LaSalle-Type Theorems for General Nonlinear Stochastic Functional Differential Equations by Multiple Lyapunov Functions

and Applied Analysis 3 Remark 1. The assumption onw 2i (t, x t ) − w 2i (t, Fr[x(t)]) can also be described in the following form theoretically ifw 2i (t, ⋅) is differentiable. Denote M i (μ) = M i (μ, t, x t ) = w 2i (t, μx t + (1 − μ) Fr [x (t)]) , M i (μ) = M i (μ, t, x t ) = dM i (μ) dμ , ?̂? i (t, x t ) = ∫ 1 0 M i (μ) dμ. (8) Assume that ?̂? i (t, x t ) ≤ ∫ t t−τ (Ĉ 1i (t) w 1i (s, x (s)) + Ĉ 2i (t) w 2i (s, x s ) + Ĉ 3i (t) w 3i (s, x (s))) ds + γ ∗ i (t) + ζ i (t, x t ,W (⋅)) . (9) Let V(t, x) ∈ C1,2(R+ × Rn, R); that is, V(t, x) is once differentiable in t and twice continuously differentiable in x. Define a differential operator L, associated with (2), acting on V(t, x) by LV (t, φ) = V t (t, φ (0)) + V x (t, φ (0)) f (t, φ) + 1 2 Tr (gT (t, φ) V xx (t, φ (0)) g (t, φ)) , (10)


Introduction
As it is well known, the Lyapunov function method is the most widely used tool to establish criteria for stability or other asymptotic properties of dynamic systems governed by differential equations or difference equations.With this method, the derivatives of the Lyapunov functions or their upper bounds are often desired to be negative definite.For some complex equations, for example, the functional differential equations, this point may be somehow difficult for us at some times.Thus a spontaneous question will arise, that is, may we weaken the negative definiteness conditions for the derivatives of the Lyapunov functions or their upper bounds?In fact, some investigations have been made in the past years in this aspect.For example, LaSalle established a very important theorem named LaSalle invariance principle or LaSalle's theorem [1], which weakened the condition of the Lyapunov function method on the negative definiteness of the derivatives of the Lyapunov functions along the solutions of the equations, and it has been widely used in the theory of ordinary differential equations.In the recent years, LaSalle's theorem has been generalized directly to the stochastic differential delay equations by Mao [2][3][4][5], and a kind of LaSalle-type theorems had been established.Then the LaSalle-type theorems for stochastic differential delay equations have also been generalized to a kind of stochastic functional differential equations with distributed delays by Shen et al. [6][7][8][9][10][11]. Limited by the derivation techniques, this kind of theorems has not been generalized to the most general nonlinear stochastic functional differential equations so far.
In this paper, we generalize the investigation by Xuerong Mao, Yi Shen, and other authors to the general nonlinear stochastic functional differential equations.With some preliminaries on lemmas and the derivation techniques, we establish three LaSalle-type theorems for the general nonlinear stochastic functional differential equations via multiple Lyapunov functions.For the typical special case with estimations involving |  |  for the derivatives of the Lyapunov functions, a theorem is established as the corollary of the main theorem of the paper.The key point of the paper lies in the treatment of the general retarded terms in the estimations for the derivatives of the Lyapunov functions or their upper bounds.At the end of the paper, an example is given to illustrate the usage of the method proposed in the paper.
For a given function () ∈ (,   ), the associated function   ∈ (  ,   ) is defined as   () = (+),  ∈   .In real applications of the results of this paper, the criteria may be used in the function space ( 2 ,   ); in this case, one can extend the norm of  ∈ ( 2 ,   ) as For the general theory of functional differential equations, the readers are referred to [12][13][14][15][16][17].
In the paper, some coefficients and functions will be involved.Assume that, for each  = 1, 2, . . ., , functions For  2 (,   ), we further assume the following.(H 2 ) Along the solution of the equation, for each  = 1, 2, . . ., , we have estimation where Ĉ (),  = 1, 2, 3 are nonnegative continuous functions on  + and with (,   , (⋅))d = 0, and, especially, where where   will be defined in Lemma 4. Remark Let (, ) ∈ The assumptions for the involved Lyapunov functions will, respectively, be as follows: 2.3.Lemmas.By the nonnegative semimartingale convergence theorem [18], with a simple variable substitution for time , we directly have the following.
Lemma 3.Under the assumption (H 2 ), along the solution of (2), one has, for each  = 1, 2, . . ., , where Proof .Firstly, we directly have and thus we have Combining with the assumption (H 2 ), this yields The proof is complete.
Lemma 4. One has the following estimation: Abstract and Applied Analysis 5 where   = ∫ 0 − k ()d and Proof .We directly have and this completes the proof.
Denote the th column of (,   ) by   (,   ); then we can rewrite (2) as Then by Itô's rule, we have where By the linear growth condition, we have With these, we then have

Example
To illustrate the usage of our results, we construct an example.Consider a 2-dimensional stochastic functional differential equation with time-varying delay where () is a scalar standard Wiener process defined on the complete probability space (Ω, F, {F  } ≥0 , P).For this equation, the Lipschitz condition is satisfied and [ 1 ,  2 ]  = [0, 0]  is the trivial solution.Besides, we have time-varying delay () = 0.2 sin 2  with 0 ≤ () ≤  = 0.2.
It is obvious that the results in the previous literature are not adapted to this kind of equations with time-varying delay.We now give a conclusion for the asymptotic property of the solution of the equation by our theorems.

Conclusion
In this paper, we have generalized a kind of LaSalle-type theorems established by Xuerong Mao, Yi Shen, and other authors to the general nonlinear stochastic functional differential equations.With some preliminaries on lemmas and the derivation techniques, we establish three LaSalletype theorems for the general nonlinear stochastic functional differential equations by the multiple Lyapunov functions.It should be pointed out that the models investigated in this paper are not desired in any kind of special form.The key point of the paper lies in the treatment of the general retarded terms in the estimations for the derivatives of the Lyapunov functions or their upper bounds.Of course, for the typical special case with estimation involving |  |  for the derivatives of the Lyapunov functions, a theorem is established as the corollary of the main theorem.At the end of the paper, an example is given to illustrate the usage of the method proposed in the paper.In fact, the asymptotic properties of the solutions of this kind of equations cannot be judged by the results in the previous literature mentioned above.
It is obvious that the results of this paper can be rewritten in another form by mathematical expectation with only minor changes for the assumptions on the stochastic derivatives of the Lyapunov functions.
At the end, we point out that one may determine the asymptotic behavior of the solutions of the equations by virtue of the integrability of the terms as ∫   0   (, ())d directly if suitable assumptions are added to the related coefficients.

Figure 1 :
Figure 1: Solution behavior of stochastic system.