0 < x 0 < δ 0 (resp. x 0 > 0), any continuous solution X of (3.1) satisfies X(t) > 0 for all t ≥ 0.

Remark.
As it was pointed out in [28], if the initial condition in equation (3.1) is a non-decreasing, continuous and non-negative function instead of a constant, we can repeat the procedure in this proof in order to obtain the same result. Indeed, suppose that 0 < ξ t < δ 0 (resp. ξ t > 0) for all t ≥ 0, first of all (3.3) becomes Secondly, instead of equallity (3.4) we have due to ξ being non-decreasing. Hence, it is not difficult to see that (3.5) is still satisfied.
An immediate consequence of the first part of the proof of Lemma 3.3 is the following.
For x 0 < 0 and X a solution of (3.1), we have −X is a solution of Now we establish the main result of this subsection.

Proposition 3.5 Let h be a function satisfiying (H2) (resp. (H1)). Then, any continuous solution of equation (3.1) is
Mittag-Leffler stable and therefore is also asymptotically stable (resp. globally stable in the large).
On the othe hand, consider the solution Z of the following linear fractional equation Then by the continuity of the solutions X and Z, there exists τ > 0 such that, for all t ∈ (0, τ ), we have 0 < X(t) < Z(t). If this inequality is satisfied for any t > 0, we can ensure that X is asymptotically stable (resp. and globally stable in the large), and that this solution also is Mittag-Leffler stable because the solution to Z of last equation is given by (see [17] or Lemma 2.2) We now suppose that there exists t 0 > 0 such that X(t 0 ) = Z(t 0 ) and X(t) < Z(t), for t < t 0 . Set Y = X − Z, then From (2.5) (see also [17]) we observe that Y also satisfies the equality For s ∈ (0, t 0 ), we have |h(X(s))| ≤ CX(s) < CZ(s). Thus h(X(s)) − CZ(s) < 0. Consequently, by the completely monotonic property of E β,β we have Y (t 0 ) < 0, and this is a contradiction because it is supposed that Y (t 0 ) = 0. Now we can conclude that X is Mittag-Leffler stable.
Finally we consider the case that −δ 0 < x 0 < 0 (resp. x 0 < 0). Note thatX = −X is such that . Hence, by the first part of this proof and the fact thath satisfies (H2) (resp. (H1)), we have that the proof is complete.

Remark. Let X be a solution to equation (3.1). Wen et al. [41] (Theorem 1) have proved that the solution to equation (3.1) is stable if lim |x|→0
|h(x)| |x| → 0. Also, Zhang and Li [46] have used a result similar to Lemma 2.2 to prove that X is asymptotically stable for the case that lim x→0 |h(x)| |x| = 0, β ∈ (1, 2) and β + 1 |A| < 2. Proposition 3.5 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.

A function as initial condition
Here we treat the case that the initial condition is a function satisfying some suitable conditions. Consider the following deterministic Volterra integral equation Here β ∈ (0, 1), A < 0, and h : R −→ R and ξ : R + −→ R are two measurable functions. Concerning the existence of a continuous solution of equation (3.6) we remark the following. For a continuous function h as in (H1) and ξ continuous, we can consider the equation which has a solution Z due to Theorem 4.2 in [20] (with g(s, x) = (|A| + C)(x + |ξ s |)) and Lemma 2.2. Therefore Z + ξ is a solution of (3.6). Similarly if ξ is "small enough" and h is either a continuous Lipschitz function on a neighbourhood of zero, or as in (H2), then we can proceed as in Remark 3.2 to see that (3.6) has at least one solution in this case. Therefore, as in Remark 3.2, we can assume that (3.6) has at least one continuous solution.
On the other hand, in this paper we analyze several stability criteria for different classes E of initial conditions. Sometimes E is a subset of a normed linear space X of continuous functions endowed with the norm || · || X . In other words we consider normed linear spaces (X , || · || X ). Mainly, in the remaining of this paper, we deal with the following classes of initial conditions. Definition 3. 6 We have the following assumptions on ξ: 1. If the initial condition ξ is continuous on [0, ∞) and there is ξ ∞ ∈ R such that, given ε > 0, there exists t 0 > 0 such that |ξ s − ξ ∞ | ≤ ε for any s ≥ t 0 , we say that ξ belongs to the family E 1 .
The stability concepts that we develop in this section are the following.

Definition 3.7 Let E ⊂ X .
A solution X of (3.6) is said to be: i) globally stable in the large for the class E (or globally E-stable in the large) if X(t) tends to zero as t → ∞, for every ξ ∈ E.
iii) asymptotically E-stable if it is E-stable and there is δ > 0 such that lim t→∞ X(t) = 0 for any ξ ∈ E such that ||ξ|| X < δ.
In the following auxiliary result, E 4 is the family of functions ξ having the form (3.7) with η = β and g is a continuous function such that lim t→∞ g(t) = 0. In this case, the involved norm is ||ξ|| X = ||g|| ∞,[0,∞) .

