Dynamic Systems and Applications 15 (2006) 193-204

ON STABILITY OF STOCHASTIC DELAY CUBIC EQUATIONS

ALEXANDRA RODKINA AND VALERY NOSOV

Department of Maths/CSci, University of the West Indies, Mona, Kingston, Jamaica SEPI-ESIME, Instituto Politecnico Nacional, Mexico-City, Mexico

ABSTRACT. We prove almost sure asymptotic stability of a trivial solution to timedelay stochastic differential equation with cubic main part and nonlinear stochastic disturbances. Disturbances contain continuous as well as jump components. All stability conditions are expressed directly in terms of the coefficients. AMS (MOS) Subject Classification. 34K20, 93E15, 60H10.

1. INTRODUCTION

The stability problems for time-delay systems have been extensively studied in recent years due to direct applicability of developed theory to various technical problems. Initiated in the background works [9, 7, 4], the stability theory for time-delay systems with the linear main part or with the main part which has sub-linear growth is now actively developed. Such modern technique as the Linear Matrix Inequalities (LMI) method seems to be especially useful in view of designed efficient computational algorithms (see [8, 3] and others). The Lyapunov-Krasovskii or Lyapunov-Razumikhin functionals have also been applied in the framework of the Lyapunov direct method. Special Liapunov-Krasovskii functionals technique has been designed for obtaining stability conditions for the stochastic difference equations of Volterra type (see for example [5, 6, 10, 18]).

In present paper we prove almost sure asymptotic stability of a trivial solution to the time-delay stochastic equation with cubic main part and nonlinear stochastic disturbances. This paper enforces the result of [13] (and also [16, 18]), generalizing it to stochastic equations with nonlinear continuous and jump type disturbances. Lyapunov-Krasovskii functional technique for continuous and discrete systems together with the convergence theorem for semimartingale inequalities ([1, 11] and also [12]) serve as the key tools for obtaining stability conditions in terms of stochastic system coefficients, without any transformation of the original system itself.

Received April 30, 2004           1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

2. SOME BASIC DEFINITIONS AND FACTS

Below we present some necessary definitions and facts from the Theory of Random Processes (for details see [1, 2, 11, 14]).

Let (w_{t})_{t≥0} be an one-dimensional Wiener process on a complete filtered probability space (Ω, F, P) with natural filtration (F_{t})_{t≥0}, where F_{t} = σ(w_{s} : 0 ≤ s ≤ t). We use the standard abbreviation ”a.s.” for the wording ”almost sure” or ”almost surely” with respect to the fixed probability measure P throughout the text.

A stochastic process (X_{t})_{t≥0} is the family of random variables X(t, ω) defined on (Ω, F). We restrict our consideration to the processes (X_{t})_{t≥0} such that, for each t ≥ 0, the random variable X_{t} is F_{t}-measurable. We also suppose that, for all ω ∈ Ω, the trajectories of X(t, ω) are right-continuous and admit left-hands limits X(t−, ω) = lim_{s→t,s<t} X(s, ω). Together with a process (X_{t})_{t≥0} we define two other processes (X_{t−})_{t≥0} and (ΔX_{t})_{t≥0} by

X_{0−} = X_{0}, X_{t−} = lim_{s→t,s<t} X_{s}, t > 0, ΔX_{t} = X_{t} − X_{t−}.

Let ℘ be the σ-algebra which is generated by all F_{t}-measurable processes that have continuous trajectories. A process that is ℘-measurable is called predictable.

Stochastic process (M_{t})_{t≥0} is said to be an F_{t}-martingale, if E|M_{t}| < ∞ and E(M_{t}|F_{s}) = M_{s} for all t > s ≥ 0.

We note that a Wiener process (w_{t})_{t≥0} is a martingale as well as the Ito integral process (∫_{0}^{t} g(s)dw_{s})_{t≥0}, where the process (g(s))_{t≥0} is predictable.

