Stability results for neutral stochastic functional differential equations via fixed point methods

Stability results for neutral stochastic functional differential equations via fixed point methods Mimia Benhadria, Tom´as Caraballob, Halim Zeghdoudic aFr`eres Mentouri Constantine University, Department of Mathematics, Algeria. emails: [email protected] bDepartamento de Ecuaciones Difererenciales y An´alisis Num´erico, Universidad de Sevilla c/ Tarfia s/n, 41012-Sevilla (Spain) email: [email protected] cLaPS Laboratory, Badji-Mokhtar University, Box 12, Annaba, 23000, Algeria. email: [email protected] July 30, 2018 Abstract In this paper we prove some results on the mean square asymptotic stability of a class of neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a necessary and sufficient condition ensuring the asymptotic stability is proved. The assumption does not require neither boundedness or differentiability of the delay functions, nor do they ask for a fixed sign on the coefficient functions. In particular, the results improve some previous ones proved by Guo et al. (2017). Finally, an example is exhibited to illustrate the effectiveness of the proposed results. AMS Subject Classifications: 34K20, 34K13, 92B20 Keywords: Fixed points theory; Asymptotic stability in mean square; Neutral stochastic differential equations; Variable delays. 1 Introduction Liapunov’s direct method has long been viewed the main classical method to study stability problems in several areas of differential equations. The success of Liapunov’s direct method depends on finding a suitable Liapunov function 1 or Liapunov functional. However, it may be difficult to look for such a good Liapunov functional for some classes of stochastic delay differential equations. Therefore, an alternative approach may be explored to overcome such difficulties. It was proposed in (Burton, 2004–2006; Burton & Zhang, 2004) to use a fixed point method to study the stability problem for deterministic systems. Luo (2007) and Appleby (2008) have applied this method to deal with stability problems for stochastic delay differential equations, and afterwards, several types of stochastic delay differential equations are investigated by using fixed point methods. For example, Sakthivel & Luo (2009a, 2009b) investigate the asymptotic stability of nonlinear impulsive stochastic differential equations and impulsive stochastic partial differential equations with infinite delays by means of the fixed point theory. On the other hand, Luo (2008, 2010) firstly considers the exponential stability for stochastic partial differential equations with delays by the fixed point method. Zhou & Zhong (2010) study the exponential p-stability of neutral stochastic differential equations with multiple delays. Pinto & Sep´ulveda (2011) deal with H-asymptotic stability by the fixed point method in neutral nonlinear differential equations with delay. It turns out that the fixed point method is becoming a powerful technique in dealing with stability problems for deterministic and stochastic differential equations with delays. Moreover, it possesses the advantage that it can yield the existence, uniqueness and stability criteria of the considered system in one step. In this paper, we address the mean square asymptotic stability for neutral stochastic functional differential equations with variable delays. