Some Fixed Point Theorems

Abstract

where f, p are continuous, g satisfies a Lipschitz condition, p(t) has period 1, and g(t)/I ? 1 for large t at any rate. Our choice of hypotheses and the main lines of our investigations have been dominated by what is significant in the theory of differential equations, but our results are concerned solely with sets of points and transformations of sets of points. In the first place if a solution of (1) for which o = 0o, = v = no1, when t= 0 has t = 0i, vq = ni, at t = 1, then the transformation 3 of the point x = (6 77o) on to the point xi = (4i, nj) is (1, 1) and continuous, and also orientation preserving in the open I, -q Cartesian plane. That is to say as the point x describes a closed Jordan curve J counter-clockwise, the point xi = 3(x) describes J1 = 3(J) counter clock-wise. Even when we consider closed invariant subsets of the plane, we continue to assume that 3 is (1, 1) continuous and orientation-preserving in the whole plane and not merely in the subset considered; for the tranformations set up by the solutions of the differential equation in which we are especially interested always satisfy these hypotheses. We use methods depending essentially on these hypotheses, and although there may be generalizations of some of our results of one kind or another, we do not attempt to discuss them. We shall also give sotne special consideration to transformations which decrease areas or leave them the same size, as this type of result is easily verified for transformations defined by certain classes of such differential equations. A complete and accurate statement of our aims and results would need various lengthy definitions and the introduction of new notation, but we shall first describe the main lines we have followed, and our reasons for doing so, in terms which to a large extent explain themselves. Precise definitions of the terms used will be given later in the appropriate places. We are indebted to Prof. L. C. Young and to Mr. H. D. Ursell for many helpful criticisms and suggestions.