The sequence of random variables (X_{n})_{n=1,2,...}, where X_{n} is F_{n}-measur-able, is said to be a stochastic sequence. A stochastic sequence (X_{n})_{n=1,2,...} is said to be an F_{n}-martingale, if E|X_{n}| < ∞ and E(X_{n}|F_{n−1}) = X_{n−1} for all n = 1, 2, . . .. A stochastic sequence (ν_{n})_{n=1,2,...} is said to be an F_{n}- martingale-difference if E|ν_{n}| < ∞ and E(ν_{n}|F_{n−1}) = 0 for any n = 1, 2, …

Let (ξ_{n})_{n=1,2,...} be a sequence of independent random variables ξ_{n}, such that ξ_{n} is an F_{n}-measurable, ξ_{n} is independent of F_{s} for all n > s, E|ξ_{n}| < ∞ and E(ξ_{n}) = 0 for any n = 1, 2, . . .. We define

(2.1) μ_{t} = 0, t ∈ [0, 1), μ_{t} = ∑_{i=1}^{n} ξ_{i} if t ∈ [n, n + 1), n = 1, 2, ….

For each t ≥ 0 the random variable μ_{t} is an F_{t}-measurable, for all ω ∈ Ω the trajectories of μ(t, ω) are right-continuous and admit left-hands limits, Δμ_{t} = ξ_{n} for t = n, n = 1, 2, …, Δμ_{t} = 0 for t ≠ n, and (μ_{t})_{t≥0} is an F_{t}-martingale.

Below we formulate the most suitable for us variant of the Ito formula (see [1, 2, 11, 14]).

Lemma 2.1. Let (X_{t})_{t≥0} be a stochastic process having the differential dX_{t} = c(t, X_{t})dt + g(t, X_{t−})dw_{t} + k(t, X_{t−})dμ_{t}. Let V (t, x) be a differentiable function with respect to the first argument and a twice differentiable function with respect to the second argument. Then

(2.2) V (t, X_{t}) = V (0, X_{0}) + ∫_{0}^{t} \frac{∂V (s, X_{s−})}{∂x}c(s, X_{s−})ds

+ ∫_{0}^{t} \frac{∂V (s, X_{s−})}{∂s}ds + \frac{1}{2} ∫_{0}^{t} \frac{∂^{2}V (s, X_{s−})}{∂x^{2}}g^{2}(s, X_{s−})ds

+ ∑_{s≤t}(V (s, X_{s}) − V (s, X_{s−}) − \frac{∂V (s, X_{s−})}{∂s}ΔX_{s})

+ ∫_{0}^{t} \frac{∂V (s, X_{s−})}{∂x}g(s, X_{s−})dw_{s}

+ ∫_{0}^{t} \frac{∂V (s, X_{s−})}{∂x}k(s, X_{s−})dμ_{s}.

The following Lemma 2.2 can be easily obtained from Doob’s decomposition of submartingales (see [1, 11, 14] and also [17] for the proof of Lemma 2.2).

Lemma 2.2. Let (ξ_{n})_{n=1,2,...} be an F_{n}-martingale-difference such that Eξ_{n}^{2} < ∞ for every n = 1, 2, . . .. Then there exists an F_{n}-martingale-difference (μ_{n})_{n=1,2,...} and a positive F_{n−1}-measurable stochastic process (η_{n})_{n=1,2,...} such that for every n = 1, 2, …,

ξ_{n}^{2} = μ_{n} + η_{n}. (2.3)

The process (η_{n})_{n=1,2,...}, as in Lemma 2.1, can be represented as η_{n} = E (ξ_{n}^{2}|F_{n−1}), and it is nonrandom for n = 1, 2, … when ξ_{n} are independent random variables. In this case η_{n} = E (ξ_{n}^{2}) , μ_{n} = ξ_{n}^{2} − E(ξ_{n}^{2}).

