η-stability of hybrid neutral stochastic differential equations with infinite delay

η-stability of hybrid neutral stochastic differential equations with infinite delay Tom´as Caraballo Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, c/ Tarfia s/n, 41012-Sevilla (Spain). [email protected] Lassaad Mchiri Department of Mathematics, Faculty of sciences of Sfax, Tunisia. lmc[email protected] BELFEKI Mohsen Department of Mathematics, Faculty of sciences of Sfax, Tunisia. [email protected] Abstract In this paper, we study the η-stability in q-th moment (η.s.q.m) of hybrid neutral stochastic differential equations with infinite delay (HNSDEID) using the Lyapunov techniques and the method of M-matrix. Finally, we apply the main result to some examples. 1 Introduction A stochastic delay differential equation (SDDE) is a specific model of stochastic functional differential equations (SFDE). SDDE and SFDE have attracted a great interest in the recent decades. Many researchers from all disciplines have been investigating SDDE and SFDE arising in problems from physics, biology, economics, logistics, computer science and others (see [3]-[16]). The stability theory was initiated at the end of the 19-th century by Lyapunov. This theory has found wide application in various fields of physics and mathematical sciences (see [8]-[9]). 1 From a mathematical point of view, the theory of stability presents a particular case of the qualitative theory of stochastic differential equations (SDE). Liapunov’s method was the ultimate object to study the stability for SDE and partial SDE. As a specific case of SDE is the neutral SFDE (NSFDE) which has been applied to model an equation whose future depends on the present state, the past state and the derivatives in the terms containing variables delays (see [8], [9] and [13]). For hybrid neutral SDDE with different structures, [17] studied p-th moment asymptotic boundedness, p-th moment and almost sure exponential stability of solutions. To the best of our knowledge, there is no existing result about the η–stability of HNSDEID. We point out that in the works [1], [2], [12], [13] and [16] the authors have investigated the stability with general decay rate of nonlinear stochastic evolution equations, SFDE with finite and infinite delay and also of NSFDE using a Razumikhin approach. Based on the work of [11], we will extend these results by studying the η–stability of HNSDEID using the method of M-matrix and the Lyapunov function. In this sense, our results generalize those in the paper [11] in the η–stability concept and by using some appropriate conditions on the neutral term of the system. Comparing with the existing articles on the literature, it is worth emphasizing that