Iterative processes: A survey of convergence theory using Lyapunov second method

Let x{sup k}, k {ge} 1, be some sequence of iterations. The diversity of motion laws governing the transition from the current iteration x{sup k} to the next iteration x{sup k+1} generates an infinite set of facts concerning convergence of the sequence x{sup k}. This richness complicates the construction of a general and at the same time substantively meaningful theory that combines all the available facts into one whole. Instead of focusing on convergence of (x{sup k}), the approach presented in this paper examines convergence to zero of a sequence of values of a nonnegative functional evaluated on the iterative process. Such sequences are found to satisfy a number of fundamental conditions that are independent of the phase space, the motion law, and the particular functional chosen, i.e., these conditions are indeed general. Assertions regarding convergence of (x{sup k}) itself require new additional restrictions each time a new class of iterative processes is considered, and in this sense are not general. The proposed approach generalizes the well-established facts and also predicts many new ones.