JournalszaaVol. 24 , No. 1DOI 10.4171/zaa/1233

Atypical Bifurcation Without Compactness

  • Pierluigi Benevieri

    Universita di Firenze, Italy
  • Massimo Furi

    Università di Firenze, Italy
  • Maria Patrizia Pera

    Universita di Firenze, Italy
  • Mario Martelli

    Claremont McKenna College, United States
Atypical Bifurcation Without Compactness cover

Abstract

We prove a global bifurcation result for an abstract equation of the type Lx+λh(λ,x)=0Lx + \lambda h(\lambda,x) = 0, where L:EFL: E \to F is a linear Fredholm operator of index zero between Banach spaces and h ⁣:R×EFh\colon \mathbb R \times E \to F is a C\sp1C\sp{1} (not necessarily compact) map. We assume that LL is not invertible and, under suitable conditions, we prove the existence of an unbounded connected set Σ\Sigma of nontrivial solutions of the above equation (i.e. solutions (λ,x)(\lambda,x) with λ0\lambda \neq 0) such that the closure of Σ\Sigma contains a trivial solution (0,xˉ)(0,\bar x). This result extends previous ones in which the compactness of hh was required. The proof is based on a degree theory for Fredholm maps of index zero developed by the first two authors.