Stability problems for non-autonomous systems are very popular in theoretical researches and applications and are difficult enough even in the deterministic case (see, for instance, [1–3, 6–13, 17]). In this paper via the general method of the Lyapunov functionals construction [14–16] some new multi-condition of stability in probability is obtained for the zero solution of a nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients. It is shown that the obtained multi-condition of stability gives for one considered equation at once a set of different complementary of each other sufficient stability conditions. Note that other approaches to analyzing stability in random systems are presented for example in [4, 18].

Consider the scalar nonlinear stochastic differential equation with discrete and distributed delays and time varying coefficients (1.1) dx(t)+( ∑k=0nak(t)x(t−hk)+ ∑k=1n∫t−hktbk(s)x(s)ds+g(t,xt))dt+σ(t)x(t−τ)dw(t)=0,x(s)=ϕ(s)∈H2,s∈[−h,0],h=max[h1,…,hn,τ],|g(t,φ)|≤∫−h0|φ(s)|αdG(s),α>1,G=∫−h0dG(s)<∞. 1,\hspace{1em}G={\int _{-h}^{0}}dG(s)<\infty .\end{aligned}\]]]>

Here ak(t) , bk(t) , σ(t) are bounded functions, w(t) is the standard Wiener process on a probability space {Ω,F,P} [5, 16], H2 is a space of F0 -adapted stochastic processes ϕ(s) , s∈[−h,0] , ‖ϕ‖0=sups∈[−h,0]|ϕ(s)|,‖ϕ‖2=sups∈[−h,0]E|ϕ(s)|2, E is the mathematical expectation, h0=0 , hk>0 0$]]>, k=1,…,n , τ≥0 , G(t) is a nondecreasing function of bounded variation, the integral with respect to dG(s) is understood in the Stiltjes sense.

Consider the stochastic differential equation [5] (1.2) dx(t)=a1(t,xt)dt+a2(t,xt)dw(t), where x(t)∈Rn , xt=x(t+s) , s≤0 , a1(t,φ)∈Rn , a2(t,φ)∈Rn×m , w(t)∈Rm , along with some functional V(t,φ):[0,∞)×H2→R+ that can be presented in the form V(t,φ)=V(t,φ(0),φ(s)) , s<0 , and for φ=xt put (1.3) Vφ(t,x)=V(t,φ)=V(t,xt)=V(t,x,x(t+s)),x=φ(0)=x(t),s<0.

Denote by D the set of functionals, for which the function Vφ(t,x) defined in (1.3) has a continuous derivative with respect to t and second continuous derivative with respect to x. For functionals from D the generator L of Equation (1.2) has the form [5, 16] (1.4) LV(t,xt)=∂Vφ(t,x(t))∂t+∇Vφ′(t,x(t))a1(t,xt)+12Tr[a2′(t,xt)∇2Vφ(t,x(t))a2(t,xt)].

If in Equation (1.2) a1(t,0)≡0 , a2(t,0)≡0 then Equation (1.2) has the zero solution and the following theorems hold.

Via Theorems 1.1, 1.2 a construction of stability conditions for a given stochastic differential equation is reduced to construction of appropriate Lyapunov functionals. Via the general method of the Lyapunov functionals construction [14–16], below some multi-condition of stability in probability for the zero solution of Equation (1.1) is obtained.

In this section sufficient conditions of exponential mean square stability are obtained for the linear part of Equation (1.1), i.e., for Equation (1.1) with g(t,xt)≡0 .

