ψ~γ-Stability Of Neutral Stochastic Functional Differential Equations

Theψγ-stability,which is a new kind of stability raised in this paper and includes exponential stability and polynomial stability,of neutral stochastic functional differential equations(NSFDE) is discussed in this paper.We got a series of discriminance methods based on the Razumikhin type theorem and new Liapunov function respectively.All these results imply and generalize those conclusions found in refences.This paper first established the Razumikhin type theorem of zero solution -th monent exponential stability for stochastic functional differential equation(SFDE).As a generalization,then we got the Razumikhin type theorem of zero solution -th monentψγ-stability for neutral stochastic functional differential equation(NSFDE).We estimated the part with time delay and the part without time delay respectively by using the dismantled estimate method.The condition which was difficult to verify in the Razumikhin type theorems had been disassembled into a series of inequalities which were more easily to verify.These inequalities became more concrete and visual through selecting a special Liapunov function.So we got the corresponding criterions of zero solution p-th monentψγ-stability for NSFDE.In the mean time,we got a new result without Razumikhin technique by adopting a new and more valid Liapunov function. Using this result,we also got the corresponding criterions of zero solution p -th monentψγ-stability for NSFDE.Finally,a few examples proofed that these criterions were of great practical value.These sufficient criterions,whether based on Razumikhin type theorem or new Liapunov function,both synthesized dispersion with the best manner,and avoided big loss of accuracy which was caused by the dismantled estimate method.Therefore,the last results still keep having decent accuracy,which were just about the advantages of these criterions.