ABOUT A NONLINEAR GENERALIZATION OF THE CONTRACTION MAPPING

Abstract

. At first, here a generalize method of iteration is proposed, presented as a
combination of the classical iteration and proportional division methods, on which the
conditions of Bolzano-Cauchy theorem are satisfied. Аn evidence of the proposed algorithm
convergence is brought. Originally, a generalize contraction mapping is considered as a
function from one variable for which a theorem about iteration convergence and its evidence
are brought. Secondly, in this article a variant of the generalize method of iteration – a
nonlinear-generalize method of iteration – is developed. A new geometrical interpretation of
the convergence of the generalize method of iteration is brought: three cases of step, spiral
and hyper-step iterations are estimated and their convergence sub-regions are considered.
An explicit formula of nonlinear-generalize contraction mapping operator as a function from
one real variable is obtained; a formulation of the nonlinear-generalize contraction mapping
as a function from complex and some real variables are also explicitly exposed. As a result,
an aggregate method of iteration is formulated. On examples of some transcendent
equations systems solution an advantage of this method compared to such known methods
as the classical iteration method and the Newton’s method is proved.