Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω ? ? ⁿ be a bounded domain, H = L ²(Ω), L: D(L) ? H → H be an unbounded linear operator, f ∈ C((bar Omega) × ?, ?) and λ ∈ ?. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem

$Lu = lambda fleft( {x,u} right), u in Dleft( L right),$

which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a secondorder ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.