可积情形下的Gronwall不等式

Remarks on Projected Gronwall Inequality

中文关键词:  微分方程  Gronwall不等式  投影  双曲性  间断性

英文关键词:differential equation  Gronwall inequality  projection  hyperbolicity  discontinu-ity.

中文摘要:

      在[6]讨论J．K．Hale指出的投影下Gronwall不等式u(t)≤ a(t)＋∫₀^tb(t-s)u(s)ds＋∫₀^∞c(s)u(t＋s)ds,?t≥ 0,基础上．指明[6]所要求的(?)a(t)=0是可以去掉的．进而,本文去掉了u(t),a(t)和c(t)的连续性以及a(t)的单调性,仅在可积性条件下得到了更一般的结论．所得的结果不仅真正包含了[5]的结果,而且新的证明方法使[5]中的疏漏得以补正．

英文摘要:

      Based on the discussion in [6] of projected Gronwall inequality u(t)≤ a(t)＋∫₀^tb(t-s)u(s)ds＋∫₀^∞c(s)u(t＋s)ds,?t≥ 0,it is indicated that the condition that (?) a(t) = 0 in [6] is not necessary. Futhermore,with-out continuity of functions u(t), a(t), and c(t), and monotonicity of function a(t),a moregeneral result is obtained. This result can be properly used in the case of [5],and the newmethod of the proof can remedy the leak in [5].