Let ni , i=1,2 , be integers such that 0≤ni≤n . Put (2.1) S(t)= ∑k=0n1ak(t+hk)+ ∑k=1n2bk(t)hk,m1=min{n1,n2},m2=max{n1,n2},Rk(t,s)=ak(s+hk)+(s−t+hk)bk(s),k=1,…,m1ak(s+hk)ifn1>n2,k=m1+1,…,m2,(s−t+hk)bk(s)ifn1<n2,k=m1+1,…,m2,R(t)= ∑k=1m2∫t−hkt|Rk(t,s)|ds,Ik(i,j)=1ifk∈[i,j],0ifk∉[i,j],Rkλ(t,s)=(S(t)−λ)Rk(t,s)Ik(1,m2)−bk(s)Ik(n2+1,n),Pλ(t)=λR(t)+ ∑i=n1+1n|ai(t)|+ ∑i=n2+1n∫t−hit|bi(θ)|dθ,Qkλ(t,s)=|Rkλ(t,s)|+Pλ(t)|Rk(t,s)|Ik(1,m2)+R(t)|bk(s)|Ik(n2+1,n),F(t,λ)=λ−2S(t)+ ∑k=1n(∫t−hkt|Rkλ(t,s)|ds+∫tt+hkeλhkQkλ(θ,t)dθ)+ ∑k=n1+1n(|ak(t)|+eλhk(1+R(t+hk))|ak(t+hk)|)+eλτσ2(t+τ). {n_{2}},& k={m_{1}}+1,\dots ,{m_{2}},\\{} (s-t+{h_{k}}){b_{k}}(s)\hspace{1em}\text{if}\hspace{1em}{n_{1}}<{n_{2}},& k={m_{1}}+1,\dots ,{m_{2}},\end{array}\right.\\{} R(t)& ={\sum \limits_{k=1}^{{m_{2}}}}{\int _{t-{h_{k}}}^{t}}|{R_{k}}(t,s)|ds,\hspace{2em}{I_{k}}(i,j)=\left\{\begin{array}{l@{\hskip10.0pt}l@{\hskip10.0pt}l}1& \text{if}& k\in [i,j],\\{} 0& \text{if}& k\notin [i,j],\end{array}\right.\\{} {R_{k}^{\lambda }}(t,s)& =\big(S(t)-\lambda \big){R_{k}}(t,s){I_{k}}(1,{m_{2}})-{b_{k}}(s){I_{k}}({n_{2}}+1,n),\\{} {P_{\lambda }}(t)& =\lambda R(t)+{\sum \limits_{i={n_{1}}+1}^{n}}|{a_{i}}(t)|+{\sum \limits_{i={n_{2}}+1}^{n}}{\int _{t-{h_{i}}}^{t}}|{b_{i}}(\theta )|d\theta ,\\{} {Q_{k}^{\lambda }}(t,s)& =|{R_{k}^{\lambda }}(t,s)|+{P_{\lambda }}(t)|{R_{k}}(t,s)|{I_{k}}(1,{m_{2}})+R(t)|{b_{k}}(s)|{I_{k}}({n_{2}}+1,n),\\{} F(t,\lambda )& =\lambda -2S(t)+{\sum \limits_{k=1}^{n}}\Bigg({\int _{t-{h_{k}}}^{t}}|{R_{k}^{\lambda }}(t,s)|ds+{\int _{t}^{t+{h_{k}}}}{e}^{\lambda {h_{k}}}{Q_{k}^{\lambda }}(\theta ,t)d\theta \Bigg)\\{} & \hspace{1em}+{\sum \limits_{k={n_{1}}+1}^{n}}\big(|{a_{k}}(t)|+{e}^{\lambda {h_{k}}}\big(1+R(t+{h_{k}})\big)|{a_{k}}(t+{h_{k}})|\big)+{e}^{\lambda \tau }{\sigma }^{2}(t+\tau ).\end{aligned}\]]]>

By virtue of S(t) and Rk(t,s) defined in (2.1) Equation (1.1) can be presented in the form of a neutral type stochastic differential equation [16] (2.2) dz(t,xt)=(−S(t)x(t)− ∑k=n1+1nak(t)x(t−hk)− ∑k=n2+1n ∫t−hktbk(s)x(s)ds−g(t,xt))dt−σ(t)x(t−τ)dw(t), where (2.3) z(t,xt)=x(t)− ∑k=1m2∫t−hktRk(t,s)x(s)ds.

For the proof it is enough to note that (2.5) is equivalent to the condition supt≥0F(t,0)<0 from which it follows that there exists small enough λ>0 0$]]> such that the condition F(t,λ)≤0 holds too.

In this section it is shown that the sufficient conditions for exponential mean square stability of the linear part of Equation (1.1) also are sufficient conditions for stability in probability of the initial nonlinear equation.

For the proof it is enough to note that (2.5) is equivalent to the condition supt≥0F(t,0)<0 from which it follows that there exist small enough λ>0 0$]]> and ε>0 0$]]> such that Condition (3.1) holds. Remark 3.1.

From 0≤ni≤n , i=1,2 , it follows that the couple ( n1,n2 ) in Equation (2.2) has (n+1)2 different values. Thus, Theorem 3.1 generally speaking gives (n+1)2 different stability conditions at once. Some of these conditions can be infeasible, from some of these conditions can follow some other conditions, the remaining conditions will complement each other.

