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Báo cáo hóa học: " EXISTENCE OF ZEROS FOR OPERATORS TAKING THEIR VALUES IN THE DUAL OF A BANACH SPACE"

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: EXISTENCE OF ZEROS FOR OPERATORS TAKING THEIR VALUES IN THE DUAL OF A BANACH SPACE | EXISTENCE OF ZEROS FOR OPERATORS TAKING THEIR VALUES IN THE DUAL OF A BANACH SPACE BIAGIO RICCERI Received 14 October 2003 and in revised form 21 April 2004 To Professor Giuseppe Pulvirenti with affection on his seventieth birthday Using continuous selections we establish some existence results about the zeros of weakly continuous operators from a paracompact topological space into the dual of a reflexive real Banach space. Throughout the sequel E denotes a reflexive real Banach space and E its topological dual. We also assume that E is locally uniformly convex. This means that for each x e E with xh 1 and each e 0 there exists 8 0 such that for every y e E satisfying IIy II 1 and x - y II e one has x yll 2 1 - 8 . Recall that any reflexive Banach space admits an equivalent norm with which it is locally uniformly convex 1 page 289 . For r 0 we set Br x e E x r . Moreover we fix a topology T on E weaker than the strong topology and stronger than the weak topology such that E T is a Hausdorff locally convex topological vector space with the property that the T-closed convex hull of any T-compact subset of E is still T-compact and the relativization of T to B1 is metrizable by a complete metric. In practice the most usual choice of T is either the strong topology or the weak topology provided E is also separable. The aim of this short paper is to establish the following result and present some of its consequences. Theorem 1. Let X be a paracompact topological space and A X E a weakly continuous operator. Assume that there exist a number r 0 a continuous function a X R satisfying a x r A x E 1 for all x e X a possibly empty closed set C c X and a T-continuous function g C Br satisfying A x g x a x 2 Copyright 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004 3 2004 187-194 2000 Mathematics Subject Classification 47H10 54C65 54C60 58K05 URL http dx.doi.org 10.1155 S1687182004310028 188 Existence of zeros for dual-valued operators for all x e c