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Some well-known results are improved and generalised. More precisely, our model contains as a particular case the one analyzed in (Guo, Chao & Jun, 2017), and therefore we ensure the validity of those results, despite the proof in (Guo et al., 2017) is not completely correct (see Remark 3.1 in Section 3 below for more details). This paper is organized as follows. In Section 2 we describe our model and recall the basic preliminary results which are necessary for our analysis. In Section 3, we prove the main result about mean-square asymptotic stability. Finally, in Section 4 an example is analyzed in order to test our abstract results as well as to highlight that the results in (Guo et al., 2017) cannot be applied to our model. 2 2 Statement of the problem and preliminaries In this paper, we consider the following class of neutral stochastic differential systems with variable delays, d ui(t)− n X j=1 qij(t)uj(t−τj(t)) =  n X j=1 aij(t)uj(t) + n X j=1 bij(t)fj(uj(t)) + n X j=1 cij(t)gj(uj(t−δj(t))) dt + n X j=1 σij(uj(t))dwj(t), t ≥t0, (1) for i= 1,2,3, ..., n,, which can be written in a vector–matrix form as follows: d[u(t)−Q(t)u(t−τ(t))] = [A(t)u(t) + B(t)f(u(t)) + C(t)g(u(t−δ(t))] dt +σ(u(t)) dw (t), t ≥t0,(2) where u(t) = [u1(t), u2(t), .., un(t)]T∈Rn,and aij, bij, cij, qij ∈C(R+,R), are continuous functions, A(t) = (aij(t))n×n, B (t) = (bij(t))n×n, Q (t) = (qij(t))n×n, are real matrices and σ(·) = (σij(·))n×nis the diffusion coefficient matrix, f(u(t)) = [f1(u1(t)) , f2(u2(t)) , ..., fn(un(t))]T∈Rn,g(u(t)) = [g1(u1(t)) , g2(u2(t)) , ..., gn(un(t))]T∈Rn. Let Ω,F,{Ft}t≥0,Pbe a complete filtered probability space and let w(t) = [w1(t), w2(t), ..., wn(t)]Tbe an n−dimensional Brownian motion on (Ω,F,{Ft}t≥0,P) such that {Ft}t≥0is the natural filtration of w(t) (i.e Ft is the completion of σ{w(s):0≤s≤t}). Here C(S1, S2) denotes the set of all continuous functions ϕ:S1→S2with the supremum norm k.k. Denote by u(t) = u(t;s, ϕ) = (u1(t;s, ϕ1), ..., un(t;s, ϕ2))T∈Rnthe solution to (1) with the intial condition ui(s) = ϕi(s) for s∈[m(t0), t0],for each t0≥0,(3) where mj(t0) = min {inf {t−τj(t), t ≥t0},inf {t−δj(t), t ≥t0}}, m(t0) = min {mj(t0),1≤j≤n}.(4) and ϕi(·)∈C([m(t0), t0],R),and s→ϕ(s)=(ϕ1(s), ..., ϕn(s))T∈Rn belongs to the space C([m(t0), t0],Rn),with the norm defined by kϕk= n X i=1 supm(t0)≤s≤t0|ϕi(s)|. Finally, Ewill denote expectation. Before proceeding, we firstly introduce some assumptions to be imposed later on (A1) The delay functions τj, δj∈C(R+,R+),with t−δj(t)→ ∞ and t−τj(t)→ ∞ as t→ ∞ for j= 1,2, ..., n. 3 (A2) there exist nonnegative constants αjsuch that for all x, y ∈R, |fj(x)−fj(y)| ≤ αj|x−y|, j = 1,2, ..., n. (5) (A3) there exist nonnegative constants βjsuch that for all x, y ∈R, |gj(x)−gj(y)| ≤ βj|x−y|, j = 1,2, ..., n. (6) (A4) there exist nonnegative constants Lij such that for all x, y ∈R, |σij (x)−σij (y)| ≤ Lij |x−y|, i, j = 1,2, ..., n. (7) Throughout this paper, we always assume that fj(0) = gj(0) = σik(0) = 0,for i, j, k = 1,2, . . . , n (8) thereby, problem (1) admits the trivial equilibrium u= 0. Very recently, Guo et al. published in (2017) related results on the solutions of a particular case of (1). More precisely, the following result was established. Theorem A. Suppose that assumptions (A1)–(A4) hold and that there exist positive scalars aisuch that, for all t≥0, n X i=1     n X j=1  |qij(t)|+ t Z0 e−ai(t−s)|aij(s)|ds + t Z0 e−ai(t−s)ai|qij(s)|ds + t Z0 e−ai(t−s)|bij(s)|αjds + t Z0 e−ai(t−s)|cij(s)|βjds   