Math. Ann. 225, 205--218 (1977) © by Springer-Verlag 1977 On Some Fixed Point Principles for Cones in Linear Normed Spaces* Gilles Fournier Mathematics Institute, Universityof Warwick, Coventry,WarwickshireCV4 7AL, England Heinz- Otto Peitgen lnstitut fiir AngewandteMathematik der Universitiit,Wegelerstr.6, D-5300Bonn, Federal Republicof Germany O. Introduction in [9] and [10] Krasnosel'ski[ established and proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. One of these, which has come to be known as the theorem about the compression and expansion of a cone has been extended by Nussbaum to an asymptotic version (cf. Theorems 1.2 and 1.3 in [13]) even for condensing maps rather than completely continuous maps: (0.1) Let P be a cone in a Banach space, T: P--,P a condensin 9 map, r, R ~IR+ and N 6 IN. Assume that (0.1.1) T"(S~) C B~ .for all n > N ; (0.t.2) there is hEP, h+-O, such that x- T"(x)+th for x6SR, n> N and t>0; (0.1.3) T({x~UIT"(x)=x})CU for n>N. Then the fixed point index of Ton U is ind(P, T, U) -- t, /f r>R. Thus, Thas a fixed point in U. (S~= {xet't Jlxll =r}, /~r= {xePl Jlxlt <r} < max {r, R }}.) and U= {xe Pl min {r, R } < Hxt] Nussbaum's proof makes an essential use of the rather involved Zabreiko and Krasnosel'skii and Steinlein (rood p)-theorem for the fixed point index (cf. [20-23]) and it is the crucial assumption (0.1.3) which makes this tool applicable (cf. Theorem 1, assumption 0.4 in [20]). However, (0.1.3) does not seem to be convenient for applications. In fact Nussbaum's fundamental applications in [13] of asymptotic fixed point theory to the existence of periodic solutions of nonlinear, autonomous functional differential equations can be reduced to the * This work was supported by the Deutsche Forschungsgemeinschaft, SFB 72 an der Universit~t Bonn and a fellowshipgivenby the NRC of Canada