A Fold Bifurcation Theorem of Degenerate Solutions in a Perturbed Nonlinear Equation

Abstract

We consider a nonlinear equation $F(\epsilon ,\lambda ,u)=0$, where the parameter$\epsilon $ is a perturbation parameter, $F$ is a differentiable mapping from $\textbf{R}\times\textbf{R}\times X$ to $\mathrm{Y}$and $X$, $Y$ are Banach spaces. We obtain an abstract bifurcation theorem by using the generalized saddle-node bifurcation theorem.