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Báo cáo hóa học: " ASYMPTOTIC FIXED POINTS FOR NONLINEAR CONTRACTIONS"

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: ASYMPTOTIC FIXED POINTS FOR NONLINEAR CONTRACTIONS | ASYMPTOTIC FIXED POINTS FOR NONLINEAR CONTRACTIONS YONG-ZHUO CHEN Received 29 August 2004 and in revised form 12 October 2004 Recently W. A. Kirk proved an asymptotic fixed point theorem for nonlinear contractions by using ultrafilter methods. In this paper we prove his theorem under weaker assumptions. Furthermore our proof does not use ultrafilter methods. 1. Introduction There are many papers in the literature that discuss the asymptotic fixed point theory in which the existence of the fixed points is deduced from the assumption on the iterates of an operator e.g. 1 6 and the references therein . Recently Kirk 5 studied an asymptotic fixed point theorem concerning nonlinear contractions. He proved the following theorem 5 Theorem 2.1 by appealing to ultrafilter methods. Theorem 1.1. Let M d be a complete metric space. Let T M M be a continuous mapping such that d Tnx Tny ộn d x y 1.1 for all x y e M where ộn 0 00 0 00 and limn o ộn ộ uniformly on the range of d. Suppose that ộ and all ộn are continuous and ộ t t for t 0. If there exists x0 e M which has a bounded orbit O x0 x0 Tx0 T2x0 . then T has a unique fixed point x e M such that limn o Tnx x for all x e M. In this paper we prove Theorem 1.1 under weaker assumptions without the use of ultrafilter methods. 2. Main results We need the following recursive inequality cf. 2 Lemmas 2.1 and 3.1 3 Lemmas 2.1 and 2.4 and 4 Lemma 1 . Lemma 2.1. Let ộ R R be upper semicontinuous that is limsupt-to ộ t ộ t0 for all t0 e R and ộ t t for t 0. Suppose that there exist two sequences of nonnegative real Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 2 2005 213-217 DOI 10.1155 FPTA.2005.213 214 Asymptotic fixed points for nonlinear contractions numbers un and n such that u2n f un n 2-1 where en 0 as n o. Then either sup un o or liminf un 0. Proof. Suppose that b sup un o- Assume that liminf un 0- Then there exist m 0 and N1 0 such that un m for all n N1- Since f is upper semicontinuous