the ηstability is closely related to the analysis of stability with general decay rate (see e.g. [6],[1] and the references therein). However, the conditions impose on the general decay rate functions are usually stronger than the ones imposed on the η–function, this is why the η–stability can be more significant and more general. Now, we will describe the main contributions of our paper: (1) Although there is an equivalence between η-stability and stability with general decay rate (see [1], [2], [12], [13] and [16]) in the sense of Definition 2.2, the main differences can be seen in Theorem 3.2, where it is only necessary to impose conditions (i) and (ii) on the function η. (2) Namely, under assumptions Theorem 3.2, the theory of stability with general decay rate cannot be applied as done in other papers in the literature due to conditions (i) and (ii). In other words, in [1] we imposed more restrictive conditions on the general decay rate function λ(t), for example, see conditions (d) and (e) in theorems 3.3 and 3.5, but in this paper we will just impose conditions (i) and (ii) in theorem 3.2 which are much weaker than the ones imposed in [1], [2], [12], [13] and [16]. (3) In [6], the class of ψ-function is more restrictive than the ones in this paper due to the condition ψ(0) = 1, condition (iii) in Definition 3.1 and condition (3.1) in assumption 3.3 (in [6]). (4) Different from the previous results in [17], taking into account the influence of the neutral term, the infinite delay, the new class of η-function and the M-matrix technique in the stability theory make our results more general. The content of our paper is as follows. After introducing some necessary preliminaries and basic notions in Section 2, we establish in Section 3 the η.s.q.m for HNSDEID. Finally, in Section 4, two examples are analyzed to show the interest of our results. 2 2 Preliminaries and basic notions Let C([ρ, 0],Ri) be the set of all continuous functions from [ρ, 0] into Ri. Let w(τ)=(w1(τ), . . . , wp(τ))Tbe an p-dimensional Brownian motion defined on a complete probability space (Ω,F,P) with the usual filtration {Fτ}τ≥0generated by {w(%) : 0 ≤%≤τ}. We denote by |y|the Euclidean norm of y∈Ri. For a matrix B, its trace norm is |B|= ptrace(BTB). Let BC((−∞,0],Ri) be the set of all bounded continuous functions µ: (−∞,0] →Riequipped with the norm ||µ|| = supζ≤0|µ(ζ)|. For q > 0, denote by Lq((−∞,0],Ri) the set of all functions ν: (−∞,0] →Risuch that R0 −∞ |ν(τ)|qdτ < ∞. Let L((−∞,0),R+) be the set of continuous bounded non-negative functions σ(·) with −∞ < σ ≤0 such that R0 −∞ σ(τ)dτ = 1. Let {l(τ), τ ∈[0,+∞)}be a right-continuous Markov chain on the probability space {Ω,F,(Fτ)τ≥0,P}taking values in a finite state space