On the generalized successive approximations method
Tóm tắt
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.
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