ABOUT A NONLINEAR GENERALIZATION OF THE CONTRACTION MAPPING - А.А, Vantsyan; H.А, Gevorgyan (IJRDO -JOURNAL OF MATHEMATICS, 2017)
. At first, here a generalize method of iteration is proposed, presented as acombination of the classical iteration and proportional division methods, on which theconditions of Bolzano-Cauchy theorem are satisfied. Аn evidence of the proposed algorithmconvergence is brought. Originally, a generalize contraction mapping is considered as afunction from one variable for which a theorem about iteration convergence and its evidenceare brought. Secondly, in this article a variant of the generalize method of iteration – anonlinear-generalize method of iteration – is developed. A new geometrical interpretation ofthe convergence of the generalize method of iteration is brought: three cases of step, spiraland hyper-step iterations are estimated and their convergence sub-regions are considered.An explicit formula of nonlinear-generalize contraction mapping operator as a function fromone real variable is obtained; a formulation of the nonlinear-generalize contraction mappingas 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 transcendentequations systems solution an advantage of this method compared to such known methodsas the classical iteration method and the Newton’s method is proved.