Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities: $$y(t)\leq E(t,\tau)||y_\tau||+\int_\tau^t K_1(t,s)||y_s||ds+\int_t^\infty K_2(t,s)||y_s||ds+\rho, \hspace{0.5cm} t\geq\tau\geq 0.$$ As a simple example of application, the retarded scalar functional differential equation $\dot x=-a(t)x+B(t,x_t)$ is considered, and the global asymptotic stability of the equation is proved under weaker conditions. Another example is the ODE system $\dot x=F_0(t,x)+\sum_{i=1}^m F_i(t,x(t-r_i(t)))$ on $R^n$ with superlinear nonlinearities $F_i$ ($0\leq i\leq m$). The existence of a global pullback attractor of the system is established under appropriate dissipation conditions. The third example for application concerns the study of the dynamics of the functional cocycle system $\frac{du}{dt}+Au=F(\theta_tp,u_t)$ in a Banach space $X$ with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous stationary solution $\Gamma$ is obtained under the hyperbolicity assumption on operator $A$ and some additional hypotheses, and the global asymptotic stability of $\Gamma$ is also addressed.

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一类延迟积分不等式解的均匀衰减估计

对于以下类型的延迟积分不等式的解，建立了一些统一的衰减估计：$$y(t)\leq E(t,\tau)||y_\tau||+\int_\tau^t K_1(t, s)||y_s||ds+\int_t^\infty K_2(t,s)||y_s||ds+\rho, \hspace{0.5cm} t\geq\tau\geq 0.$$ 作为一个简单的例子应用中，考虑了延迟标量泛函微分方程$\dot x=-a(t)x+B(t,x_t)$，证明了方程在较弱条件下的全局渐近稳定性。另一个例子是 $R^n$ 上的 ODE 系统 $\dot x=F_0(t,x)+\sum_{i=1}^m F_i(t,x(t-r_i(t)))$ 具有超线性非线性 $F_i$ ($0\leq i\leq m$)。在适当的耗散条件下，系统的全局回拉吸引子的存在成立。第三个应用实例涉及在具有亚线性非线性的 Banach 空间 $X$ 中研究功能共循环系统 $\frac{du}{dt}+Au=F(\theta_tp,u_t)$ 的动力学。特别地，在算子$A$的双曲假设和一些附加假设下，获得了非自治平稳解$\Gamma$的存在唯一性，并解决了$\Gamma$的全局渐近稳定性。