To establish asymptotic stability we shall make use of a certain application of well-known martingale convergence theorems (cf. [1] [11]) in terms of inequalities. Such an application is expressed by Lemma 2.3, which is originally proved in [12].

Lemma 2.3. Let (A_{t}^{1})_{t≥0}, (A_{t}^{2})_{t≥0}, (B_{t}^{1})_{t≥0}, (B_{t}^{2})_{t≥0} with A_{0}^{1} = A_{0}^{2} = B_{0}^{1} = B_{0}^{2} be a.s. non-decreasing predictable processes with B_{t}^{1} ≤ A_{t}^{1}, B_{t}^{2} ≥ A_{t}^{2} and A_{t} = B_{t}^{1} − B_{t}^{2} for all t ≥ 0. Let (M_{t})_{t≥0} be a martingale and let (Z_{t})_{t≥0}, Z_{t} = Z_{0} + A_{t} + M_{t}, be a non-negative stochastic process. Then {ω : A_{∞}^{1} < ∞} ⊆ {Z →} ∩ {ω : A_{∞}^{2} < ∞} a.s., where {Z →} denotes the set of all ω ∈ Ω for which Z_{∞} = lim_{t→∞} Z_{t} exists and is finite.

Remark 2.4. Further below in this paper, we use the notation x(t) for a value of any stochastic process (other than a Wiener process (w_{t})_{t≥0}) at a moment t, although the notation x_{t} is commonly accepted for this purpose in the literature on stochastic processes. This enables one to distinguish between the pointwise value x(t) of a stochastic process and its history up to the moment t, which is denoted by x_{t}, x_{t}(s) = x(t + s), s ∈ [−h, 0], as commonly accepted in the literature on time delay systems.

Let m be a nonnegative integer, h_{i} ≥ 0, i = 1, …, m, h = max{h_{i}, i = 1, …, m}. For all t > 0 and x : [t − h, t] → ℝ^{1} we define x_{t}(s) = x(t + s) for s ∈ [−h, 0]. We define x_{t−}(s) = x(t + s) for s ∈ [−h, 0) and x_{t−}(0) = lim_{s→0,s<0} x(t + s). We put [t] = max{i : i ≤ t, i is an integer}.

We suppose that functions a_{l,3−l}^{i} : [0, ∞) → ℝ^{1} are nonrandom and continuous, i = 1, …, m, l = 0, 1, 2, 3. For all t ≥ 0 and x : [t − h, t] → ℝ^{1} we define

(2.4) .c_{m} (t, x_{t}) = a_{3,0}(t)x^{3}(t) + x^{2}(t) ∑_{i=1}^{m} a_{2,1}^{i}(t)x(t − h_{i})

+ x(t) ∑_{i=1}^{m} a_{1,2}^{i}(t)x^{2}(t − h_{i}) + ∑_{i=1}^{m} a_{0,3}^{i}(t)x^{3}(t − h_{i})

We consider the scalar stochastic delay cubic equation

(2.5)  dx(t) = c_{m} (t, x_{t}) dt + g (t−, x_{t−}) dw_{t}
+ k ([t − 1], x([t − 1]), …, x(0)) dμ_{t}, t ≥ 0,

with the initial condition

x(t) = φ(t), t ∈ [−h, 0), (2.6)

where ϕ(t), t ∈ [−h, 0), is a nonrandom initial function, right continuous and having left limits. We assume that the functions g and k in Eqn (2.5) are nonrandom and continuous in their domain of definition, g(t, 0) ≡ 0, k([t], 0, …, 0) ≡ 0, and μ_{t} is from (2.1).

Remark 2.5. In this paper we do not discuss the existence of solution of Eqn (2.5)-(2.4) on [0, ∞). We just suppose that it exists. However we note that the local existence of a right continuous solution of Eqn (2.5) can be proven in the case when some additional conditions of local Lipschitz type are imposed on the functions g and k. Then the global existence can be proven under the conditions of Theorem 3.1 below (see, for example, [15]).

