Stability analysis of stochastic fractional-order competitive neural networks with leakage delay

Abstract: This article, we explore the stability analysis of stochastic fractional-order competitive neural networks with leakage delay. The main objective of this paper is to establish a new set of sufficient conditions, which is for the uniform stability in mean square of such stochastic fractionalorder neural networks with leakage. Specifically, the presence and uniqueness of arrangements and stability in mean square for a class of stochastic fractional-order neural systems with delays are concentrated by using Cauchy-Schwartz inequality, Burkholder-Davis-Gundy inequality, Banach fixed point principle and stochastic analysis theory, respectively. Finally, four numerical recreations are given to confirm the hypothetical discoveries.


Introduction
The fresh concept of fractional order calculus and differential equations has three hundred years old of branch. For long period, the theory of fractional calculus is developed only on pure mathematics. In 1695, the establishment of non-integer order math, which is a speculation integer order differential and integrals, was most importantly talked about through Guillaume de Leibnitz and Gottfried Wilhelm Leibnitz, furthermore, its advancement were continuous for extensive stretch [1]. Owing to lack of solution methods, the development of fractional order calculus has not much attracted more mathematicians in those periods. In recent years, fractional order dynamical system has aroused reported in [26][27][28][29][30][31][32][33][34]. A large number of stochastic financial models appeared in the literature, see, for example, [35][36][37][38][39][40] and the references therein. Also, the important effect of noise disturbances should be taken into account in studying the dynamics of a financial system by means of the neural network approach.
Stochastic differential equations becomes a extraordinary interest due to its applications in characterizing numerous issues in physics, biology, mechanics, etc. Qualitative properties such as existence, uniqueness, controllability and stability for various stochastic differential systems have been extensively studied by many researchers, see for instance [41][42][43][44][45][46][47][48] and the references therein. Around 1960, for obvious mathematical reasons, systems of ordinary stochastic differential equations of Ito type [49][50][51], stochastic partial differential equations [52,53], stochastic fractional differential equations [54,55]. The effects of random environmental fluctuations are characterized by normalized Wiener process [56]. So it is characteristic and important to explore dynamical properties of the solutions for SDEs to discover the impacts of random perturbations in the relating realistic systems. The numerical models got have been extraordinarily produced for SDEs under an irregular disturbance of the Gaussian white noise, namely, the examinations concerning SDEs driven by Brownian movement have been extremely plentiful up to now.
To the best of our knowledge, stability analysis of stochastic fractional-order competitive neural networks with leakage delay has not been fully investigated, and there is still much room left for further investigation. Motivated by the above discussions, this paper devotes to presenting a sufficient criterion for stability analysis of stochastic fractional-order competitive neural networks with leakage delay model. Meanwhile, the existence, uniqueness, and uniform stability in mean square are proved.
The main aim and contribution of our paper are highlighted as follows: (1) We get stochastic fractional-order competitive neural networks with leakage delay model by use fractional-order instead of integer-order.
(2)Our main theme of our paper is to design both the stability analysis of stochastic fractional-order competitive neural networks with leakage delay by using Cauchy-Schwartz inequality, Burkholder-Davis-Gundy inequality, some sufficient condition for guarantee the stability.
(3) We establish a new set of sufficient criterion ensuring the uniform stability in mean square of the system and existence and uniqueness of solutions also proved by using contraction mapping principle.
(4) Various lemma's and fractional-order theory are applied to derive the main results. This paper is organized as follows. In Section 2, we introduce the definitions and lemmas and stochastic fractional-order competitive neural networks with leakage delay model.In Section 3, we shall establish a new set of sufficient criterion ensuring the uniform stability in mean square of the system and the existence, uniqueness, and uniform stability in mean square. In Section 4, we give a numerical examples which confirm the theoretical results. Finally, the paper is concluded in Section 5.
Notations:The Caputo fractional derivative operator D p is chosen for fractional-order derivative with order p; R n and R n×n denote the n-dimensional Euclidean space and the set of all n × n real matrices, respectively; C the complex number set; R, R + and Z + are the set of all real numbers, the set of all nonnegative numbers and the set of al nonnegative integer numbers, respectively; E(·) stands for the mathematical expectation with some probability measure; Ω = (L 2 F 0 ([0, T ], R n ), || · ||); for any z = (z 1 , ..., z n ) T ∈ R n , we define the vector.