S={1,2, . . . , N}with a generator Λ = (λmn)N×Ngiven by P(l(τ+ ∆) = n|l(τ) = m) =    λmn∆ + o(∆),if m6=n 1 + λmm∆ + o(∆),if m=n where ∆ >0. Here λmn ≥0 is the transition rate from mto n, if m6=n, while λmm =−X n6=m λmn. For a vector z > 0 or a matrix D > 0, we mean that all elements in this vector or matrix are positive. A Z-matrix is a square matrix D= (dmn)N×Nthat has non-positive off-diagonal entries (namely dmn ≤0 for all m6=n) and all positive diagonal entries. Definition 2.1. A square matrix D= (dmn)N×Nis said to be a nonsingular M-matrix if Dcan be expressed in the form D=tI −Awith t > ρ(A) while all elements of Aare nonnegative, where Iis the identity matrix and ρ(A) the spectral radius of A. Lemma 2.1. If Dis a Z-matrix, the following statements are equivalent: (i) Dis a nonsingular M-matrix. (ii) Dis semi-positive; that is, there exists z > 0in RNsuch that Dz > 0. (iii) D−1exists and its elements are all nonnegative. We assume that the Markov chain l(·) is independent of the Brownian motion w(·). 3 Consider the following HNSDEID: d(α(τ)−G(τ, ατ)) = f1(α(τ), ατ, τ, l(τ)) dτ +f2(α(τ), ατ, τ, l(τ)) dw(τ), τ ≥0 α0=χ, (2.1) where the initial condition χ∈Lq((−∞,0],Ri), l(0) = m0∈S,α(τ) = (α1(τ), α2(τ), . . . , αi(τ))T and ατ={α(τ+θ) : θ∈(−∞,0]}is a BC((−∞,0],Ri)-valued stochastic process, with θ∈ (−∞,0]. In particular when θ=−∞, the interval [τ+θ, τ] be replaced by (−∞, τ]. We suppose that f1:Ri×BC((−∞,0],Ri)×R+×S→Ri, f2:Ri×BC((−∞,0],Ri)×R+×S→Ri×p, G :R+×BC((−∞,0],Ri)→Ri. Assume that G(τ, 0) = f1(0,0, τ, r) = f2(0,0, τ, r) = 0 for all (τ, r)∈R+×S. Then, system (2.1) possesses the trivial solution α(τ) = 0 with the initial condition χ= 0. Denote by C1,2(BC((−∞,0],Ri)×R+×S, R+) the family of all non-negative functions V(α, τ, j) on BC((−∞,0],Ri)×R+×S, which are twice continuously differentiable with respect to αand once continuously differentiable with respect to τ. Define the operator LV:Ri×BC((−∞,0],Ri)×R+×S→Rby (see [10]) LV(α, ψ, τ, j) = Vτ(α−G(τ, ψ), τ, j) + Vα(α−G(τ, ψ), τ, j)f1(ψ, τ, j) +1 2trace fT 2(ψ, τ, j)Vαα(α−G(τ, ψ), τ, j)f2(ψ, τ, j) + N X k=1 γjkV(α−G(τ, ψ), τ, k), where Vτ(α, τ, j) = ∂V (α, τ, j) ∂τ , Vα(α, τ, j) = ∂V (α, τ, j) ∂α1 ,...,∂V (α, τ, j) ∂αi, Vαα(α, τ, j) = ∂2V(α, τ, j) ∂αm∂αni×i . Definition 2.2. System (2.1) is said to be η-stable in q-th moment (η-s.q.m), if there exists an increasing differentiable function η:R→R∗+, such that η(τ)↑ ∞ as τ→ ∞, and a positive constant δsuch that, for any χ∈Lq((−∞,0],Ri), lim sup τ→∞ ln E(|α(τ, χ)|q) ln η(τ)≤ −δ, (2.2) when q= 2, it is said to be η-stable in mean square. 4 Definition 2.3. System (2.1) is said to be almost surely η-stable (a.s.