Nonlinear boundary value problems with parameters are called parametrized nonlinear boundary problems. This paper studies a priori error estimates of finite element solutions of second-order parametrized strongly nonlinear boundary value problems in divergence form on one-dimensional bounded intervals. The Banach space W01,∞ is chosen in formulation of the error analysis so that the nonlinear differential operators defined by the differential equations are nonlinear Fredholm operators of index 1. Finite element solutions are defined in a natural way, and several a priori estimates are proved on regular branches and on branches around turning points. In the proofs the extended implicit function theorem due to Brezzi et al. (1980) plays an essential role.
Tsuchiya, T., & Babuška, I. (1997). A priori error estimates of finite element solutions of parametrized strongly nonlinear boundary value problems. Journal of Computational and Applied Mathematics, 79(1), 41–66. https://doi.org/10.1016/S0377-0427(96)00149-5