Preliminaries
In this section we present some definitions, lemma and recall the well-known results about fractional differential equations.
Definition 2.1. [57] The Caputo fractional-order derivative with order p for a differential function z(t) is defined as where t ≥ 0 and m − 1 < p < m ∈ Z + . Peculiarly, when p ∈ (0, 1), Definition 2.2. [57] The e Riemann-Liouville fractional integral of order p ∈ (0, 1) for a function z(t) is defined as where Γ(·) is the gamma function, Definition 2.3. The solution of system Eq (3.1) is said to be stable if for any > 0 there exists and v(t, t 0 , φ). It is uniformly stable in mean square if the above δ is independent of t 0 .
Assumption 1. [60] We assume that the non-linear functions g j (·) and σ i j (·, ·) satisfy the following conditions: There exist positive constants L j and η i j such that for any x, y,x,ȳ ∈ R, i, j = 1, ..., n. For convenience, we introduce the following notation related to model Eq (3.1).

Main results
In this paper, the stochastic fractional-order competitive neural networks with leakage delays is defined by where D p denotes Caputo fractional derivative of order p with 0 < p < 1, z i (t) is the corresponding state variable of the number of n neurons at time t, b i > 0 and v i > 0, represent the self feedback connection weight matrices, a ik , d ik are represents the synaptic connection weight matrix and delayed synaptic connection weight matrix, respectively to ith and kth neurons, p l denotes the constant external stimulus, c i denotes the external strengths of the stimulus, r il denotes the synaptic efficiency, µ > 0, δ > 0 denote the leakage delays, g k (z k (t)) and g k (z k (t − η)) are referred the bounded neuron output activation, where the time varying delay η is bounded and differentiable. After setting s i (t) = m l=1 r il (t)ξ l = r T i (t)ξ, where r i (t) = (r i1 , ......., r im ) T , ξ = (ξ 1 , ............., ξ m ) T . Without loss of generality, ξ is is assumed to normalized with unit magnitude |ξ| 2 = 1 , where ξ is the input stimulus. Then the model (3.1) can be simplified the following state-space form such as Initial conditions of the model(3.1) is described as: Now we applying stochastic terms in the above equation we get, σ(·, ·) = (σ(·, ·)) n×n is the diffusion coefficient matrix and ω(·) = (ω 1 (·), ..., ω n (·)) T is an n-dimensional Brownian motion defined on a complete probability space (Ω, F, P) with a natural filtration 0], R) denotes the family of all C-valued random processes γ(s) such that γ(s) is F 0 -measurable and 0 −τ E||γ(s)|| 2 ds < ∞. Theorem 3.1. If assume Assumption 1 hold, then the system Eq (3.1) has a unique solution.
Proof. According to the properties of the fractional calculus, one can obtain that system Eq (3.4) is equivalent to the following Volterra fractional integral with memory where t ∈ [0, T ]. We consider a mapping φ : R n → R n , defined by: For any two different functions (z 1 (t), ..., z n (t)) T , (y 1 (t), ..., y n (t)) T , we have [a ik g k (y k (s)) − a ik g k (z k (s))] Then, applying elementary inequality, one sees that First, we have a tendency to valuate the primary term of the right hand side of the above inequality by using Cauchy inequality to obtain Next, we evaluate the second term by using Assumption 1, we have However, by using the Burkholder-Davis-Gundy inequality and Assumption 1, we get that Thus, by combining the above inequalities together, one obtains Similarly, by the same procedures as the above inequality, we obtains By using Cauchy inequality From (3.14) and (3.15) in (3.13) From (3.12) and (3.16) By combining the both the equations we get Therefore the mapping φ is a contraction mapping. As a consequence of the Banach fixed point theorem, the problem Eq (3.5) has a unique fixed point, so that we conclude that system Eq (3.1) has a unique solution, which complete the proof of the theorem.
Theorem 3.2. If Assumption 1 hold, the solution of system given by Eq (3.1) satisfying initial condition is uniformly stable in mean square.
Proof. Assume that For any two different functions (z 1 (t), ..., z n (t), s 1 (t), ......, s n (t)) T , (y 1 (t), ..., y n (t), u 1 (t), ......., u n (t)) T , solutions of Eq (3.1) with the different initial conditions Based on Lemma 2.4, the solution of the system Eq (3.21) can be expressed in the following form Then we have Firstly, we observe that Secondly, we can get from Assumption 1 However, by using the Burkholder-Davis-Gundy's inequality and Assumption 1, we get that Consequently, by combining the above inequalities together, we have where, Similarly, by the same procedures as the above inequality, we obtains Substituting the above (3.30) and (3.31) in the above equation (3.29) we get From Eqs (3.28) and (3.32) .
by the above two equations we get which means that the solution of system Eq (3.1) is uniformly stable in mean square.

