ABOUT THE GENERAL POINT OF OPERATORS

Abstract

The work continues the study of the classical principle of contraction mapping. This principle has numerous theoretical and practical applications in various fields of mathematics. The results obtained indicate that classical results can be extended to the case of several operators, bearing in mind the existence of a common point for them. In addition, in some cases, the condition that the operator must make a contraction mapping can be weakened. To do this, it is enough to break it into several simple operators and look for common points of these operators. In particular, we prove a theorem on the existence of a common point of two operators that reflect a complete metric space on itself. In this case, between the images that these operators create, a certain relationship must be satisfied, which is analogous to the condition of the contracting mapping. A similar result is established for the case when the condition opposite to the condition of the contracting mapping is fulfilled between the images of the operators. In applications of the method of successive approximation, a situation often arises when the operator carrying out the reflection of the total space on himself does not satisfy the classical condition for the compression operator. Sometimes you can use the inverse operator if it exists. In some cases, this inconvenience can be bypassed by conducting certain analytical transformations. In particular, majorant operators of contraction can be used. In the work, a sufficient condition is established for the use of such operators to search for a fixed point of an operator that is not a compression operator. This result can also be used in the case where the operator is not continuous. The graphic diagrams of application of the method of successive approximation in the above cases are presented in the work. The results obtained in the work can be used to search for fixed points of individual operators, which are not compression operators, but allow for their replacement by simpler operators that or either have an inverse operator or are compression operators.