2 +2 ai n X j=1 L2 ij  ≤γ < 1 2,(9) where aij(t) = aij(t)(i6=j),aii(t) = aii (t)+ai.Then, for any ϕ∈C([m(0) ,0] ,Rn), there exists a unique global solution u(t, 0, ϕ). Moreover, the zero solution is mean-square asymptotically stable. Our objective here is to generalize Theorem A to the general case of equation (1) by proving a necessary and sufficient condition for the asymptotic stability of the zero solution. We also provide an example to illustrate our results. For each t0≥0 and ϕ∈C([m(t0), t0],Rn) fixed, we define Xli ϕi,t0as the following space of stochastic processes Xli ϕi,t0={ui(t, ω):[m(t0),∞)×Ω→R/ ui(t, .) = ϕi(t) for t∈[m(t0), t0], kuikXli ϕi,t0≤lifor t≥t0and E|ui(t)|2→0 as t→ ∞}, where kui(t, ω)kXli ϕi,t0 =Esupt≥m(t0)|ui(t)|21/2. 4 Now, we denote Xl ϕ,t0=Xl1 ϕ1,t0×Xl2 ϕ2,t0... ×Xln ϕn,t0,which can be rewritten as Xl ϕ,t0={u(t, ω):[m(t0),∞)×Ω→Rn/ u(t, .) = ϕ(t) for t∈[m(t0), t0], kukX≤lfor t≥t0and E n X i=1 |ui(t)|2→0 as t→ ∞}, where kukX:= (n X i=1 Esupt≥m(t0)|ui(t)|2)1 2 .It is easy to check that Xl ϕ,t0is a complete metric space with metric induced by the norm k·kX. When no confusion is possible we will not write Xl ϕ,t0,Xli ϕi,t0but Xl ϕ,Xli ϕi respectively, and we will also omit the random parameter ω. Let us know recall the definitions of stability that will be used in the next section. Definition 2.1: The zero solution of the system (1) is said to be : i) stable if for any ε > 0 and t0≥0,there exists a δ=δ(ε, t0)>0 such that ϕ∈C([m(t0), t0],Rn) and kϕk< δ imply E n X i=1 |ui(t, t0, ϕ)|2< ε for t≥t0. ii) asymptotically stable if the zero solution is stable and for any ε > 0 and t0≥0,there exists a δ=δ(ε, t0)>0 such that ϕ∈C([m(t0), t0],Rn) and kϕk< δ imply E n X i=1 |ui(t, t0, ϕ)|2→0 as t→ ∞. To prove our main result we will use a classical contraction mapping principle. We recall it below for the readers convenience. Theorem 2.1 (see Smart, 1974) Let Hbe a contraction operator on a complete metric space X, then there exists a unique point x∗∈Xsuch that H(x∗) = x∗. 3 Main Results Our purpose here is to extend the work carried out in (Guo et al., 2017) by providing a necessary and sufficient condition for asymptotic stability of the zero solution of equation (1). Zhang (2004, 2005) was the first to establish necessary and sufficient condition for the stability of solutions of functional differential equation by the fixed point theory. The necessity of condition (12) below for the main stability result was first established in (Zhang, 2004). It is well known that studying the stability of an equation using a fixed point technique involves the construction of a suitable fixed point mapping. This can be an arduous task. Thus, to construct our mapping P, we begin by transforming (1) into a more tractable, but equivalent, equation, which we will then invert to obtain an equivalent integral equation from which we derive a fixed point mapping. After that, we use a suitable complete metric space Xl ϕdefined above, which may 5 depend on the initial condition ϕ. Using the contraction mapping principle, we obtain a fixed point for this mapping and hence a solution for (1), which in addition is mean square asymptotically stable. Now, we can state our main result. Theorem 3.1. Suppose that assumptions (A1)–(A4) hold, and there exist continuous functions ai: [t0,∞)→Rsuch that for t≥t0 lim inf t→∞ Zt t0 ai(s)ds > −∞, i = 1, ..., n (10) n X i=1     n X j=1 |qij(t)|+Zt t0 e−Rt sai(ξ)dξ |aij(s)|ds +Zt t0 e−Rt sai(ξ)dξ |ai(s)||qij(s)|ds + t Zt0 e−Rt sai(ξ)dξ |bij(s)|αjds +Zt t0 e−Rt sai(ξ)dξ |cij(s)|βjds2 + 4 n X j=1 t Zt0 L2 ije−2Rt sai(ξ)dξ  ≤γ < 1 2,(11) where aij(t) = aij(t)(i6=j),aii(t) = aii (t) + ai(t).Then for any ϕ∈ C([m(t0), t0],Rn)there exists a unique global solution u(t, t0, ϕ). Moreover, the zero solution is mean-square asymptotically stable if and only if Zt t0 ai(s)ds → ∞ as t→ ∞.