Quasilinearization method for nonlinear differential equations with causal operators

Employing quasilinearization technique coupled with the method of upper and lower solutions, we construct monotone sequences whose iterates are solutions to corresponding linear problems and show that the sequences converge uniformly and monotonically to the unique solution of the nonlinear problem with causal operator. Especially, instead of assuming convexity or concavity assumption on the nonlinear term that is demanded by the method of quasilinearization, we impose weaker conditions to be more useful in applications. The results obtained include several special cases and extend previous results.