To construct Lyapunov-Krasovskii functionals we use the following auxiliary functions:

(2.7) L(u) = L_{ε}(u) = ∏_{i=0}^{m} ln_{i}(u + e_{m} + ε), Φ(u) = ∫_{0}^{u} L^{−1}(v)dv, u ≥ 0,

where e_{0} = 0, e_{1} = 1, e_{m} = exp e_{m−1}, ln_{0}(x) = x, ln_{i}(x) = ln_{i−1}(ln x) for i = 1, 2, …, m, x > 0, and ε > 0. The appropriate references can be found in [12].

3. STABILITY RESULT IN THE CASE OF ONE DELAY

Let the coefficients λ_{i,j} be nonrandom for i ≥ 1 and 1 ≤ j ≤ i, e_{j} > 0 for j ≥ 1, γ_{1}, γ_{2}, γ_{3}, α, δ > 0 be some constants, the function R(s) be continuous, bounded and non-decreasing, the function α_{1}(t) be positive, Δ_{j}(t) ≥ 0, 1 − Δ′_{j}(t) ≥ ε_{j} > 0 for t ≥ 0, j ≥ 1. Let also,

e := ∑_{j=1}^{∞} e_{j}/ε_{j} < ∞, R := R(∞) − R(0) + 1.

Let the following conditions hold for t ≥ 0 and for every function x(t) which is right continuous and admits left-hands limits

(3.1) |g(t, x_{t})|^{2} ≤ γ_{1}x^{4}(t) + γ_{2}x^{2}(t)x^{2}(t − h) + γ_{3}x^{4}(t − h)

+ ∫_{0}^{∞}L (x^{2}(t − s)) α_{1}(t − s)dR(s)

+ ∑_{j=1}^{∞} e_{j}L (x^{2}(t − Δ_{j}(t))) α_{1}(t − Δ_{j}(t)),

∫_{0}^{∞}α_{1}(s)ds<∞, (3.2)

|k [i, x(i), x(i − 1), …, x(0)]|^{2} ≤ ∑_{j=0}^{i} λ_{i,j}^{2}L(x^{2}(i − j)), (3.3)

∑_{i=0}^{∞} ∑_{l=0}^{∞} η_{i+1+l}λ_{i+l,l}^{2} ≤ ∞. (3.4)

In this section we consider the case of one delay, m = 1. It means that in the Eqn (2.5) instead of c_{m} we have

(3.5)  c_{1} (t, x_{t}) = a_{3,0}(t)x^{3}(t) + a_{2,1}(t)x^{2}(t)x(t − h)
+ a_{1,2}(t)x(t)x^{2}(t − h) + a_{0,3}(t)x^{3}(t − h).

Let the following conditions be fulfilled for some positive numbers γ_{1}, γ_{2}, γ_{3} and t ≥ 0:

(3.6)  2a_{3,0}(t) + |a_{2,1}(t)| + γ_{1} + α ≤ −δ < 0,
2a_{1,2}(t) + |a_{2,1}(t)| + |a_{0,3}(t)| + γ_{2} ≤ 0, |a_{3,0}(t)| + γ_{3} ≤ α.

Theorem 3.1. Let conditions (3.1-3.6) be fulfilled. Then P { lim_{t→∞} x^{2}(t) = 0} = 1 for any solution x(t) to Eqn. (2.5).

Proof. Let R̃(t) = R(t) + l(t), where l(t) = 0, t < 0 and l(t) = 1, t ≥ 0. We define a Liapunov-Krasovskii functional W by the formula

(3.7)  W = Φ (V) , V (t, x_{t}) = ∑_{j=1}^{4} V_{j}(t, x_{t})
V_{4}(t, x_{t}) ≡ V_{4}(i, x_{i}), if i ≤ t < i + 1,
(3.8)  V_{1}(x) = x^{2},  V_{2}(t, x_{t}) = α ∫_{t−h}^{t} x^{4}(s)ds,

