The LaSalle-type Theorems For Neutral Stochastic Functional Differential Equations

Stability theory has come to play an important role in the research of stochastic differential equations for a long time.There were lots of useful results about stability, which were obtained by different kinds of methods and techniques by many authors. Many researchers tried their best to extend some theories and skills about ordinary differential equations to stochastic differential equations,and obtained a great number of available results for stochastic differential equations.Naturally,we hope that LaSalle theorem can be very helpful for the discussion of stochastic differential equations as it does in the study of ordinary differential equations.However,there is little work about the application of LaSalle theorem for stochastic differential equations until the end of last century by professor Xuerong Mao.He established the stochastic LaSalle theorem for stochastic differential equations and obtained a lot of results about the asymptotic properties of stochastic differential equations.Mao's results can be used in the studying of stochastic differential delay equations and stochastic functional differential equations.Because of the special neutral part of the neutral stochastic functional differential equations,his conclusion is disabled for the neutral equations.Therefore,we add some specific conditions for the neutral part and take Mao's mind to establish LaSalle-type theorems for neutral stochastic functional differential equations.With the help of our new LaSalle-type theorems,several new results on stochastic asymptotic stability,attractor and boundedness are given.Many books about the asymptotic properties of solutions of stochastic differential equations said that the solutions of equations would tend to some specific area.However, they did not tell us the speed of solutions approaching this specific area,which makes trouble for our research.Then,we introduce a new kind ofψfunction to our work in order to resolve this problem.We consider the product ofψfunction and the solution function as one part and discuss the asymptotic properties of the product.If we know the asymptotic properties of the product,with the help of the specificψfunction,we can discover the speed of solutions approaching the special area.In other words,ψfunction gives us a chance to master the asymptotic properties of solutions of stochastic differential equations.Meanwhile,we establishψfunctional stability which is more general than the classical exponential stability and polynomially stability.We also establish several criteria onψfunctional stability for different kinds of differential equations.Moreover, the diverse asymptotic stabilities can be described by a kind ofψfunctional stability. Motivated by the newψfunctional stability,we can establish some new kinds of stabilities which are different from the exponential stability.With the help of semimartingale convergence theorem,stochastic calculous,moment inequality,Burkholder-Davis-Gundy inequality,Gronwall inequality etc.,we find that the product ofψfunction and the solutions of neutral equations will approach a nonempty set if some specific conditions are satisfied.We also establish stochastic LaSalle-type theorems for these neutral equations.The same technique can be applied in the discussion of the asymptotic properties of solutions of neutral equations,which means that the new results of these asymptotic properties can be considered as some stochastic version LaSalle theorems.At the end of this paper,we consider two kinds of stochastic differential equations with Markovian switching.We establish some LaSalle theorems for neutral stochastic functional differential equations with Markovian switching.Therefore,we can extend our results to a more general kind of stochastic differential equations.We also obtain the pth momentψfunctional stability for stochastic differential delay equations with Markovian switching,which can help us comprehend the concept ofψfunctional stability. Furthermore,we can deduce the almost surelyψfunctional stability from the pth moment stability.Obviously,it is another method to get asymptotic stability and it is different of the LaSalle method.