关于对偶映象连续性的几个问题

Several Problems aboud the Continuity of Duality Mapping

令X是实Banach空间,由X到X~*(X的对偶)的非空子集族中的对偶映象由F(x)={X~*∈X~*;(x,x~*)=‖x‖~2=‖x~*‖~2}来定义。本文的主要结果如下: 1~0假设对偶映象F:X→X~*是单值的,u_n∈X(n=0,1,2,…),如果:(ⅰ){U_n}弱收敛于u_0,(ⅱ)lim(u_n—u_m,F(u_n)—F(u_m))≤0;则F(u_n)弱~*收敛于F(u_0) 2~0令Z是一致凸的实Banach空间,u_n∈(n=0,1,2,…)则{u_n}强收敛于u_0,当且仅当:(i){u_n}弱收敛于u_0,(ⅱ)存在v_n∈F(u_n)(n=1,2,…)使lim(u_n—u_m,v_n—v_m)≤0。3~0 假若Z有Fr'echet可微范数,u_n∈X(n=0,1,2,…),如果:(ⅰ){u_n}弱收敛于u_0,(ⅱ)lim(u_n—u_m,F(u_n)—F(u_m))≤0;则{F(u_n)}在X~*中强收敛于F(u_0)。

Let X be a real Banach Space. The duality map form X into the family of nonempty subsets of X^(?) (the dual of X)is defined by F(x)={x^(?)∈X^(?); (x,x^(?))=||x||~2=||x^(?)||~2}. The main results of this paper are as follows: 1° Suppas that duality mapping F:X→X~■ is single-valued,u_n∈X(n=0,1,2,…), If: (i) {u_n} converges weakly to U_o. (ii) ■(u_n-u_m,F(u_n)-F(u_m))≤0; Then F(u_n) converges weak-star to F(u_o). 2° Let X be a uniformly convex real Banach Space,u_n∈X(n = 0,1,2,…), then {u_n} converges strongly to u_o if and only if: (i) {u_n} converges weakly to uo, (ii)There exists v_n∈F(u_n)(n=1,2,…) so that (?)(u_n—u_m,v_n—v_m)≤0. 3° Assume that Xhas Fr' echet differentiable norm,u_n∈X(n=0,1,2, …). If:(i) {u_o) converges weakly to u_o,(ii)■(u_n—u_m,F(u_n)—F(u_m))≤0; then {F(u_n)} converges strongly to f (u_o) in X.