(3.9)  V_{3}(t, x_{t}) = ∫_{0}^{∞}dR̃(s) ∫_{t−s}^{t} α_{1}(τ)L (x^{2}(τ)) dτ

+ ∑_{j=1}^{∞} e_{j}/ε_{j} ∫_{t−Δ_{j}(t)}^{t} α_{1}(τ)L (x^{2}(τ)) dτ, t ≥ 0,

(3.10) V_{4}(i, x_{i}) = ∑_{l=0}^{∞} η_{i+1+l} ∑_{j=1+l}^{i+l} λ_{i+l,j}^{2}L(x^{2}(i + l − j)), i = 1, 2, ….

We apply the Ito formula to W (t, x(t)) = Φ (V (t, x_{t})) where x is a solution to Eqn(2.5). For the simplicity everywhere in the proof we put W (t) = W (t, x(t)), V (t) = V (t, x_{t}), V_{j}(t) = V_{j}(t, x_{t}), g_{t} = g (t, x_{t}), k_{i} = k [i, x(i), x(i − 1), …, x(0)]. We obtain

(3.11)  W (t) ≤ W (0) + ∫_{0}^{t} L^{−1}(V (s−))(2x(s−)c_{1} (s, x_{s−})

+ g_{s−}^{2} + \frac{∂V_{2}(s−)}{∂s} + \frac{∂V_{3}(s−)}{∂s})ds

+ ∑_{s≤t} (W (s) − W (s−) − \frac{∂W (s−)}{∂x}Δx(s)) + m_{t}^{(1)},

where Δx(i) = x(i)−x(i−), Δx(t) = x(t)−x(t−) ≠ 0 only for t = 1, 2, …, and

m_{t}^{(1)} = 2 ∫_{0}^{t} L^{−1}(V (s−))x(s−)g_{s−}dw_{s} + 2 ∑_{i+1≤t} L^{−1}(V (i))x(i)k_{i}ξ_{i+1}

is a martingale. Since L^{−1}(v) is decreasing, then ΔW ≤ L^{−1}(V)ΔV and W (s) − W (s−) ≤ L^{−1}(V (s−))ΔV (s). From (2.5) we have

(Δx(i + 1))^{2} = k_{i}^{2}η_{i+1} + k_{i}^{2}μ_{i+1}.

ON STABILITY OF STOCHASTIC DELAY CUBIC EQUATIONS 199

Further,

\frac{∂W (s−)}{∂x}Δx(s) = L^{−1}(V (s−))2x(s−)Δx(s),

and

ΔV (s) = 2x(s−)Δx(s) + (Δx(s))^{2} + ΔV_{4}(s).

We can show, that

∑_{s≤t} (W (s) − W (s−) − \frac{∂W (s−)}{∂x}Δx(s)) (3.12)

= ∑_{i+1≤t} (W (i + 1) − W (i) − \frac{∂W (i)}{∂x}Δx(i + 1)))

≤ ∑_{i+1≤t} L^{−1}(V (i)) ((Δx(i + 1))^{2} + ΔV_{4}(i + 1))

and

(3.13)  .2x(t−)c_{1} (t, x_{t−}) + g_{t−}^{2} + \frac{∂V_{2}(t−)}{∂t} + \frac{∂V_{3}(t−)}{∂t}
≤ −δx^{4}(t−) + (R + e) α_{1}(t)L(x^{2}(t−))

We can prove (see, for example [18], [16]) that

k_{i}^{2}η_{i+1} + ΔV_{4}(i + 1) ≤ L(x^{2}(i)) ∑_{l=1}^{∞} η_{i+1+l}λ_{i+l,l}^{2}.

