Non-linear Analysis, Theory, Methods &Applications. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Vol. 3. No. I. pp. 35-44 0 Pergamon Press zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Ltd. 1979 Prmted m Great Britain 0362-546X/79/0101-0035 102.0010 POSITIVE EIGENVECTORS OF R-SET CONTRACTIONS IVAR MASSABO* Dipartimento di Matematica, Universita della Calabria, Cosenza, Italy and CHARLES A. STUART Department de Mdthematiques, Ecole Polytechnique Fed&ale, Lausanne, 61 avenue de Cour, CH-1007 Lausanne, Switzerland zyxwvutsrqponmlkjihgfedcbaZYXWVUTS (Received 17 January 1978) Key words: k-set contraction, cone, positive eigenvector, component of solutions, essential spectrum, elliptic equation. INTRODUCTION THIS paper begins with a version of a theorem due to Krasnoselskii (Theorem 1.1, p. 243 of [l]) which is itself closely related to an early result of Birkhoff and Kellog [2]. Krasnoselskii’s theorem is given in the context of a compact mapping T: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK P -+ P where P is a cone in a real Banach space. It asserts the existence of an eigenvector of Ton the boundary of an open set Q containing 0, provided that I/ Txll > 0 for all x E 80. Our result (Theorem 1.1) allows T to be a k-set contrac- tion, but we suppose that P is normal. Other results related to the Krasnoselskii and Birkhoff- Kellog theorem are contained in [3-61. Let Y = {(x, 2) E P x [0, co]: x = ATx} and let %? be the component of Y containing (0,O). We use Theorem 1.1 to formulate an alternative concerning the global behaviour of %‘. This result (Theorem 1.2) is in the spirit of a theorem of Rabinowitz [7] concerning non-linear elliptic eigenvalue problems on bounded domains. In the context of abstract Banach space and T: P + P a compact mapping, the unboundedness of G?? has been established by Dancer [S] and Amann [9]. The case where T: X + X is a k-set contraction is treated in [lo]. Our proof of Theorem 1.2 is based on Theorem 1.1 via a trick used by Kuiper [ll]. As is shown by the example in Section 6 of [lo], our alternative cannot, in general, be sharpened. In Section 2, we reformulate the alternative in the context of unbounded operators in an ordered Hilbert space. Cast in this form, the result is most easily applied to non-linear elliptic eigenvalue problems on unbounded domains. A typical application of this kind is given in Section 3. The results given in this article are primarily concerned with non-linear operators and some (Theorem 1.2, for example) are trivial in the case of linear operators since 0 E P is accepted as a positive eigenvector. For linear operators a positive eigenvector should be an element of P\(O). For compact linear operators the theorems of Krein and Rutman [18] deal with this situation, and some extensions to linear k-set contractions T: X -+ X with T(P) c P are given in [19]. In Section 4, we show how Theorem 3 of [ 193 can be obtained very easily from Theorem 1.1. Actually we need the additional assumption that the norm in X is monotone, but our result is more general in so far as T: X -+ X is only required to be homogeneous and monotone. * Partially supported by a C.N.R. (Italy) Fellowship. 35