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Existence and uniqueness theorems for a differential equation with a discontinuous right-hand side
M. G. Magomed-Kasumov Southern Mathematical Institute VSC RAS, 53 Vatutin St., Vladikavkaz 362027, Russia
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.
Key words:
initial value problem, Cauchy problem, discontinuous right-hand side, Sobolev orthogonal system, existence and uniqueness theorem, Caratheodory solution.
Received: 23.05.2021
Citation:
M. G. Magomed-Kasumov, “Existence and uniqueness theorems for a differential equation with a discontinuous right-hand side”, Vladikavkaz. Mat. Zh., 24:1 (2022), 54–64
Linking options:
https://www.mathnet.ru/eng/vmj801 https://www.mathnet.ru/eng/vmj/v24/i1/p54
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Abstract page: | 114 | Full-text PDF : | 43 | References: | 23 |
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