非线性Volterra方程零解的全局渐近稳定性

Global Asymptotic Stability of Zero Solutions for Nonlinear Volterra Equation

利用不动点理论,研究具有可变时滞的非线性Volterra方程x′(t)=-a(t)x(t)+q(t,x(t-τ1(t)),x′(t-τ1(t)))+∫t-τ2(t)t k(t,s)f(t,x(s),x′(s))ds,给出了该方程在C1空间上零解全局渐近稳定的新条件。这些新条件不需要时滞τ可微,也不要求τ′≠1。所得结论推广了已有文献中的相应结果,并给出了一个实例验证了所得结论的有效性。

In this paper,the following nonlinear Volterra equation with variable delays x′(t)=-a(t)x(t)+q(t,x(t-τ1(t)),x′(t-τ1(t)))+∫t-τ2(t)t k(t,s)f(t,x(s),x′(s))ds,was studied by using the fixed point theory.Some new conditions were given to ensure that the zero solutions are globally asymptotically stable in C1.Previously,almost all scholars used fixed point theory to study the asymptotic stability of zero solutions of nonlinear neutral differential equations with variable delays,it requiredτquadratic differentiability andτ′≠1.Unlike most research methods,these conditions do not require a quadratic differentiability of delayτandτ′≠1.The results obtained generalize the corresponding results in the literatures.An example is given to verify the validity of the conclusions.