Following Remark 3.1 let us consider some of possible values of the couple (n1,n2) and obtain appropriate different stability conditions.

If n1=n2=0 then via (2.1) m1=m2=0 , S(t)=a0(t) , Rk(t,s)=0 , R(t)=0 , Qk0(t,s)=|Rk0(t,s)|=|bk(s)| , and Condition (2.5) takes the form (4.1) supt≥01a0(t)[ ∑k=1n(|ak(t)|+|ak(t+hk)|+|bk(t)|hk+∫t−hkt|bk(s)|ds)+σ2(t+τ)]<2.

If n1=n , n2=0 then via (2.1) m1=0 , m2=n , and Condition (2.5) gives (4.2) supt≥01S0(t)[ ∑k=1n(∫t−hkt|S0(t)ak(s+hk)−bk(s)|ds+∫tt+hk|S0(θ)ak(t+hk)−bk(t)|dθ+|bk(t)|∫tt+hkA0(θ)dθ+|ak(t+hk)|∫tt+hkB0(θ)dθ)+σ2(t+τ)]<2, where S0(t)= ∑k=0nak(t+hk),A0(t)= ∑k=1n∫tt+hk|ak(s)|ds,B0(t)= ∑k=1n∫t−hkt|bk(s)|ds.

If n1=0 , n2=n then m1=0 , m2=n , and from Condition (2.5) we obtain (4.3) supt≥0[1S1(t)( ∑k=1n|bk(t)|∫tt+hk(S1(θ)+ ∑i=1n|ai(θ)|)(t−θ+hk)dθ+ ∑k=1n(|ak(t)|+(1+B1(t+hk))|ak(t+hk)|)+σ2(t+τ))+B1(t)]<2, where S1(t)=a0(t)+ ∑k=1nbk(t)hk,B1(t)= ∑k=1n∫t−hkt(s−t+hk)|bk(s)|ds. If at last n1=n2=n then m1=m2=n , and Condition (2.5) takes the form (4.4) supt≥0[ ∑k=1n∫t−hkt|ak(s+hk)+(s−t+hk)bk(s)|ds+1S2(t)( ∑k=1n∫tt+hkS2(θ)|ak(t+hk)+(t−θ+hk)bk(t)|dθ+σ2(t+τ))]<2, where S2(t)=a0(t)+∑k=1n(ak(t+hk)+bk(t)hk) .

Using different other combinations of n1 and n2 , one can get different other stability conditions. Example 4.1.

To demonstrate a possible connection between the obtained different stability conditions consider, for the sake of simplicity, the equation with constant coefficients and without a non-delay term (4.5) dx(t)+(ax(t−h)+b∫t−htx(s)ds+cx2(t−h))dt+σx(t−τ)dw(t)=0. For Equation (4.5) n=1 , so, via Remark 3.1 there are 4 possible stability conditions. Since in Equation (4.5) a0=0 Condition (4.1) does not hold.

Put p=12σ2 . Condition (4.2) gives p+|a2−b|h+a|b|h2<a and can be presented in the form (4.6) b>p−a(1−ah)h(1+ah)ifb≤0,b>p−a(1−ah)h(1−ah)ifb∈(0,a2),0<ah<1,b<a(1+ah)−ph(1+ah)ifb≥a2. \frac{p-a(1-ah)}{h(1+ah)}\hspace{1em}\text{if}\hspace{1em}b\le 0,\\{} b& >\frac{p-a(1-ah)}{h(1-ah)}\hspace{1em}\text{if}\hspace{1em}b\in \big(0,{a}^{2}\big),\hspace{1em}0 Conditions (4.3) take the form (4.7) |a|<bh(1−12bh2)−p1+12bh2,0<bh2<2. Calculating the integrals in (4.4) separately for a≥0 and a<0 , from Condition (4.4) we obtain (4.8) p<(a+bh)(1−ah−12bh2)ifa≥0,(a+bh)(1−ah−12bh2−a2b)ifa<0,a+bh>0. 0.\]]]> So, if at least one of Conditions (4.6)–(4.8) holds then the zero solution of Equation (4.5) is stable in probability and the zero solution of the linear part ( g(t,xt)≡0 ) of this equation is exponentially mean square stable.

Fig. 1.

Stability regions (1), (2), (3) for Equation (4.5), defined by Conditions (4.6), (4.7), (4.8) respectively, for the values of the parameters h=0.5 , p=0.2