Remark 1.
In the proof of Theorem 3.2, we investigated stochastic fractional-order competitive neural networks with leakage delay without constructing Lyapunov function and by using Cauchy inequality, analysis method and Burkholder DavisGundy inequality.

Remark 2.
If there are no stochastic disturbance, then system (3.4) becomes the following fractionalorder competitive neural networks with leakage delay:

38)
Theorem 3.3. If assume Assumption 1 hold, then the system Eq (3.38) has a unique solution.
Proof. According to the properties of the fractional calculus, one can obtain that system Eq (3.38) is equivalent to the following Volterra fractional integral with memory where t ∈ [0, T ]. We consider a mapping φ : R n → R n , defined by: where φ(u) = (φ 1 (u), φ 2 (u), ...., φ n (u)) T . For any two different functions z(t) = (z 1 (t), ..., z n (t)) T , y(t) = (y 1 (t), ..., y n (t)) T , we have [a ik g k (y k (s)) − a ik g k (z k (s))] Then, applying elementary inequality, one sees that First, we evaluate the first term of the right hand side of the above inequality by using Cauchy's inequality to obtain Next, we evaluate the second term of Eq (3.41) by using Assumption 1, we have Thus, by combining the above inequalities together, one obtains Similarly, by the same procedures as the above inequality, we obtains By using Cauchy inequality

From (3.45) and (3.49)
By combining the both the equations we get Therefore the mapping φ is a contraction mapping. As a consequence of the Banach fixed point theorem, the problem Eq (3.40) has a unique fixed point, so that we conclude that system Eq (3.38) has a unique solution, which complete the proof of the theorem.

Remark 3.
If there are no stochastic disturbance, then system (3.4) becomes the following fractionalorder competitive neural networks with leakage delay:

54)
Theorem 3.4. If Assumption 1 hold, the solution of system given by Eq (3.54) satisfying initial condition is uniformly stable in mean square.
Proof. Assume that z(t) = (z 1 (t), ..., z n (t), s 1 (t), ........., s n (t)) T and y(t) = (y 1 (t), ..., y n (t), u 1 (t), ....., u n (t)) T are solutions of Eq (3.54) with the different initial conditions Based on Lemma 2.4, the solution of the system Eq (3.55) can be expressed in the following form Then we have Firstly, we observe that Secondly, we can get form Assumption 1 Consequently, by combining the above inequalities together, we have where, ] ] Similarly, by the same procedures as the above inequality, we obtains Substituting the above (3.63) and (3.64) in the above equation (3.62) we get From Eqs (3.61) and (3.65) .
By the above two equations we get which means that the solution of system Eq (3.54) is uniformly stable in mean square.
Remark 4. The author derived the existence and uniqueness results using Banach contraction fixed point theorem, sufficient conditions for uniform stability of equilibrium point for the networks. But, it is more complicated to study the stochastic fractional-order competitive neural networks with leakage delays. This interesting problem competitive neural networks model.

Remark 5.
In this paper we investigated the mean square stability of stochastic fractional-order competitive neural networks with leakage delays with 1 2 ≤ α < 1 by using Cauchy Schwarz inequality. Many authors have focused on studying the stability analysis of fractional-order neural networks which depends on the orders a of fractional derivatives. Unlike the previous works, we analyzed the stability of stochastic fractional-order neural networks with delays and leakage terms which are dependent on the orders α and β different fractional derivatives and reflect the close relation between neuron activation functions, time-delay of network parameters, and coefficient terms. This was motivated by the long range delay dependent dynamic process that is part of our current project. However, we propose to investigate the proposed problem for not only 0 < α < 1, but also more general set of linearly independent multi time-scales.

Conclusion
In this paper, we investigate the stability analysis of stochastic fractional-order competitive neural networks with leakage delay. As is well known, there are many stability results about integer-order neural networks in the past few decades, most of which are obtained by constructing Lyapunov function, but these results and methods could not be extended and applied to fractional-order case. According to the Cauchy-Schwartz inequality, Burkholder-Davis-Gundy inequality, analysis techniques, some sufficient conditions were derived to guarantee the existence and uniqueness and the uniform stability in mean square. Furthermore, the main tools used in this paper are stochastic analysis techniques, fractional calculations and Banach contraction principle. Finally, four numerical examples are given to illustrate the effectiveness of the proposed theories.