9ACADEMIC RESEARCH IN EDUCATIONAL SCIENCES VOLUME 2 | ISSUE 9 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-9-515-522
SOME PROPERTIES OF FIXED POINTS FOR NONLINEAR OPERATORS
Sirojiddin Masharipov
Basic doctoral (PhD) student, National University of Uzbekistan, Uzbekistan siroiiddinmasharipov1995@gmail.com
ABSTRACT
In this main goal is study nonlinear operators and find fixed points. Especially, Volterra operators very important for us. In this article given three-dimensional matrix and some properties nonlinear operators. From matrices used nonlinear operators and founded fixed points. For each class studied fixed points and trajectories. 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 for bistochastic operators not opened yet. This all works related to quadratic stochastic operators.
Keywords: Volterra operator, fixed points, simplex, three - dimensional matrices, trajectories, nonlinear operators, dynamical systems.
INTRODUCTION
Three dimensional matrices Pijk for us to study theories and research of some problems very important. We use them to study quadratic stochastic operators and relate some dynamical systems. The main connection between them is as follows:
V{x) < x,
where V(x) operator called Volterra operator. The operator is written as follows:
m
(Vx)k = ^ Pij^Xj
i=i
Three-dimensional matrix of the nth order Pijk will be wrote in simple form by the symbol IPiJikl (i,j, k = 1,2, ... ,n) , and other form this matrix, simply by P.
The matrix elements for each way denoted orientation. Thus for every one matrix denoted in this way. [2]
ACADEMIC RESEARCH IN EDUCATIONAL SCIENCES VOLUME 2 | ISSUE 9 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-9-515-522
^iMl
I ;,fc1
I 1
(j,k=1,2,... .,n)
For (j) and (k) also we can see similarly. Matrix elements called direction row (i) or (jk) orientation. This matrix can be written one simple form, because each one orientation parallel each other.
1 ;fc' yfc' ■■■' ^n;fc 1
Where j, k are fixed values of indices j, k. Lines of directions (j) and (k) are defined similarly. In a cubic matrix of the nth order, two sections of different orientations have n common elements located in one row, while three sections of different orientations have only one common element. Each section of any orientation and each line of the direction perpendicular to this section have one and only one common element. Corresponding elements of two parallel sections are elements belonging to the same line perpendicular to these sections. Corresponding elements of two parallel lines are elements belonging to the same section perpendicular to these lines. All these concepts related to a cubic matrix can be easily extended to a spatial matrix of any number of dimensions. A spatial matrix in which all elements located outside the main diagonal are equal to zero is called diagonal if there are nonzero among the elements of the main diagonal, and zero, otherwise, when all elements of the matrix are zero. Spatial matrix is called symmetric with respect to several indices if it is symmetric with respect to any pair of them. If symmetry takes place with respect to all indices, then the matrix will be called simply symmetric. A cubic matrix will be symmetric if
Pijk ~ Pikj ~ Pjik ~ Pkij ~ Pkji ~ Pjki-
A spatial matrix is called skew-symmetric with respect to two indices if every two elements of it, obtained from one another by permuting these indices, differ from each other only in sign.
Pijk ~ ~Pikj-
Elements of this matrix with the same values of the indices j, k are equal to zero. Elements that differ from each other only in sign are symmetrically located with respect to the main diagonal section corresponding to direction (i). This diagonal section consists entirely of zeros. A spatial matrix is called skew-symmetric with respect to several indices if it is skew-symmetric with respect to Any pair of them. If skew symmetry takes place with respect to all indices, then the matrix will be called simply skew-symmetric. Elements of a skew-symmetric matrix, for which not all
ACADEMIC RESEARCH IN EDUCATIONAL SCIENCES VOLUME 2 | ISSUE 9 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-9-515-522
indices have different values, are obviously equal to zero. Thus, a cubic matrix will be skew-symmetric if
Pijk = — Pik j = Pjki = — Pjik = Pkij = — Pkj i. Its elements, which differ from each other only in sign, are symmetrically located relative to the main diagonal, and all three main diagonal sections corresponding to directions (i), (j), (k) consist entirely of zeros. RESEARCH METHODS
Definition. The nonlinear stochastic operators are defined on a simplex and the dimensional of the simplex is (m — 1 ) , as
m
Sm~1 = {xi = (x1,x2,.. .,xm) E Rm,y^xi = 1, and xi > 0 }
¿=i
Definition. The simplex interior is a set where int while the simplex vertices (extreme points) is a set where xk = (0,0,... ,1,... ,0),(k = 1,m). However, the simplex center is a set x =
AA..).
mm
Definition. The evaluation of the nonlinear stochastic operators as
m
(Vx)k = ^ Pij^ixj i=1
Where the Pijk is the transaction matrix under the condition:
m
Pij,k = Pji,k > 0, ^ Pij,k = 1 k=1
Results and discussions.
We find some points of V2(x) = V(x) from these dynamic systems. We write the dynamic system as follows:
x' = xy + y2 + z2 z' = xy + 2yz z' = x2 + 2xz
According to the definition of the trajectory we find V2(x):
x'' = (xy + y2 + z2)y + y2 + z2 y'' = x(xy + 2yz) + 2(2xy + 2yz)z z'' = x2 + 2x(x2 + 2xz) We solve this equation. V2(x) = V(x):
xy2 + y3 + yz2 + y2 + z2 = xy + y2 + z2
ACADEMIC RESEARCH IN EDUCATIONAL SCIENCES VOLUME 2 | ISSUE 9 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-9-515-522
x2y + 2 xyz + 2 xyz + 4 yz2 = xy + 2 yz x2 + 2x3 + 4x2z = x2 + 2xz
1
X = y = z = -17 3
Also, this equation considered but this point not belongs to
simplex.
If we look only considered this condition
. Then we have various
equations. One of them:
x' = xz + z2 + y2 y' = 2 yz + 2xy z' = xz + x2
Again, we find trajectories and after that we find fixed points for this equation:
x" = (xz + x2 + y2)z + z2 + y2 y" = 2(2yz + 2xy) + 2x(2yz + 2xy) z" = x(xz + x2) + x2 After a few steps we get the following result:
x = y = z = 0 We can see that only one solution there. But it is not also considered simplex.
We construct an equation similar to the previous equation.
x' = 2 xz + z2 + y2 y' = 2 yz + x2 z' = 2xz
If we find next generation of trajectories:
x" = 2(2 xz + z2 +y2) +z2 y" = 2(2 yz + x2) + x2 z" = 2x ■ 2xz
Consequently, fixed points not belong to simplex again. y 2 = — z2 — 1z . That
is way it is not considered simplex. Thus, it is difficult to look at simplex for dynamic systems, and it may or may not belong to simplex. There will always be a zero solution in the equations. Moreover, if the system satisfies the simplex and satisfies some conditions, a 1/3 solution is formed. This fixed point will be the fixed point for all trajectories.
CONCLUSION
ACADEMIC RESEARCH IN EDUCATIONAL SCIENCES VOLUME 2 | ISSUE 9 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-9-515-522
The theory of quadratic stochastic operators spaces has its origins in discoveries made forty or fifty years ago by such mathematicians . Most of this early work is concerned with the properties of individual functions of inequalities and is classical in spirit. In recent years, the development of functional analysis has stimulated new interest in the Volterra operators for bistochastic matrices classes as linear spaces. This point of view has suggested a variety of natural problems and has provided new methods of attack, leading to important advances in the theory.
Simplex an important part in the classical theory of dynamical systems and in multidimensional matrices. Various theories on simplex have been found. Tournament theory is also accociated with simplex. Thus, there are many unresolved issues in this class at the moment.
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