Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations
Abstract
:1. Introduction
2. On One Two-Term Linear Operator
3. Solvability Conditions for the Linear Equation
4. Conditions for the —Maximal Regularity of Solution
5. The Solvability of the Nonlinear Equation
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Greguš, M. Third Order Linear Differential Equations; Reidel: Dordrecht, The Netherlands, 1982. [Google Scholar]
- Padhi, S.; Pati, S. Theory of Third-Order Differential Equations; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Tunç, C.; Mohammed, S.A. On the asymptotic analysis of bounded solutions to nonlinear differential equations of second order. Adv. Differ. Equ. 2019, 2019, 1–19. [Google Scholar] [CrossRef]
- Qian, C. On global stability of third-order nonlinear differential equations. Nonlinear Anal. 2000, 42, 651–661. [Google Scholar] [CrossRef]
- Beldjerd, D.; Remili, M. Boundedness and square integrability of solutions of certain third-order differential equations. Math. Bohemica 2018, 143, 377–389. [Google Scholar] [CrossRef]
- Kunstmann, P.C.; Weis, L. Maximal Lp -regularity for Parabolic Equations, Fourier Multiplier Theorems and H1-functional Calculus. Lect. Notes Math. 2004, 1855, 65–311. [Google Scholar]
- Arendt, W.; Duelli, M. Maximal Lp -regularity for parabolic and elliptic equations on the line. J. Evol. Equ. 2006, 6, 773–790. [Google Scholar] [CrossRef]
- Everitt, W.N.; Giertz, M. Some properties of the domains of certain differential operators. Proc. Lond. Math. Soc. 1971, 23, 301–324. [Google Scholar] [CrossRef]
- Otelbaev, M. Coercive estimates and separability theorems for elliptic equations in Rn. Proc. Steklov Inst. Math. 1984, 161, 213–239. [Google Scholar]
- Muratbekov, M.B.; Muratbekov, M.M. Sturm-Liouville operator with a parameter and its usage to spectrum research of some differential operators. Complex Var. Elliptic Equ. 2019, 64, 1457–1476. [Google Scholar] [CrossRef]
- Muratbekov, M.B.; Muratbekov, M.M. Estimates of the spectrum for a class of mixed type operators. Differ. Equ. 2007, 43, 143–146. [Google Scholar] [CrossRef]
- Ospanov, K.N. Qualitative and approximate characteristics of solutions of Beltrami-type systems. Complex Var. Elliptic Equ. 2015, 60, 1005–1014. [Google Scholar] [CrossRef]
- Ospanov, K.N.; Akhmetkaliyeva, R.D. Separation and the existence theorem for second order nonlinear differential equation. Electron. J. Qual. Theory Differ. Equ. 2012, 1, 1–12. [Google Scholar] [CrossRef]
- Muratbekov, M.B.; Muratbekov, M.M.; Ospanov, K.N. Coercive solvability of odd-order differential equations and its applications. Dokl. Math. 2010, 82, 909–911. [Google Scholar] [CrossRef]
- Akhmetkaliyeva, R.D.; Persson, L.-E.; Ospanov, K.; Woll, P. Some new results concerning a class of third order differential equations. Appl. Anal. 2015, 94, 419–434. [Google Scholar] [CrossRef]
- Ospanov, K.N.; Yeskabylova, Z.B.; Beisenova, D.R. Maximal regularity estimates for higher order differential equations with fluctuating coefficients. Eurasian Math. J. 2019, 10, 65–74. [Google Scholar] [CrossRef]
- Ospanov, K.N.; Yeskabylova, Z.B.; Bekjan, T.N. The solvability results for the third-order singular non-linear differential equation. Eurasian Math. J. 2019, 10, 85–91. [Google Scholar]
- Yosida, K. Functional Analysis; Springer: Berlin/Heidelberg, Germany, 1966. [Google Scholar]
- Kato, T. Perturbation Theory for Linear Operators; Springer: Berlin/Heidelberg, Germany, 1995. [Google Scholar]
- Schauder, J. Zur Theorie stetiger Abbildungen in Funktionalräumen. Math. Z. 1927, 26, 47–65. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Ospanov, M.; Ospanov, K. Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations. Mathematics 2022, 10, 1717. https://doi.org/10.3390/math10101717
Ospanov M, Ospanov K. Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations. Mathematics. 2022; 10(10):1717. https://doi.org/10.3390/math10101717
Chicago/Turabian StyleOspanov, Myrzagali, and Kordan Ospanov. 2022. "Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations" Mathematics 10, no. 10: 1717. https://doi.org/10.3390/math10101717