η-s), if there exists an increasing differentiable function η:R→R∗ +, such that η(τ)↑ ∞ as τ→ ∞, and a positive constant δsuch that, for any χ∈Lq((−∞,0],Ri), lim sup τ→∞ ln |α(τ, χ)| ln η(τ)≤ −δ. (2.3) Lemma 2.2. Let q > 1,ε > 0and (a1, a2)∈R2. Then, |a1+a2|q≤1 + ε 1 q−1q−1|a1|q+|a2|q ε. Proof. See ([8]). Remark 2.3. Let q > 1 and (a1, a2)∈R2. By taking ε= 1, in Lemma 2.2, we obtain |a1+a2|q≤2q−1|a1|q+|a2|q. Assumption 2.4. For each a > 0, there exists Ka>0such that |f1(ψ1(0), ψ1, τ, j)−f1(ψ2(0), ψ2, τ, j)|∨|f2(ψ1(0), ψ1, τ, j)−f2(ψ2(0), ψ2, τ, j)| ≤ Ka||ψ1−ψ2|| (2.4) ∀ψ1, ψ2∈BC((−∞,0],Ri)with ||ψ1||∨||ψ2|| ≤ aand ∀(τ, j)∈R+×S. Assumption 2.5. Assume that there exists a constant κ∈(0,1) such that for all ψ1, ψ2∈ BC((−∞,0],Ri), we have |G(ψ1)−G(ψ2)| ≤ κ|ψ1(0) −ψ2(0)|.(2.5) Assumption 2.6. Let q≥2and σ∈ L((−∞,0),R+)be the same as in Assumption 2.5. Assume that for each m∈Sthere exist a constants ξm3∈R,ξm1,ξm2and ξm4such that ξm2≥ξm1and |ψ(0) −G(τ, ψ)|q−2(ψ(0) −G(τ, ψ))Tf1(ψ(0), ψ, τ, m) + q−1 2|f2(ψ(0), ψ, τ, m)|2 ≤ξm4|ψ(0) −G(τ, ψ)|q−2+ξm3|ψ(0) −G(τ, ψ)|q+ξm1Z0 −∞ |ψ(θ)|qσ(θ)dθ −ξm2|ψ(0)|q.(2.6) Assume that B=−diag (qξ13, qξ23, ..., qξN3)−Λ,(2.7) and define (φ1, φ2, ..., φN)T=B−1(β, β, ..., β)T,(2.8) where {φj}1≤j≤Nare positive constants and βis a given positive number. 5 3 Main results Under the assumption that Bis a non singular M-matrix, for all φj,j∈S, which are positive, we can establish now our first main result about the existence and uniqueness of solution. Theorem 3.1. Assume that Assumptions 2.4-2.6 hold. Assume also that κ∈(0,21 q−1). Then, for any initial condition χ∈BC((−∞,0],Ri)∩Lq((−∞,0],Ri)and l(0) = m0∈S, there is a unique global solution α(τ)to the HNSDEID (2.1). Proof. We will split the proof into two steps. Step 1: Let V:BC((−∞,0],Ri)×S→R+be the Lyapunov function: V(α, j) = φj|α|q,(3.1) where φjis defined in (2.8). It is easy to see that ϕ1|α|q≤V(α, j)≤ϕ2|α|q,(3.2) where ϕ1= min j∈Sφjand ϕ2= max j∈Sφj. Let e ψ=ψ(0) −G(τ, ψ). Applying the generalized Itˆo formula to V(e ψ, l(τ)), we have V(e ψ, l(τ)) = V(eα0, m0) + Zτ 0 LV (ψ(0), ψ, t, l(t)) dt +M(τ),(3.3) where M(τ) is a continuous local martingale with M(0) = 0. Then, LV (ψ(0), ψ, τ, j) = qφj|ψ(0) −G(τ, ψ)|q−2(ψ(0) −G(τ, ψ))Tf1(ψ(0), ψ, τ, j) +1 2qφj|ψ(0) −G(τ, ψ)|q−2|f2(ψ(0), ψ, τ, j)|2 +1 2q(q−2)φj|ψ(0) −G(τ, ψ)|q−4|(ψ(0) −G(τ, ψ))Tf2(ψ(0), ψ, τ, j)|2 + N X n=1 λjnφn|ψ(0) −G(τ, ψ)|q.(3.4) Using the inequality |(ψ(0) −G(τ, ψ))Tf2(ψ(0), ψ, τ, j)|2≤ |ψ(0) −G(τ, ψ)|2|f2(ψ(0), ψ, τ, j)|2,(3.5) we deduce LV (ψ(0), ψ, τ, j) 6 =qφj|ψ(0) −G(τ, ψ)|q−2(ψ(0) −G(τ, ψ))Tf1(ψ(0), ψ, τ, j) + q−1 2|f2(ψ(0), ψ, τ, j)|2 + N X n=1 λjnφn|ψ(0) −G(τ, ψ)|q ≤qφjξj4|ψ(0) −G(τ, ψ)|q−2+qφjξj3|ψ(0) −G(τ, ψ)|q+ N X n=1 λjnφn|ψ(0) −G(τ, ψ)|q +qφjξj1Z0 −∞ |ψ(θ)|qσ(θ)dθ −qφjξj2|ψ(0)|q.