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DOI: 10.46698/p7919-5616-0187-g

# Existence and Uniqueness Theorems for a Differential Equation with a Discontinuous Right-Hand Side

Magomed-Kasumov, M. G.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 1.
Abstract:
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.
Keywords: initial value problem, Cauchy problem, discontinuous right-hand side, Sobolev orthogonal system, existence and uniqueness theorem, Caratheodory solution