An Iteration Method for Bifurcation Problem Involving Fredholm Mappings

Using the idea of the Liapunov‐Schmidt procedure and the modification of the Seidel‐Newton methods we show the existence of bifurcation solutions of equations of the form\documentclass{article}\pagestyle{empty}\begin{document}$$T\left( u \right) = L\left( {\lambda,u} \right) + H\left( {\lambda,u} \right).$$\end{document}Let λ be a characteristic value of the pair ( T, L ) (i.g. T(u) = L (λ, u .) for some u ± 0). Under sufficient conditions on T, L, H we prove by a constructive method that (λ, 0) is a bifurcation solution of the above equations. Furthermore, we obtain an analytical form of a parametric family of nontrivial solutions in a neighborhood of (λ, 0).