Banach压缩映象定理的逆与非线性收敛原理

A New Converse to Banachs Contraction Theorem: Nonlinear Convergence Principle

设Ｌｉｐ（Ｔ）表示非线性算子Ｔ的Ｌｉｐ数。本文证明了在常规条件下，Ｌｉｐ（Ｔ）＜１是逐次迭代ｘｎ＋１＝Ｔｘｎ对任何初值收敛到Ｔ之唯一不动点的充要条件。推广了熟知的线性收敛原理，并给出了Ｂａｎａｃｈ压缩映象原理的一个新的逆形式。

Abstract Let Lip (T) denote the Lipschitz number of a nonlinear operator T(i.e., Lip (T)= lim n→∞ 1/n , L(T)= sup x≠y‖Tx-Ty‖‖x-y‖ ) in a Banach space X . It is proved that if DX is a bounded closed subset of X, T: D→D is a Lipschitz operator, then Lip (T)<1 if and only if (a) T has a unique fixed point x * , (b) {T nx} converges to x * for any x∈D , and the convergence is uniform on a neighborhood of x * , and (c) T is locally contractable on x * . The obtained theorem generalized the well known linear convergence principle: if A is a linear operator, then the sequence {A nx} converges to x * for any x∈D iff ρ(A)<1 , where ρ(A) is the specture radius of A , and offered a new converse to Banachs contraction theroem.