Geodesic
Existence and uniqueness theorems for a differential equation with a discontinuous right-hand side
Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 1, pp. 54-64 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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M. G. Magomed-Kasumov. Existence and uniqueness theorems for a differential equation with a discontinuous right-hand side. Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 1, pp. 54-64. http://geodesic.mathdoc.fr/item/VMJ_2022_24_1_a5/

[1] Filippov A. F., “Differential Equations with Discontinuous Right-Hand Side”, Matematicheskii Sbornik. Novaya Seriya, 51(93):1 (1960), 99–128 (in Russian) | MR | Zbl

[2] Filippov A. F., Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988 | MR

[3] Krasovskij N. N., Game-Theoretic Problems of Capture, Nauka, M., 1970 (in Russian) | MR

[4] Krasovskij N. N., Subbotin A. I., Positional Differential Games, Nauka, M., 1974 (in Russian) | MR | Zbl

[5] Hermes H., Discontinuous Vector Fields and Feedback Control, Differential Equations and Dynamical Systems, Academic Press, New York, 1967 | MR

[6] Cid J. N., Pouso R. L., “Ordinary Differential Equations and Systems with Time-Dependent Discontinuity Sets”, Proceedings of the Royal Society of Edinburgh: Section a Mathematics, 134:4 (2004), 617–637 | DOI | MR | Zbl

[7] Hajek O., “Discontinuous Differential Equations, I”, Journal of Differential Equations, 32:2 (1979), 149–170 | DOI | MR | Zbl

[8] Cortes J., “Discontinuous Dynamical Systems”, IEEE Control Systems Magazine, 28:3 (2008), 36–73 | DOI | MR | Zbl

[9] Ceragioli F., Discontinuous Ordinary Differential Equations and Stabilization, PhD thesis, Universita di Firenze, 2000 | Zbl

[10] Sansone G., Equazioni Differenziali Nel Campo Reale, v. 2, Consiglio nazionale consiglio nazionale delle ricerche. Monografie di matematic applicata, Nicola Zanichelli, 1949 | MR

[11] Sharapudinov I. I., “On the Existence and Uniqueness of Solutions of ODEs with Discontinuous Right-Hand Sides and Sobolev Orthogonal Systems of Functions”, Daghestan Electronic Mathematical Reports, 9 (2018), 68–75 (in Russian) | DOI | MR

[12] Sharapudinov I. I., “Sobolev-Orthogonal Systems of Functions and the Cauchy Problem for ODEs”, Izvestiya: Mathematics, 83:2 (2019), 391–412 | DOI | MR | Zbl

[13] Sharapudinov I. I., “Sobolev Orthogonal Polynomials Associated with Chebyshev Polynomials of the First Kind and the Cauchy Problem for Ordinary Differential Equations”, Differential Equations, 54:12 (2018), 1602–1619 | DOI | MR | Zbl

[14] Sharapudinov I. I., “Approximation of the Solution of the Cauchy Problem for Nonlinear ODE Systems by Means of Fourier Series in Functions Orthogonal in the Sense of Sobolev”, Daghestan Electronic Mathematical Reports, 7 (2017), 66–76 (in Russian) | DOI | MR

[15] Stein E., Shakarchi R., Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2009 | MR

[16] Walter W., Ordinary Differential Equations, Graduate Texts in Mathematics, Springer, New York, 1998 | DOI | MR | Zbl

[17] Sharapudinov I. I., “Sobolev-Orthogonal Systems of Functions and Some of their Applications”, Russian Mathematical Surveys, 74:4 (2019), 659–733 | DOI | MR | Zbl

[18] Sharapudinov I. I., “Sobolev-Orthogonal Systems of Functions Associated with an Orthogonal System”, Izv. Math., 82:1 (2018), 212–244 | DOI | MR | Zbl

[19] Sharapudinov I. I., “Approximation Properties of Fourier Series of Sobolev Orthogonal Polynomials with Jacobi Weight and Discrete Masses”, Mathematical Notes, 101:3 (2017), 718–734 | DOI | MR | Zbl

[20] Sharapudinov I. I., “Sobolev Orthogonal Polynomials Generated by Jacobi and Legendre Polynomials, and Special Series with the Sticking Property for their Partial Sums”, Sbornik: Mathematics, 209:9 (2018), 1390–1417 | DOI | MR | Zbl