(3.6) By the definition of φj, we can derive that qφjξj3+ N X n=1 λjnφn=−β, j ∈S. Hence, LV (ψ(0), ψ, τ, j)≤ −β|ψ(0)−G(τ, ψ)|q+ϕ3|ψ(0)−G(τ, ψ)|q−2+qφjξj1Z0 −∞ |ψ(θ)|qσ(θ)dθ −|ψ(0)|q, (3.7) where ϕ3= max j∈Sqφjξj4. Note that ϑ(t) = ϕ3tq−2−βtqwill have a finite supremum value over R+denoted by ϕ4= sup t∈[0,∞) ϑ(t). Therefore, LV (ψ(0), ψ, τ, j)≤ϕ4+qφjξj1Z0 −∞ |ψ(θ)|qσ(θ)dθ −|ψ(0)|q.(3.8) Step 2: Since coefficients of HNSDEID (2.1) are locally Lipschitz continuous, for any given initial condition χ∈BC((−∞,0],Ri)∩Lq((−∞,0],Ri), by the standard truncation method, there is a unique maximal local strong solution of equation (2.1) on τ∈(−∞, σe), where σeis the explosion time (see, e.g., [9, Theorem 3.2.2, p.95] and [15, Theorem 3.3]). Let i0>0 be sufficiently large such that ||χ|| < i0. For each i∈N∗with i≥i0, we define the stopping time: σi= inf{τ∈(−∞, σe); |α(τ)| ≥ i}. It is obvious that σiis increasing. Define σ∞= lim i→∞ σi. It is clear that σ∞≤σea.s. We will show that σ∞=∞a.s, which implies that σe=∞a.s. 7 Let eατ=α(τ)−G(τ, ατ). By the Itˆo formula and (3.8), EV(eατ∧σi, l(τ∧σi)) = EV(eα0, m0) + EZτ∧σi 0 LV (α(t), αt, t, l(t)) dt, ≤EV(eα0, m0) + ϕ4τ, +qφjξj1EZτ∧σi 0Z0 −∞ |α(t+θ)|qσ(θ)dθ −|α(t)|qdt.(3.9) Moreover, by the Fubini theorem and the fact that θ≤0, we have EZτ∧σi 0Z0 −∞ |α(t+θ)|qσ(θ)dθ −|α(t)|qdt =EZ0 −∞ Zτ∧σi 0|α(t+θ)|qdtσ(θ)dθ−EZτ∧σi 0|α(t)|qdt, ≤EZ0 −∞ Zτ∧σi −∞ |α(t)|qdtσ(θ)dθ−EZτ∧σi 0|α(t)|qdt, =EZ0 −∞ |χ(t)|qdt. (3.10) Therefore, by (2.5), we obtain EV(eατ∧σi, l(τ∧σi)) ≤EV(eα0, m0) + ϕ4τ+qφjξj1EZ0 −∞ |χ(t)|qdt, ≤ϕ2E|eα0|q+ϕ4τ+qφjξj1EZ0 −∞ |χ(t)|qdt, ≤2q−1ϕ2E|α0|q+ 2q−1ϕ2E|G(0, α0)|q+ϕ4τ+qφjξj1EZ0 −∞ |χ(t)|qdt, ≤2q−1ϕ2E||χ||q(1 + κq) + ϕ4τ+qφjξj1EZ0 −∞ |χ(t)|qdt. (3.11) Thus, E|eατ∧σi|q≤L+ϕ4 ϕ1 τ, (3.12) where L= 2q−1ϕ2 ϕ1 E||χ||q(1 + κq) + 1 ϕ1 qφjξj1EZ0 −∞ |χ(t)|qdt. On the other hand, for −∞ < θ ≤0, we derive E|α(τ∧σi)|q≤2q−1E|eατ∧σi|q+ 2q−1E|G(τ∧σi, ατ∧σi)|q, ≤2q−1E|eατ∧σi|q+ 2q−1κqE|α(τ∧σi)|q.(3.13) 8 Then, we have E|α(τ∧σi)|q≤2q−1L+ϕ4 ϕ1 τ+ 2q−1κqE|α(τ∧σi)|q.(3.14) Hence, sup 0≤t≤τ E|α(t∧σi)|q≤1 1−2q−1κq2q−1L+ϕ4 ϕ1 τ.(3.15) Therefore, E|α(τ∧σi)|q≤1 1−2q−1κq2q−1L+ϕ4 ϕ1 τ.(3.16) Since, E|α(τ∧σi)|q≥iqP(σi≤τ), then, P(σi≤τ)≤Q(τ) iq, where Q(τ) = 1 1−2q−1κq2q−1L+ϕ4 ϕ1 τ. Letting i→ ∞, we get P(σ∞≤τ) = 0. Thus, P(σ∞> τ) = 1, ∀τ≥0, which implies that P(σ∞=∞) = 1, as desired. Now, we will discuss the η.s.q.m and the a.s.η-s of solutions for system (2.1). Theorem 3.2. Let Assumption (2.5) holds. Assume that there exists an increasing and twice differentiable continuous function η:R+→R∗ +satisfying η(τ)↑ ∞ as τ→ ∞ and a constant cη>0such that (i) 0<supτ>0 η0(τ) η(τ)≤cη. (ii) η00(τ)η(τ)< η0(τ)2, for all τ > 0. Assume that there exists σ∈ L((−∞,0),R+)and a constant 0≤p2cη≤p1such that ∀τ≥0 and ∀ψ∈BC((−∞,0],Ri) |ψ(0) −G(τ, ψ)|q−2(ψ(0) −G(τ, ψ))Tf1(ψ(0), ψ, τ, m) + q−1 2|f2(ψ(0), ψ, τ, m)|2 ≤ −p1|ψ(0)|q+ξm3|ψ(0) −G(τ, ψ)|q+p2 η0(τ) η(τ)Z0 −∞ |ψ(θ)|qσ(θ)dθ. (3.17) Then, system (2.1) is η-s.q.m and a.s.