Global Bifurcation Theory for Fredholm Operator via Odd Crossing Number and Its Application

We establish the Rabinowitz-type global bifurcation theorem and the Dancer-type unilateral global bifurcation theorem for F ( λ , u ) = 0 , ( λ , u ) ∈ R × X , where F : R × X → Y is a C 1 map and locally proper with F ( λ , 0 ) = 0 for λ ∈ R , X and Y are real Banach spaces with X ⊆ Y . Let S be the closure of the set of nontrivial solutions of F ( λ , u ) = 0 , ( λ , u ) ∈ R × X . We shall show that, if D u F ( λ , u ) is a Fredholm operator with index 0 for all ( λ , u ) ∈ R × X and D u F ( λ , 0 ) has an odd crossing number at λ = μ , then S possesses a maximal component C