Nonlinear Analysis 41 (2000) 787 – 801 www.elsevier.nl/locate/na A global bifurcation theorem with applications to nonlinear Picard problems Jacek Gulgowski Institute of Mathematics, University of Gda nsk, ul. Wita Stwosza 57, 80-952 Gda nsk, Poland Received 9 July 1998; accepted 29 July 1998 Keywords: Nonlinear eigenvalue problems; Bifurcation points; Rabinowitz bifurcation theorem; Picard problem; Bernstein conditions In [5] Rabinowitz has established sucient conditions for the existence of closed and connected set of nontrivial solutions for nonlinear eigenvalue problem (;x)=0 with a completely continuous map  : R × E → E given by (∗) (;x)= x − Lx − g(;x); (;x) ∈ R × E; where L : E → E is linear and completely continuous, and g : R × E → E is a completely continuous map such that lim ‖x‖→ 0 g(;x)= ‖x‖ = 0 uniformly on bounded  intervals. In this paper we will generalize this theorem for maps f(;x)= x − F (;x) where F : R × E → E is a completely continuous map and need not be dierentiable at 0 (Theorem 1.2). Then by means of Theorem 1.2 we will prove some theorems for nonlinear Picard problem. In Theorems 1:3 and 1:4 the only conditions stated for right side of the dierential equation are those of local nature. Granas et al. [1, 2] studied problems (P) u ′′ (t ) − p(t;u(t );u ′ (t )) = 0 for t ∈ [0;]; u(0) = u()=0; where p : [0;] × R × R → R is continuous. They have proved ([1]) that if p satises the following Bernstein conditions: (B1) ∃ M ≥0 ∀ |s|¿M p(t;s; 0)s ≥ 0 0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S0362-546X(98)00310-1