强一致收敛下的初值敏感性与等度连续性

The sensitive dependence on initial conditions and the equicontinuity under strongly uniform convergence

首先,举例指出了《Nonlinear Anglgsis》文中定理3.2的条件下并不能使函数序列的初值敏感性遗传至极限函数,并证明了若函数序列的敏感常数的上极限为某一正数,则在强一致收敛下,函数序列的极限函数也具有初值敏感性.其次,证明了在强一致收敛下,序列系统的等度连续性和一致几乎周期性能被极限系统所继承.

First,give an example to show that the sensitive dependence on initial conditions of the dynamical systems sequence can't be inherited by the limit system under the condition showd in theorem 3.2 in 《Nonlinear Analysis》,and prove that if the upper limit of the sensitivity constants in the dynamical systems sequence is a positive number,then the sensitive dependence on initial conditions can be inherited by the limit system under strongly uniform convergence.Second,the equicontinuity of the dynamical systems sequence can be inherited by the limit system under strongly uniform convergence is proved.