Lemma 3.8 Let B < 0 and ξ ∈ E 4 . Then the solution to the equation
is E 4 -stable and globally E 4 -stable in the large.
Proof. We observe that, by Lemma 2.2, we have So, the completely monotone property of E β,β , (2.1) and (2.3) lead us to establish Thus, Y is E 4 -stable. Also, by (2.3) we are able to write Therefore, using (2.1) and the proof of Proposition 3.3.1 in [10] again, together with the facts that B < 0 and g is a continuous function such that lim t→∞ g(t) = 0, we obtain Y (t) → 0 as t → ∞.

Now we give a general result.
Theorem 3.9 Let (H2) (resp. (H1)) be true, and E a family of continuous functions of a normed linear space X such that the solution of the equation is asymptotically E-stable (resp. globally E-stable in the large). Then any continuous solution of equation (3.6) is also asymptotically E-stable (resp. globally E-stable in the large).

Proposition 3.10 Let
Proof. By previous remark we only need that equation (3.8) is E i -stable and E i -stable in the large, for i = 1, 2, 3. To prove this, let Y be the solution to equation (3.8). The global E i -stability in the large has already been considered in [10] (Theorem 3.3). Now we divide the proof in three steps.
Step 1. Here we consider the case i = 1. Then Lemma 2.2 and (2.3) give that, for t ≥ 0, which implies that the solution of (3.8) is ξ (1) -stable.
Step 2. For i = 2, we get Consequently, [10] (proof of Theorem 3.2.2) yields where C > 0 is a constant and we have utilized that υ < β.
Step 3. Finally we consider the case i = 3. In this scenario, from Lemma 2.2, we obtain 2 (t), t ≥ 0.
For I 1 we can apply Hölder inequality to write, for q −1 = 1 − p −1 and C > 0 and for I 2 we use the fact that η − 1 − β < 0. Thus
The following result is an immediate consequence of Theorem 3.9 and Proposition 3.10.

Definition 4.1
Let E ⊂ X be a family of continuous functions. We say that a solution X of (4.1) is ii) asymptotically (E, p)-stable if it is (E, p)-stable and there is δ > 0 such that lim t→∞ X(t) = 0 for any (ξ, f, θ) satisfiying (4.2).
An extension of Theorem 3.9 is the following.

Theorem 4.2 Let (H2) (resp. (H1)) be satisfied and E a class of continuous functions such that the solution of the equation
is asymptotically (E, p)-stable (resp. globally E-stable in the large). Then, any continuous solution of (4.1) is also asymptotically (E, p)-stable (resp. globally E-stable in the large).
Proof. Observe X(0) = ξ 0 . Consequently the proof is similar to that of Theorem 3.9.
Now we state a consequence of Theorem 4.2.
Observe that, in the previous proof, the inequality is still true for β + 1 ≥ α, wich is used in the proof of Theorem 4.9 below.

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 (Ω, F , P ).
The last remark motivate the following: Here, in order to finish the paper, A, h, β, γ and f are as in equation (4.1) such that β + γ > 1, and ξ is a continuous stochastic process. We remark that we interprete equation (4.6) path by path (i.e. ω by ω).
The following definition is also inpired by Remark 4.4.
In other words, we have E (|X(t)|) ≤ E X (1) (t) + E X (2) (t) , t ≥ 0. (4.9) Finally, observe that (4.7), (4.8), the fact that A is a negative number and Jensen inequality give, for θ s = s γ , Hence by (4.9), Lemma 2.3, Hypothesis (H2), and the proofs of Proposition 3.10 and Theorem 4.3 we get that the result holds. Indeed, for i = 1, 2, where u (i) is the unique solution to the equation

Example 4.10 A function h that satisfies the conditions of Theorem 4.9 is
where C > 0. Indeed, we have that Thus, given ε > 0 there is δ > 0 such that |h(x)| ≤ (C + ε)|x| for |x| ≤ δ.

Example 4.11
Here we give a function that satisfies Assumption 2 on Definition 3.6. Let ξ t = g(t) sin 1 t , t ≥ 0. The function g is bounded and satisfies g(t) = ψ(t)c 0 t 3−υ + ϕ(t) c1 1+t , where ψ, ϕ ∈ C ∞ (R + ) are such that Thus Now it is easy to verify our claim is true using straightforward calculations.

Aknowledgements:
The authors thank Cinvestav-IPN and Universitat de Barcelona for their hospitality and economical support.