Fixed point theorems in ordered Banach spaces via quasilinearization Original Research Article

In this paper we prove abstract fixed point results in ordered Banach spaces based on the sub–supersolution method combined with the idea of quasilinearization. Appropriately associated linear iteration schemes involving the Frechet derivative of the fixed point operator are established that allow us to approximate fixed points in a constructive and monotone way. Moreover, a characteristic feature of the approximation scheme is its quadratic convergence rate. Applications to nonlinear ordinary and partial differential equation problems are given which demonstrate that the abstract results developed here provide a proper theoretical framework for the method of quasilinearization applicable also to numerous other concrete problems in ODE and PDE.