RENDICONTI DEL SEMINARIO MATEMATICO

130 D. Papini - F. Zanolinat the u-components and the s-components as the unstable-expansive and the stablecompressiveones, respectively), a reasonable choice of assumptions to put on the mapψ along the s-component will be that of taking conditions that reduce to those of theBrouwer or of the Rothe fixed point theorems (or to analogous ones) in the special casewhen u = 0 and s = N. On the other hand, it seems perhaps less evident which could bethe best choice of assumptions to express the expansive effect along the s-components.With this respect, both conditions on the norm (like in [2, 3]) and componentwise conditions(like in [80, 81]) have been assumed. As we have already explained with somedetails in the first part of this Introduction, motivated by the stretching property (H ± )discovered in [55] for equation (2) we obtained in [56] a fixed point theorem for planarmappings where the main hypothesis requires that the map expands the paths connectingtwo opposite sides of a topological rectangle. Further generalizations were thengiven in [59, 60], but still for a setting which is basically two-dimensional in nature.We recall that an expansive condition for paths connecting the opposite faces of a N-dimensional rectangle was also considered by Kampen in [32], allowing an arbitrarynumber of expansive directions (see [32, Corollary 4]). However, when reduced to thespecial case N = 2, Kampen’s result and ours seem to differ in some relevant points.In particular, a crucial assumption of our fixed point theorem in [56] allows the mapto be defined only on some subsets of the rectangle and, moreover, even when the themapping is defined on the whole rectangle, the assumptions in [32] and those in [56]about the compressing direction are basically different. One of the main features thatwe ask to a fixed point theorem for expansive-compressive mappings is to depend onhypotheses that can be easily reproduced for compositions of maps. This, in turns,permits to apply the theorem to the iterates of ψ and thus obtain results about the existenceof nontrivial periodic points. Since our path-stretching property well fits alsowith respect to this requirement (of course, it is not the only one; in fact, nice alternativeapproaches are available in literature), we want to address our investigations toward asuitable extension of such property to the case N > 2.1.4. ContentsAfter such a long introduction in which we surveyed some of our preceding resultsfor the two-dimensional case, we are ready to present some new developments in thehigher dimensional setting. Then the rest of this paper is organized as follows. InSection 2 we present our main result (Theorem 6) which is a fixed point for a compactmap defined on a subset of a cylinder in a normed space. In order to simplifythe exposition, we confine ourselves to the idealized situation in which we split ourspace as a product R× X and indicate its elements as pairs (t, x), so that we can easilyexpress our main assumption as an hypothesis of expansion of the paths contained inthe cylinder B[a, R] = [−a, a] × B[0, R] along the t-direction. The principal tool forthe proof of our basic fixed point theorem is the Leray−Schauder continuation theoremin its strongest form asserting the existence of a continuum of solution-pairs fora nonlinear operator equation depending on a real parameter (Théorème Fondamental[42]). Such result, with its variants and extensions, is one of the main theorems of theLeray−Schauder topological degree theory and it has found several important applica-