Since L(x^{2}(s)) ≤ L(V (s)) we obtain that

(3.14) W (t) ≤ W (0) − δ ∫_{0}^{t} L^{−1}(V (s−))x^{4}(s−)ds + ∫_{0}^{t} (R + e) α_{1}(s)ds

+ ∑_{i+1≤t} ∑_{l=1}^{∞} η_{i+1+l}λ_{i+l,l}^{2} + m_{t},

where m_{t} = m_{t}^{(1)} + ∑_{i+1≤t} L^{−1}(V_{i})k_{i}^{2}η_{i+1} is a martingale. Inequality (3.14) can be rewritten in the following way:

W (t) ≤ W (0) + A^{(1)}(t) − A^{(2)}(t) + m_{t}.

Here

A^{(2)}(t) = δ ∫_{0}^{t} L^{−1}(V (s−))x^{4}(s−)ds,

A^{(1)}(t) = ∑_{i+1≤t} ∑_{l=1}^{∞} η_{i+1+l}λ_{i+l,l}^{2} + ∫_{0}^{t} (R + e) α_{1}(s)ds

are a.s. increasing predictable processes, m_{t} is a martingale, and

P {A^{(1)}(∞) < ∞} = 1

due to conditions (3.2) and (3.4). Applying Lemma 2.3, we have

P {A^{(1)}(∞) < ∞} ⊂ P {W → } ∩ P {A^{(2)}(∞) < ∞} (3.15)

and therefore P {W → } = 1. Since function Φ is continuously invertible, we get P {V → } = 1 and hence, there exist two a.s. finite random variables K_{1} = K_{1}(ω) < ∞ a.s. and K = K(ω) < ∞ a.s. such that P {sup_{t>0} V (t) < K_{1}} = 1 and also P {sup_{t>0} |x(t)| < K} = 1.

From (3.7) we have: V = Ṽ + V^{∗}, where Ṽ = V_{1} + V_{2}, V^{∗} = V_{3} + V_{4}. We estimate

(3.16)  V_{4}(i + 1)−V_{4}(i)=(ΔV_{4})(i + 1)≤L(K^{2}) (∑_{l=1}^{∞} η_{i+1+l}λ_{i+l,l}^{2}) ,

where ∑_{l=1}^{∞} η_{i+1+l}λ_{i+l,l}^{2} is a common term of some convergent series due to condition (3.4). We also get

(3.17)  |V_{3}(t_{1}) − V_{3}(t_{2})| ≤ 2L(K^{2}) (R + e) ∫_{t_{2}}^{t_{1}} α_{1}(τ)dτ

≤ 2L(K^{2}) (R + e) ∫_{t_{2}}^{∞}α_{1}(τ)dτ,

where t_{1} > t_{2} and the integral ∫_{0}^{∞}α_{1}(τ)dτ converges due to condition (3.2).

Therefore

(ΔV_{4})(i + 1) → 0

when i → ∞ and

|V_{3}(t_{1}) − V_{3}(t_{2})| → 0

when t_{1}, t_{2} → ∞.

Using (3.16) and (3.17) we can prove by contradiction that P {V^{∗}(t) → } = 1. Then P {Ṽ (t) → } = 1, since P {V (t) → } = 1. We note that a.s.

(3.18) |V_{2}(t_{1}) − V_{2}(t_{2})| = |∫_{t_{2}}^{t_{1}} x^{4}(τ)dτ − ∫_{t_{2}−s}^{t_{1}−s}x^{4}(τ)dτ| ≤ 2K^{4}|t_{2} − t_{1}|.

Suppose now P {lim sup_{t→∞} x^{2}(t) = ζ_{0}(ω) > 0} = p_{0} > 0. Then there exists

a sequence t_{k} = t_{k}(ω) such that P (Ω_{1}) = p_{0}, where Ω_{1} = {ω: x^{2}(t_{k}, ω) > ζ_{0}(ω)/2 > 0}. From x^{2}(t) = Ṽ (t) − V_{2}(t) we have

|x^{2}(t) − x^{2}(s)| ≤ |Ṽ (t) − Ṽ (s)| + |V_{2}(t) − V_{2}(s)| . (3.19)

