Open AccessТеоремы существования и единственности для дифференциального уравнения с разрывной правой частью
Author(s) -
М. Г. Магомед-Касумов
Publication year - 2022
Publication title -
vladikavkazskij matematičeskij žurnal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.126
H-Index - 2
eISSN - 1814-0807
pISSN - 1683-3414
DOI - 10.46698/p7919-5616-0187-g
Subject(s) - mathematics , uniqueness , initial value problem , sobolev space , operator (biology) , mathematical analysis , fourier series , lipschitz continuity , series (stratigraphy) , pure mathematics , paleontology , biochemistry , chemistry , repressor , biology , transcription factor , gene
We consider new conditions for existence and uniqueness of a Caratheodory solution for an initial value problem with a discontinuous right-hand side. The method used here is based on:1) the representation of the solution as a Fourier series in a system of functions orthogonal in Sobolev sense and generated by a classical orthogonal system;2) the use of a specially constructed operator $A$ acting in $l_2$, the fixed point of which are the coefficients of the Fourier series of the solution.Under conditions given here the operator $A$ is contractive. This property can be employed to construct robust, fast and easy to implement spectral numerical methods of solving an initial value problem with discontinuous right-hand side.Relationship of new conditions with classical ones (Caratheodory conditions with Lipschitz condition) is also studied.Namely, we show that if in classical conditions we replace $L^1$ by $L^2$, then they become equivalent to the conditions given in this article.