η-s. Proof. Consider χ∈BC((−∞,0],Ri)∩Lq((−∞,0],Ri) and l(0) = m0∈S. Set ν(τ) = ln(η(τ)), then ν0(τ)≤cη. Let δ∈(0,1) be arbitrary such that δcηmaxj∈Sφj≤βand η00(τ)η(τ)<(1 −δ)η0(τ)2. Applying the Itˆo formula, we have Eφjeδν(τ)|α(τ)−G(τ, ατ)|q 9 Example 4.2. We consider the following HNSDEID: dα(τ)−1 2 + τ|sin α(τ)|=f1(α(τ), ατ, τ, l(τ)) dτ +f2(α(τ), ατ, τ, l(τ)) dw(τ),(4.8) where the initial condition is χ=3e0.03τ,if τ∈(−120,0] 0,if τ∈(−∞,−120] and ω(t)is a one dimensional Brownian motions. Let f1(α(τ), ατ, τ, 1) = −32 + τ 3 + τα(τ), f2(α(τ), ατ, τ, 1) = r2 + τ 1 + τZ0 −∞ |α(τ+θ)|eθdθ, f1(α(τ), ατ, τ, 2) = −42 + τ 3 + τα(τ), f2(α(τ), ατ, τ, 2) = s2 + τ 3(1 + τ)Z0 −∞ |α(τ+θ)|eθdθ. Let S={1,2}and the matrix Λ=(λmn)1≤m,n≤2define by −1 1 1−1 Then, for m= 1 (ψ(0) −G(τ, ψ))Tf1(ψ(0), ψ, τ, 1) + 1 2|f2(ψ(0), ψ, τ, 1) |2 =−32 + τ 3 + τ|ψ(0)|2+3 3 + τψ(0)|sin ψ(0)|+1 2 2 + τ 1 + τZ0 −∞ |ψ(θ)|eθdθ2 ≤ −32 + τ 3 + τ|ψ(0)|2+3 3 + τ|ψ(0)|2+1 2 2 + τ 1 + τZ0 −∞ |ψ(θ)|eθdθ2 ≤ −|ψ(0)|2+Z0 −∞ |ψ(θ)|eθdθ2 .(4.9) For m= 2 (ψ(0) −G(τ, ψ))Tf1(ψ(0), ψ, τ, 2) + q−1 2|f2(ψ(0), ψ, τ, 2) |2 16 =−42 + τ 3 + τ|ψ(0)|2+4 3 + τψ(0)|sin ψ(0)|+1 2 2 + τ 3(1 + τ)Z0 −∞ |ψ(θ)|eθdθ2 ≤ −42 + τ 3 + τ|ψ(0)|2+4 3 + τ|ψ(0)|2+1 2 2 + τ 3(1 + τ)Z0 −∞ |ψ(θ)|eθdθ2 ≤ −|ψ(0)|2+Z0 −∞ |ψ(θ)|eθdθ2 .(4.10) Substituting (4.4) into (4.9) and (4.10), we can derive that for all m∈ {1,2} (ψ(0) −G(τ, ψ))Tf1(ψ(0), ψ, τ, m) + 1 2|f2(ψ(0), ψ, τ, m)|2 ≤ −|ψ(0)|2+|ψ(0) −G(τ, ψ)|2+1 2 η0(τ) η(τ)Z0 −∞ |ψ(θ)|2eθdθ. (4.11) Thus assumption is satisfied with p1= 1,p2=1 2,ξm3= 1 (∀m∈ {1,2}),η(τ) = (1 + τ)eτ,q= 2 and cη= 2. Using the fact that |ψ(0) −G(τ, ψ)|2−2= 1, it is easy to check that ∀m∈ {1,2} (ψ(0) −G(τ, ψ))Tf1(ψ(0), ψ, τ, m) + 1 2|f2(ψ(0), ψ, τ, m)|2 ≤1 + |ψ(0) −G(τ, ψ)|2+Z0 −∞ |ψ(θ)|2eθdθ −|ψ(0)|2.(4.12) Which proves that assumptions 2.6 is satisfied with ξm1=ξm2=ξm4= 1. Moreover, the parameter κin theorem 3.1 equal 1 2. We have B=−diag(2ξ13,2ξ23)−Λ = 2 0 0 2 −−1 1 1−1=3−1 −1 3 . Then, B−1=  3 8 1 8 1 8 3 8  . Then, Bis a nonsingular M-matrix. It is easily to check that all conditions of theorems 3.1 and 3.2 are satisfied. Therefore system (4.8) is η-stable in mean square and a.s.η-stable. References [1] T. Caraballo, M. A. Hammami, L. Mchiri, Practical asymptotic stability of nonlinear stochastic evolution equations, Stochastic Analysis and Applications, 32 (2014), 77-87. 17 [2] T. Caraballo, Mar´ıa J. Garrido-Atienza, Jos´e Real, Asymptotic Stability of Nonlinear Stochastic Evolution Equations. Stochastic Analysis and Applications, Vol .21, No. 2, 301- 327, 2003. [3] L. Hu, X. 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