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Existence of solutions of a system of two ordinary differential equations with a modular–cubic type nonlinearity

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Abstract

We use asymptotic analysis to study the existence of solutions of a one-dimensional nonlinear system of ordinary differential equations with different powers of a small parameter at higher derivatives. A specific feature of the problem is the presence of a discontinuity of the first kind in the right-hand side of the equation \(\varepsilon^4u''=f(u,v,x,\varepsilon)\) in the unknown variable \(u\) at the level \(u=0\), while the right-hand side of the second equation \(\varepsilon^2v''=g(u,v,x,\varepsilon)\) is assumed to be smooth in all variables. We define a generalized solution of the problem is in terms of differential inclusions. Conditions under which generalized solutions turn into strong ones are proposed, and the possibility that the \(u\)-component of the solution intersects zero only once is studied. The existence theorems are proved by using the asymptotic method of differential inequalities.

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Notes

  1. We note that the domain of \(f^{(-)}\) is extended to \(\overline{I^{-}_u}\).

  2. The intersection of classes \(C^1[0,1]\cap W^2_2(0,1)\), as well as \(C[0,1]\,{\cap}\,W^1_2(0,1)\) that appears in what follows, redundant at first glance, is justified by a decrease in the number of additional stipulations to be made, because the argument is based specifically on continuous and smooth representatives of functions from the corresponding Sobolev spaces.

  3. According to the nomenclature in [4], this type of quasimonotonicity is called the NN type; reversing the signs of the inequalities gives the PP type. The PN and NP types are introduced similarly.

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Acknowledgments

The author thanks N. T. Levashova for the valuable remarks on the text of the paper.

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  1. Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia

    B. V. Tishchenko

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  1. B. V. Tishchenko
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Correspondence to B. V. Tishchenko.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 215, pp. 318–335 https://doi.org/10.4213/tmf10411.

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Tishchenko, B.V. Existence of solutions of a system of two ordinary differential equations with a modular–cubic type nonlinearity. Theor Math Phys 215, 735–750 (2023). https://doi.org/10.1134/S0040577923050124

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  • DOI: https://doi.org/10.1134/S0040577923050124

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