Banach空间中的平均非扩张映象:不动点的存在定理

<正> 设X是Banach空间,E是X中的集合,T是映集合E到自身的映象.若T满足条件(称为平均非扩张条件)其中x,y∈E,a,b,c≥0且a+2b+2c≤1,则称T是平均非扩张映象. 文1]概括了近年来研究关于平均非扩张映象不动点的一些主要结果.本文进

EXISTENCE THEOREMS OF FIXED POINT OF AVERAGE NONEXPANSIVE MAPPINGS IN BANACH SPACE

Let E be a set in Banach space X. Let T be an average nonexpansive mapping of E into itself (i.e. ‖Tx- Ty‖≤ a‖x- y‖ + b(‖x - Tx‖ + ‖y - Ty‖) + e(‖x-Ty‖ + ‖y - Tx‖), where x, y ∈ E, a, b, e ≥ 0 and a + 2b + 2c ≤ 1). The present paper deals with existence problems of fixed point for this mapping. The main results are as follows.Theorem 1. Suppose E is a closed nonempty set. If b > 0 and c > 0, then for the mapping T, there exists a unique fixed point x in E, and for any x∈E,Theorem 2. Suppose E is a closed convex nonempty set, and the mapping T is also Lipschitzian in E for a constant A > 0.If b>0, then for any x_o∈E and 0<θ< min {2b/A(1-c), 1}, the successive sequence {x_n} determined by x_o ∈ E, x_(n+1)=θT~2x_n+(1-θ)Tx_n, (n = 0, 1,…) converges to a unique fixed point of the mapping T in E.Theorem 3. Suppose E is a quasi-weakly compact convex nonempty set (i.e. E is closed, and any bounded closed covex subset of the set E is weakly compact). If b > 0 and one of the following conditions is satisfied, then the mapping T has a unique fixed point in E:(1) The mapping T is continuous in .E.(2) 25 < 1.(3) The set E has close-to-normed structure.(4) In any subset K of E with T(K)K and δ(K) > 0, there exists a point x∈K and a weakly limit point y of the {T~nx} such that lim sup‖Ty-T~nx‖<δ(K).Theorem 4. Suppose E is a weakly compact convex nonempty set, and the mapping T satisfies the asymptotic normed condition (i.e. for any closed convex subset K of E with T(K)K and σ(K) >0, there exist x, y∈K such that lim sup‖y-T~nx‖<σ(K)). If b=0, then the mapping T has a fixed point in E.