Persistence and Bifurcation of Degenerate Solutions

We consider a nonlinear equation F(ε, λ, u)=0, where F is a differentiable mapping from R×R×X to Y and X, Y are Banach spaces. When ε varies from a fixed ε=ε0, bifurcation occurs to the solution curve (λ(s), u(s)). We study the degenerate solutions of the equation, and we obtain several bifurcation theorems on the degenerate solutions, which can be applied in many nonlinear problems to obtain precise global bifurcation diagrams.