Due to convergence of Ṽ (t) there exists T = T (Ṽ , ω) such that |Ṽ (t) − Ṽ (s)| ≤ ζ_{0}(ω)/8 for t, s ≥ T. From (3.18) we can find δ = δ(ζ_{0}) such that |V_{2}(t) − V_{2}(s)| ≤ ζ_{0}(ω)/8, when |t − s| ≤ δ. Then

|x^{2}(t_{k}) − x^{2}(s)| ≤ ε = ζ_{0}(ω)/4

for ω ∈ Ω_{1}, |s − t_{k}| ≤ δ, t_{k} > T and

(3.20) |x^{2}(s)| ≥ |x^{2}(t_{k})| − |x^{2}(t_{k}) − x^{2}(s)| ≥ ζ_{0}(ω)/2 − ζ_{0}(ω)/4 = ζ_{0}(ω)/4.

Let k(n) = k(n, ω) be a number of elements of the sequence {t_{k}} in the interval [T, n]. Applying inequality (3.20) we obtain for ω ∈ Ω_{1}

∫_{0}^{n} L^{−1}(V (s))x^{4}(s)ds ≥ ∫_{T}^{n} L^{−1}(V (s))x^{4}(s)ds

≥ ∑_{k: T ≤t_{k}+δ≤n} ∫_{t_{k}−δ}^{t_{k}+δ} L^{−1}(V (s))x^{4}(s)ds

≥ \frac{ζ_{0}}{4L(K_{1})} ∑_{k: T ≤t_{k}+δ≤n} ∫_{t_{k}−δ}^{t_{k}+δ} ds = \frac{2δζ_{0}^{2}}{16L(K_{1})} ∑_{k≤k(n)} 1

= \frac{k(n)δζ_{0}^{2}}{8L(K_{1})} → ∞

as n → ∞, because k(n) → ∞ as n → ∞. Hence P {A^{2}(∞) = ∞} ≥ p_{0} > 0. This contradicts (3.15) proving the result.

4. STABILITY RESULT IN THE CASE OF MANY DELAYS

In this section we consider Eqn.(2.5), where c_{m} is defined in (2.4) and k_{i} satisfies (3.3)-(3.4). Let

(4.1)  |g(t, x_{t})|^{2} ≤ γ_{1}x^{4}(t) + x^{2}(t) ∑_{i=1}^{m} γ_{2}^{i}x^{2}(t − h_{i}) + ∑_{i=1}^{m} γ_{3}^{i}x^{4}(t − h_{i})

+ ∫_{0}^{∞}L (x^{2}(t − s)) α_{1}(t − s)dR(s)

+ ∑_{j=1}^{∞} e_{j}L (x^{2}(t − Δ_{j}(t))) α_{1}(t − Δ_{j}(t)),

and

2a_{3,0}(t) + γ_{1} + ∑_{i=1}^{m} [|a_{2,1}^{i}(t)| + α_{i}] ≤ −δ < 0, (4.2)

∑_{i=1}^{m} [2a_{2,1}^{i}(t) + |a_{2,1}^{i}(t)| + |a_{0,3}^{i}(t)| + γ_{2}^{i}] ≤ 0,

∑_{i=1}^{m} [|a_{3,0}^{i}(t)| + γ_{3}^{i} − α_{i}] ≤ 0.

Theorem 4.1. Let conditions (3.2-3.4), (4.1), (4.2) be fulfilled. Then P { lim_{t→∞} x^{2}(t) = 0} = 1 for any solution x(t) to Eqn. (2.5).

The proof of this theorem is analogous to that of Theorem 3.1. We apply Lyapunov-Krasovskii functional similar to (3.7-3.10). The only difference is that in (3.8) we put

V_{2}(t, x_{t}) = ∑_{i=1}^{m} ∫_{t−h_{i}}^{t} α_{i}x^{4}(s)ds.

Acknowledgement. The authors are grateful to comments of anonymous referee.

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