(12) Proof: Set Mi= sup t≥t0ne−Rt t0ai(s)dso,(13) which is well defined thanks to (10). Suppose also that (12) holds. We now re-write equation (1) in an equivalent form. For this end, we use the variation of parameter formula and rewrite the equation in an integral from which we derive a contracting mapping. 6 We rewrite (1) as d ui(t)− n X j=1 qij(t)uj(t−τj(t))  = −ai(t)ui(t) + n X j=1 aij(t)uj(t) + n X j=1 bij(t)fj(uj(t)) + n X j=1 cij(t)gj(uj(t−δj(t))) dt + n X j=1 σij(uj(t))dwj(t), t ≥t0,(14) with the intial condition ui(t) = ϕi(t) for t∈[m(t0), t0]. Multiplying both sides of (14) by eRt 0ai(ξ)dξ and integrating from t0to t, Rt t0heRs 0ai(ξ)dξui(s)i0 ds =Rt t0eRs 0ai(ξ)dξ    d  n X j=1 qij(s)uj(s−τj(s))  + n X j=1 aij(s)uj(s) + n X j=1 bij(s)fj(uj(s)) + n X j=1 cij(s)gj(uj(s−δj(s)))   ds +Rt t0eRs 0ai(ξ)dξ n X j=1 σij(uj(s))dwj(s). As a consequence, we arrive at eRt 0ai(ξ)dξui(t)−eRt0 0ai(ξ)dξui(t0) = Rt t0eRs 0ai(ξ)dξ    d  n X j=1 qij(s)uj(s−τj(s))  + n X j=1 aij(s)uj(s) + n X j=1 bij(s)fj(uj(s)) + n X j=1 cij(s)gj(uj(s−δj(s)))   ds +Rt t0eRs 0ai(ξ)dξ n X j=1 σij(uj(s))dwj(s). Dividing both sides of the above equation by eRt 0ai(ξ)dξ, we obtain ui(t) = e−Rt t0ai(ξ)dξui(t0) + Rt t0e−Rt sai(ξ)dξ    d  n X j=1 qij(s)uj(s−τj(s))  + n X j=1 aij(s)uj(s) + n X j=1 bij(s)fj(uj(s)) + n X j=1 cij(s)gj(uj(s−δj(s)))   ds +Rt t0e−Rt sai(ξ)dξ n X j=1 σij(uj(s))dwj(s). 7 Performing now an integration by parts, we have for t≥t0, i = 1,2, ..., n, ui(t) = ϕi(t0)−  n X j=1 qij(t0)ϕj(t0−τj(t0))  e−Rt t0ai(ξ)dξ +  n X j=1 qij(t)uj(t−τj(t)) − t Zt0 ai(s)e−Rt sai(ξ)dξ   n X j=1 qij(s)uj(s−τj(s)) ds + t Zt0 e−Rt sai(ξ)dξ n X j=1 aij(s)uj(s)ds + t Zt0 e−Rt sai(ξ)dξ n X j=1 bij(s)fj(uj(s)) ds + t Zt0 e−Rt sai(ξ)dξ n X j=1 cij(s)gj(uj(s−δj(s))) ds + t Zt0 e−Rt sai(ξ)dξ n X j=1 σij(uj(s))dwj(s).(15) Use (15) to define the operator P:Xl ϕ→Xl ϕby (Pu)(t) := [(P1u1) (t),(P2u2) (t), ..., (Pnun) (t)]T∈Xl ϕ, where Pi:Xli ϕi→Xli ϕiby (Piui) (t) = ϕi(t) for t∈[m(t0), t0] and for t≥t0, where Pi(ui):[m(t0),+∞)→R(i= 1,2, ..., n) is defined as follows: (Piui) (t) = ϕi(t0)−  n X j=1 qij(t0)ϕj(t0−τj(t0))  e−Rt t0ai(ξ)dξ +  n X j=1 qij(t)uj(t−τj(t)) − t Zt0 ai(s)e−Rt sai(ξ)dξ   n X j=1 qij(s)uj(s−τj(s)) ds + t Zt0 e−Rt sai(ξ)dξ n X j=1 aij(s)uj(s)ds + t Zt0 e−Rt sai(ξ)dξ n X j=1 bij(s)fj(uj(s)) ds + t Zt0 e−Rt sai(ξ)dξ n X j=1 cij(s)gj(uj(s−δj(s))) ds + t Zt0 e−Rt sai(ξ)dξ n X j=1 σij(uj(s))dwj(s) = 7 X m=1 Qim (t),(16) where, 8 Qi1(t) =  ϕi(t0)−  n X j=1 qij(t0)ϕj(t0−τj(t0))  e−Rt t0ai(ξ)dξ, Qi2(t) = n X j=1 qij(t)uj(t−τj(t)), Qi3(t) = Zt t0 ai(s)e−Rt sai(ξ)dξ   n X j=1 qij(s)uj(s−τj(s)) ds, Qi4(t) = Zt t0 e−Rt sai(ξ)dξ n X j=1 aij(s)uj(s)ds, Qi5(t) = Zt t0 e−Rt sai(ξ)dξ n X j=1 bij(s)fj(uj(s)) ds, Qi6(t) = Zt t0 e−Rt sai(ξ)dξ n X j=1 cij(s)gj(uj(s−δj(s))) ds, Qi7(t) = Zt t0 e−Rt sai(ξ)dξ n X j=1 σij(uj(s))dwj(s). Now we split the rest of our proof into three steps. First step: We prove that PXl ϕ⊂Xl ϕ. First we show the mean square continuity of Pon [t0,∞).For ui∈Xli ϕi, it is necessary to show that Pi(ui)∈ Xli ϕi. It is clear that Piis continuous on [m(t0), t0]. For fixed time t≥t0,each i∈ {1,2,3, ..., n}, ui∈Xli ϕi,and |ε|be sufficiently small, we then have E|(Pi(ui)) (t+ε)−(Pi(ui)) (t)|2≤7 7 X m=1 E|Qim (t+ε)−Qim (t)|2.