On the generalized successive approximations method

Abstract

In this paper, denoted a continuous function from [0,1]² into [0,1] by f, we consider the iterative process:

$$f(y,x) = x \ne y = f(x,y),$$

, and we dealt with the question of global convergence of above iterative process, i.e. the question of convergence, for each point(x₁,x₀) of [0,1]², of sequence ℕ of points of [0,1] obtained from (I).

In sections 4,5,6, relatively to above convergence, necessary conditions, necessary and sufficient conditions, sufficient conditions are separately explained (theorem (5.4) and the results contained in section 6 are already known).

In section 3 it is proved that, if f is decreasing with respect to first variable and there does not exist a point (x,y) of [0,1]² such that:

$$f(y,x) = x \ne y = f(x,y),$$

, then there does not exist a point (x,y) of [0,1]² such that:

$$f(y,x) \leqslant x< y \leqslant f(x,y).$$

.

Such result and other propositions, proved in smae section 3, have been utilized in sections 4 and 5.