Constructing model of nonlinear operators with three-dimensional matrices.

The main goal is to study nonlinear operators and find majorization for operators. Especially, Volterra operators are very important for us. This article has given a three-dimensional matrix and some properties nonlinear operators. Nonlinear operators from matrices are used, and trajectories are founded. The theory of boundary properties made considerable advances in the 20th century, owing to the work of several scientists; it resumed its rapid advance in the second half of that century, accompanied by the appearance of new ideas and methods, novel directions, and objects of study. This class also studied Lyubich. He found some results, but bistochastic operators have not opened yet. This all works related to quadratic stochastic operators. Three-dimensional matrices Pijk for us to study theories and research of some problems are very important. We use them to study quadratic stochastic operators and relate them to majorization. The main connection between them is as follows: V (x) ≺ x where V(x) operator called Volterra operator. The operator is written as follows: (V x) = ∑ i = 1 m P i j , k x i x j This article gave some properties for majorization and showed some connection with nonlinear operators. The article also gave some examples for nonlinear operators. The main goal is to find an operator and a corresponding matrix that satisfies the definition of majorization. This paper used the theory of majorization and some classes. This article's main result is, constructs nonlinear operators. It is also shown that the operators are bistochastic. The initial points are given from the simplex, and the result shows whether the operator belongs to the simplex or not. The theory of boundary properties made considerable advances in the 20th century, owing to the work of several scientists; it resumed its rapid advance in the second half of that century, accompanied by the appearance of new ideas and methods, novel directions, and objects of study. Its development is closely connected with various fields of mathematical analysis and mathematics in general, first and foremost with probability theory, the theory of real functional analysis, the theory of majorization, and Volterra operators. On the other hand, one of the main purposes of this article is to connect with the Volterra operators. The theory of boundary properties of majorization, which grew out of the works of the Moscow Mathematical School (G.G.Hardy, D.E.Littlewood), was developed in the further works of E. Beccenbach and R.Belman as well as in the works of and other Russian scientists. We will extend this class by constructing a nonlinear operators class for the Volterra operators. Not everything goes exactly without an analog. In such cases, calculations are carried out in other ways. [ABSTRACT FROM AUTHOR]