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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.
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
Language: English Download the full text
For citation: Magomed-Kasumov, M. G. Existence and Uniqueness Theorems for a Differential Equation with a Discontinuous Right-Hand Side, Vladikavkaz Math. J., 2022, vol. 24, no. 1, pp. 54-64. DOI 10.46698/p7919-5616-0187-g
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