(17) We must prove the mean square continuity of Pion [t0,∞[. It is easy to obtain that E|Qim (t+r)−Qim (t)|2→0,as r→0, i = 1,2, ..., 6. As for the last term, E|Qi7(t+ε)−Qi7(t)|2 =EZt t0 e−Rt sai(ξ)dξ e−Rt+ε tai(ξ)dξ −1n X j=1 σij(uj(s))dwj(s) +Zt+ε t e−Rt+ε sai(ξ)dξ n X j=1 σij(uj(s))dwj(s) 2 ≤2E n X j=1 Zt t0 e−Rt sai(ξ)dξ e−Rt+ε tai(ξ)dξ −1σij(uj(s))dwj(s) 2 9 (Guo et al., 2017, p. 1557), but this statement is not correct. However, in our investigation we use a different space which is indeed a complete metric space. Remark 3.2: It follows from the first part of the proof of Theorem 3.1 that the zero solution of (1) is stable under (11). Moreover, Theorem 3.1 still holds if (11) is satisfied for t≥tρfor some tρ∈R+. 4 Example In this section, we analyze an example to illustrate two facts. On the one hand, we will show how to apply our main result in this paper, Theorem 3.1. On the other hand and most importantly, we will highlight the real interest and importance of our result because the previous theory developed by Guo et al. (2017) cannot be applied to this example. Exemple: 4.1 Consider the following two-dimensional stochastic delay differential equation d[x(t)−Q(t)x(t−τ(t))] = [A(t)x(t) + B(t)x(t−τ(t)] dt +G(t)x(t−τ(t))dw (t), t ≥0,(27) Where Q(t) =    −sin t 80 0.025 3 sin t 20   , A (t) =    −0.112 t+ 1 0 0−0.125 t+ 1    B(t) =     −0.112 20 (t+ 1)20 −5×10−3 (t+ 1)2−3×10−2 4 (t+ 1)2     , G (t) = r0.01 2 (t+ 1)I. By straightforward computations, we can check that condition (11) in Theorem 3.1 holds true, where τ∈C(R+,R+) is an arbitrary continuous functions which satisfies t−τ(t)→ ∞ and t→ ∞ and choosing a1(t) = 0.112 t+1 , a2(t) = 0.125 t+1 , we obtain that 2 X i=1     2 X j=1  |qij(t)|+ t Z0 e−Rs tai(ξ)dξ |aij(s)|ds + t Z0 e−Rs tai(u)du |ai(s)||qij(s)|ds + t Z0 e−Rs tai(u)du |bij(s)|ds   2 + 4 2 X j=1 t Z0g2 ij (s)e−2Rs tai(ξ)dξds   <0.2925 + 0.17 <1 2,(28) 16 and since Zt 0 a1(s)ds =Zt 0 0.112 s+1 ds = 0.112 ln (t+ 1) → ∞ and Zt 0 a2(s)ds =Zt 0 0.125 s+1 ds = 0.125 ln (t+ 1) → ∞ as t→ ∞.It is easy to see that all the conditions of Theorem 3.1 hold for γ≃0.4625 <0.5.Thus, Theorem 3.1 implies that the zero solution of (27) is asymptotically stable. Remark 4.1: Observe that Example 4.1 cannot be analyzed by applying Theorem A (see also Theorem 3.1 in Guo et al., 2017). Indeed, in order to apply Theorem A, we need to check that there exist positive constants a1, a2 such that (9) holds. However, notice that, for any (fixed) a1>0, if we set a11(t) = a1−0.112 t+1 , we have that there exists T0>0 such that a11(t)>3a1 4for all t≥T0. Consequently for one of the integrals appearing in (9) we deduce, for t>T0, Zt 0 e−a1(t−s) a1−0.112 s+ 1  ds ≥Zt T0 e−a1(t−s) a1−0.112 s+ 1  ds >3 4a1Zt T0 e−a1(t−s)ds =3 4h1−e−a1(t−T0)i. Then, it is clear that there exists T1≥T0such that for t≥T1, 3 4h1−e−a1(t−T0)i>1 2, which implies that (9) cannot hold true. Acknowledgements. The authors would like to thank the referee for the helpful and interesting suggestions and comments which allowed to improve the presentation of our paper. The research of the second author has been partially supported by grant MTM2015-63723-P (MINECO/FEDER, EU), and Junta de Andaluc´ıa (Spain) under the Proyecto de Excelencia P12-FQM-1492. 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