We prove a global bifurcation result for an equation of the type Lx + λ(h(x) + k(x)) = 0, where L : E → F is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset Ω of E, h, k : Ω × [0,+ ∞) → F are CM1 and continuous, respectively. Under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation, that is, solutions (x,λ) with λ ≠ 0, whose closure contains a trivial solution (x̄, 0). The proof is based on a degree theory fora special class of noncompact perturbations of Fredholm maps of index zero, called α-Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.
Bifurcation results for a class of perturbed Fredholm maps / P. Benevieri; A. Calamai. - In: FIXED POINT THEORY AND APPLICATIONS. - ISSN 1687-1820. - ELETTRONICO. - Art. ID 752657:(2008), pp. 0-0. [10.1155/2008/752657]
Bifurcation results for a class of perturbed Fredholm maps
BENEVIERI, PIERLUIGI;
2008
Abstract
We prove a global bifurcation result for an equation of the type Lx + λ(h(x) + k(x)) = 0, where L : E → F is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset Ω of E, h, k : Ω × [0,+ ∞) → F are CM1 and continuous, respectively. Under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation, that is, solutions (x,λ) with λ ≠ 0, whose closure contains a trivial solution (x̄, 0). The proof is based on a degree theory fora special class of noncompact perturbations of Fredholm maps of index zero, called α-Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.