Báo cáo hóa học: " A LERAY-SCHAUDER ALTERNATIVE FOR WEAKLY-STRONGLY SEQUENTIALLY CONTINUOUS WEAKLY COMPACT MAPS"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: A LERAY-SCHAUDER ALTERNATIVE FOR WEAKLY-STRONGLY SEQUENTIALLY CONTINUOUS WEAKLY COMPACT MAPS | A LERAY-SCHAUDER ALTERNATIVE FOR WEAKLY-STRONGLY SEQUENTIALLY CONTINUOUS WEAKLY Compact maps RAVI P. AGARWAL DONAL O REGAN AND XINZHI LIU Received 23 June 2004 and in revised form 17 November 2004 A new applicable Leray-Schauder alternative is presented for weakly-strongly sequentially continuous maps. This result is then used to establish a general existence principle for operator equations. 1. Introduction This paper presents new fixed point results for weakly sequentially upper semicontinu-ous maps defined on locally convex Hausdorff topological spaces which are angelic when furnished with the weak topology. Moreover we establish an applicable Leray-Schauder alternative Theorem for a certain subclass of these maps. Our alternative combines the advantages of the strong topology . the sets are open in the strong topology with the advantages of the weak topology . the maps are weakly-strongly sequentially continuous and weakly compact . In Section 3 we illustrate how easily Theorem can be applied in practice. Finally we recall the following definition from the literature 9 . Definition . A Hausdorff topological space X is said to be angelic if for every relatively countably compact set C Q X the following hold i C is relatively compact ii for each x e C there exists a sequence xn n 1 Q C such that xn x. Remark . All metrizable locally convex spaces equipped with the weak topology are angelic see the Eberlein-Smulian theorem . 2. Fixed point theory We begin with some fixed point results which will be needed to obtain our applicable nonlinear alternative of Leray-Schauder type see Theorem . Theorem . Let E be a locally convex linear Hausdorff topological space which is angelic when furnished with the weak topology and let C be a weakly compact convex subset of E. Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 1 2005 1-10 DOI 2 Leray-Schauder alternative Then any weakly .