PROXIMITY MAPS, BEST APPROXIMATIONS AND FIXED POINTS  

作　　者：Lin Tzuchu (University of Wisconsin-Milwaukee, USA) 

出　　处：《Analysis in Theory and Applications》2000年第4期1-16,共16页分析理论与应用（英文刊）

摘　　要：In [Canada Math. Bull. 22, pp. 513–515, 1979], the author extended an interesting (best approximation) theorem of Ky Fan [see Theorem 2, Math. Z. 112, pp. 234–240] to condensing maps defined on a closed ball in a Banach space and defined on a closed convex subset in a Hilbert space. In [J. Approximation Theory 52, pp. 141–148, 1988], the author and Yen further extended it to 1-set-contractive maps in a Hilbert space. In this paper, we will address several related questions. First, we extend Lin and Yens’ result for 1-set-contractive maps but to satisfy weaker condition, which allows us to include generalized contraction maps. Second, we give ways to construct the best approximation (not just the existence results) for nonexpansive maps. Third, we give conditions on when the best approximation {un|n=1,2,…} of a sequence of contractions {fn|n=1,2,…} will converge to the best approximation u of a contraction f. Fourth, common best approximation theorems for nonexpansive maps are proved. Applications to fixed point theorems for weakly inward maps and others are given for all those four topics.In [Canada Math. Bull. 22, pp. 513–515, 1979], the author extended an interesting (best approximation) theorem of Ky Fan [see Theorem 2, Math. Z. 112, pp. 234–240] to condensing maps defined on a closed ball in a Banach space and defined on a closed convex subset in a Hilbert space. In [J. Approximation Theory 52, pp. 141–148, 1988], the author and Yen further extended it to 1-set-contractive maps in a Hilbert space. In this paper, we will address several related questions. First, we extend Lin and Yens’ result for 1-set-contractive maps but to satisfy weaker condition, which allows us to include generalized contraction maps. Second, we give ways to construct the best approximation (not just the existence results) for nonexpansive maps. Third, we give conditions on when the best approximation {un|n=1,2,…} of a sequence of contractions {fn|n=1,2,…} will converge to the best approximation u of a contraction f. Fourth, common best approximation theorems for nonexpansive maps are proved. Applications to fixed point theorems for weakly inward maps and others are given for all those four topics.

关 键 词：oscillatory integral operator MULTIPLIER SINGULARITY 

分 类 号：O174.41[理学—数学]