一类随机积分微分方程解的稳定性

Solution Stability for a Class of Integral-differential Equations

研究具有变时滞r(t)的非线性随机积分-微分方程dx(t)=-∫(t-r(t))ta(t,s)f(x(s)))dsdt+g(t,x(t))dB(t),t≥0的解的稳定性问题,其中在x=0的某邻域内满足xg(,x)>0(x≠0).不仅使用不动点定理给出了方程解的均方渐近稳定的充分必要条件,同时给出了一个例子说明了主要结果.

In this paper, the authors investigate the solution stability issues concerning nonlinear stochastic integral-differential equations given as dx（t）=-（∫t-r（t）a（t,s）f（x（s）））dsdt＋g（t,x（t））dB（t）,t≥0 with variable delay r（t）, where xg（.,x） 〉 0（x ≠0） in a neighborhood of x = 0. Using the fixed point theorem, the sufficient and necessary conditions are given to ensure the solution of stochastic integral-differential equation to be mean square asymptotically stable. Meanwhile, one example is offered to help explain the obtained results.