次线性泛函微分方程的Hartman-Wintner增长结果
Hartman-Wintner growth results for sublinear functional differential equations
摘要：
本文研究了一类标量自治非线性泛函和Volterra微分方程的增长率。在这些方程中，右边是状态的非线性函数的正连续线性泛函。我们假设非线性增长在无穷远，导致亚指数增长的解决方案。我们的主要结果表明，当非线性增长慢于临界速率时，泛函微分方程的解渐近于辅助自治常微分方程的解。如果非线性比这个速率增长得更快，则ODE支配FDE。如果非线性以这个速率趋于无穷大，则FDE和ODE以相同的速率增长，模为常数非单位因子。最后，如果非线性增长比临界速率慢，则常微分方程和频微分方程以相同的速率渐近增长。我们还证明了最后一个结果的部分逆。在增长率比常微分方程慢的情况下，我们计算了解的精确界。
英文摘要：
In this paper, we determine rates of growth to infinity of scalar autonomous nonlinear functional and Volterra differential equations. In these equations, the right-hand side is a positive continuous linear functional of a nonlinear function of the state. We assume the nonlinearity grows sublinearly at infinity, leading to subexponential growth in the solutions. Our main results show that the solutions of the functional differential equations are asymptotic to those of an auxiliary autonomous ordinary differential equation when the nonlinearity grows more slowly than a critical rate. If the nonlinearity grows more rapidly than this rate, the ODE dominates the FDE. If the nonlinearity tends to infinity at exactly this rate, the FDE and ODE grow at the same rate, modulo a constant non-unit factor. Finally, if the nonlinearity grows more slowly than the critical rate, then the ODE and FDE grow at the same rate asymptotically. We also prove a partial converse of the last result. In the case when the growth rate is slower than that of the ODE, we calculate sharp bounds on the solutions.
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