Bifurcation of operator equations with unbounded linearized part
David Westreich · Pacific Journal of Mathematics · 1975
The bifuraction problem for the operator equation x = λLx + G(λ, x) is considered, where L is a closed linear operator with characteristic value λ 0 , and G(λ, x) is a continuous higher order term.If I -λ 0 L is a closed Fredholm operator and either L is self-adjoint and 6 is a continuously differentiable gradient operator or λ 0 is of odd algebraic multiplicity, then λ 0 is shown to be a bifurcation point.Introduction* Several authors have considered the bifurcation problem for nonlinear operator equations with closed linearized part.J. MacBain [7] considered the case where the nonlinear term is compact and obtained global results, similar to those gotten by P. H. Rabinowitz [8] for compact operator equations.Other results in specialized instances were obtained, among others, by M. G. Grandall and P. H. Rabinowitz [3], M. Reeken [10], and R. Bohme [2] who also considered gradient operator equations.In this note we extend known local bifurcation results, to nonlinear operator equations with linearized parts closed Fredholm operators and continuous higher order terms, dependent on λ, and where characteristic value is of odd algebraic multiplicity.Bifurcation results are also obtained for variational equations, though except for the dependence of the higher order term on λ they are not as strong as those of Bohme in [2].