APPLICATION OF THE PRINCIPLE OF CONTRACTION MAPPING TO STUDY THE SOLUTION OF NONLINEAR FUNCTIONAL EQUATIONS IN BANACH SPACES

Abstract

Processes of applied problems are often modeled by differential, integral and integro-differential equations, inequalities and inclusions with corresponding initial and boundary conditions determined by the environment.Functional models, where the unknown function and / or its derivatives have nonlinear dependences, often adequately describe the processes of the objects under study.The variety of mathematical models of nonlinear dependence requires a special scientific approach to their study and methods for finding solutions to problems. In applications, for an adequate course of the process, it becomes necessary to take into account the real circumstances in the essence of the varieties of nonlinear dependence.The questions investigated are related to the quasilinear differential equations solution.The theorem on the sufficient condition for the applicability of the principle of contraction mappings in space , , with reduction to a nonlinear integral equation, using the Green's function, to find the solution of the initial-boundary conditions of quasi-linear ordinary differential equations. The way is given in the proof of the theorem using the initial condition is determined, then with the same procedure of finding a sequence of functions , enables the approach to solving the problem with the desired accuracy. The above theorem and other as a byproduct, the results can be applied to research and find practical solutions of problems.This makes it possible to obtain a fairly wide application of the theorem, for example, when increasing the accuracy of the operation of automatic control and measuring devices.