Abstract:
We give some conditions for solvability in $L_2(\mathbb{R})$ ($\mathbb{R}=(-\infty,+\infty)$) of the following
singular non-linear differential equation:
$$
ly\equiv-y'''(x)+q(x,y,y')y'+s(x,y,y')y=h(x).
$$
We assume that $q$ and $s$ are real-valued unbounded functions and $q$ does not obey the “potential” $s$.
For the solution $y$ we prove that
$$
||y'''||_2+||q(\cdot,y,y')y'||_2+||s(\cdot,y,y')y||_2<\infty,
$$
where $||\cdot||_2$ is the norm in $L_2$. To establish these facts, we use coercive solvability results for the
corresponding linear third-order differential equation obtained by us earlier.
Keywords and phrases:
non-linear differential equation, intermediate term, solvability, estimates of solutions.
Citation:
Zh. B. Yeskabylova, K. N. Ospanov, T. N. Bekjan, “The solvability results for the third-order singular non-linear differential equation”, Eurasian Math. J., 10:4 (2019), 85–91
\Bibitem{YesOspBek19}
\by Zh.~B.~Yeskabylova, K.~N.~Ospanov, T.~N.~Bekjan
\paper The solvability results for the third-order singular non-linear differential equation
\jour Eurasian Math. J.
\yr 2019
\vol 10
\issue 4
\pages 85--91
\mathnet{http://mi.mathnet.ru/emj350}
\crossref{https://doi.org/10.32523/2077-9879-2019-10-4-85-91}
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Linking options:
https://www.mathnet.ru/eng/emj350
https://www.mathnet.ru/eng/emj/v10/i4/p85
This publication is cited in the following 6 articles:
Kordan Ospanov, Myrzagali Ospanov, “Estimates for the Diameters of the Set of Solutions to a Nonlinear Differential Equation With Unbounded Coefficients”, Math Methods in App Sciences, 2025
R. D. Akhmetkaliyeva, T. D. Mukasheva, K. N. Ospanov, “Correct and coercive solvability conditions for a degenerate high order differential equation”, Eurasian Math. J., 14:4 (2023), 9–14
N. T. Orumbayeva, A. T. Assanova, A. B. Keldibekova, “On an algorithm of finding an approximate solution of a periodic problem for a third-order differential equation”, Eurasian Math. J., 13:1 (2022), 69–85
Myrzagali Ospanov, Kordan Ospanov, “Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations”, Mathematics, 10:10 (2022), 1717
K. N. Ospanov, “Correctness conditions for high-order differential equations with unbounded coefficients”, Bound. Value Probl., 2021:1 (2021), 47
A. N. Yesbayev, M. N. Ospanov, “The solvability conditions for the second order nonlinear differential equation with unbounded coefficients in l-2(r)”, Bull. Karaganda Univ-Math., 101:1 (2021), 104–110
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