Multiple positive fixed points of asymptotically linear maps

Abstract

In particular, we are interested in the set A, of those X E R, for which f(h, a) has at least two distinct fixed points. We shall assume that for every X > O,f(h, *) is asymptotically linear and thatf(h, 0) # 0. Then, by imposing some additional technical hypotheses, we shall show that A, contains an interval which will be characterized by the asymptotical behaviour of the map f. Simple one-dimensional examples show that our result is optimal, in the sense that there are maps for which A, coincides with the given interval. In a forthcoming paper we shall study asymptotically linear orderconvex maps. In that case we shall be able to characterize completely the set A,. It should be mentioned that the case where for every h E R, , f(h, 0) = 0 is much easier to investigate. In this case, the existence of at least two distinct fixed points off (A, -), for given h E 08, , can be shown by employing the results of [S] and [I 51. This paper was stimulated by a recent publication by J. P. Keener and H. B. Keller [13]. In that paper, the authors study nonlinear elliptic eigenvalue problems of the form