On the convergence of an iterative sequence to the solution of a system of ordinary differential equations with deviated arguments

Abstract

In tha theory of optimal control, very frequently there appears the problem of solving a system of ordinary differential equations of second order with deviated arguments. This problem is of great importance for many branches of this theory. Therefore it has been dealt with by many authors. A large number of references can be found e.g. in [3j and [8]. This paper is a generalization of [4]. We consider a system of ordinary differential equations of second order with deviated arguments (1), similarly to the way it was done in [5]» but with one difference. Namely, our boundary value conditions are different from those of [5] wbejre authors supposed that initial functions were given. Here, together with equation (1) we shall consider a linear two point boundary value condition. A similar condition was considered in [2] p. 50. In this note initial functions, which authors of paper [5] supposed to be given, are constructed in order that the solution of system (1) fulfil condition (4). We shall prove existence and uniqueness of the solution of problem (1), (4) in certain interval provided that this interval is small. This task is more general than that from paper [1], where the arguments were not deviated. This, however, does not yield better results (Fragment tekstu).