Mixed generalized iterative method for scalar first order non-linear initial value problem with applications

Abstract

In this paper, we provide the mixed generalized iterative method for scalar first order non-linear initial value problem, where the forcing function is the sum of increasing, decreasing, convex and concave functions. We develop the generalized iterative coupled lower and upper solutions of mixed type. Our method will yield generalized monotone method and generalized quasilinearization method as special cases. In these situations, we obtain linear and quadratic convergence respectively. In all other situations, we obtain super linear convergence. As an application, we provide some numerical results.