Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access August 14, 2015

Linear and nonlinear abstract differential equations of high order

  • Veli B. Shakhmurov
From the journal Open Mathematics

Abstract

The nonlocal boundary value problems for linear and nonlinear degenerate abstract differential equations of arbitrary order are studied. The equations have the variable coefficients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding differential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established. In application, the separability and spectral properties of nonlocal boundary value problem for the system of degenerate differential equations of infinite order is derived.

References

Search in Google Scholar

[1] Amann H., Maximal regularity for non-autonomous evolution equations. Advanced Nonlinear Studies, (2004) 4, 417-430. 10.1515/ans-2004-0404Search in Google Scholar

[2] Agranovich M. S., Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces Hpơ p and Bơ; Funct. Anal. Appl., (2008) 42(4), 249–267. 10.1007/s10688-008-0039-xSearch in Google Scholar

[3] Arendt W., Duelli, M., Maximal Lp- regularity for parabolic and elliptic equations on the line, J. Evol. Equ. (2006), 6(4), 773-790. 10.1007/s00028-006-0292-5Search in Google Scholar

[4] Agarwal R., Bohner M., Shakhmurov V. B., Linear and nonlinear nonlocal boundary value problems for differential operator equations, Appl. Anal., (2006), 85(6-7), 701-716. 10.1080/00036810500533153Search in Google Scholar

[5] Ashyralyev A, Cuevas. C and Piskarev S., On well-posedness of difference schemes for abstract elliptic problems in spaces, Numer. Func. Anal. Opt., (2008)29, (1-2), 43-65. 10.1080/01630560701872698Search in Google Scholar

[6] Bourgain, J., Some remarks on Banach spaces in which martingale difference sequences are unconditional, Arkiv Math. (1983)21, 163-168. 10.1007/BF02384306Search in Google Scholar

[7] Burkholder D. L., A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions, Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund, Chicago, 1981,Wads Worth, Belmont, (1983), 270-286. Search in Google Scholar

[8] Dore, G., Lp-regularity for abstract differential equations. In: Functional Analysis and Related Topics, H. Komatsu (ed.), Lecture Notes in Math. 1540. Springer, 1993. Search in Google Scholar

[9] Denk R., Hieber M., Prüss J., R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. (2003), 166 (788), 1-111. 10.1090/memo/0788Search in Google Scholar

[10] Favini A., Shakhmurov V., Yakubov Y., Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces, Semigroup Form, (2009), 79 (1), 22-54. 10.1007/s00233-009-9138-0Search in Google Scholar

[11] Favini, A., Yagi, A., Degenerate Differential Equations in Banach Spaces, Taylor & Francis, Dekker, New-York, 1999. 10.1201/9781482276022Search in Google Scholar

[12] Goldstain J. A., Semigroups of Linear Operators and Applications, Oxford University Press, Oxfard, 1985. Search in Google Scholar

[13] Krein S. G., Linear Differential Equations in Banach space, American Mathematical Society, Providence, 1971. Search in Google Scholar

[14] Lunardi A., Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, 2003. Search in Google Scholar

[15] Lions J. L., Peetre J., Sur one classe d’espases d’interpolation, IHES Publ. Math. (1964)19, 5-68. 10.1007/BF02684796Search in Google Scholar

[16] Shklyar, A.Ya., Complete second order linear differential equations in Hilbert spaces, Birkhauser Verlak, Basel, 1997. 10.1007/978-3-0348-9187-5Search in Google Scholar

[17] Sobolevskii P. E., Coerciveness inequalities for abstract parabolic equations, Dokl. Akad. Nauk, (1964), 57(1), 27-40. Search in Google Scholar

[18] Shahmurov R., On strong solutions of a Robin problem modeling heat conduction in materials with corroded boundary, Nonlinear Anal. Real World Appl., (2011),13(1), 441-451. 10.1016/j.nonrwa.2011.07.047Search in Google Scholar

[19] Shahmurov R., Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annuals, J. Differential Equations, 2010, 249(3), 526-550. 10.1016/j.jde.2010.03.029Search in Google Scholar

[20] Shakhmurov V. B., Estimates of approximation numbers for embedding operators and applications, Acta. Math. Sin., (Engl. Ser.), (2012), 28 (9), 1883-1896. 10.1007/s10114-012-9547-ySearch in Google Scholar

[21] Shakhmurov V. B., Degenerate differential operators with parameters, Abstr. Appl. Anal., (2007), 2006, 1-27. 10.1155/2007/51410Search in Google Scholar

[22] Shakhmurov V. B., Regular degenerate separable differential operators and applications, Potential Anal., (2011), 35(3), 201-212. 10.1007/s11118-010-9206-9Search in Google Scholar

[23] Shakhmurov V. B., Shahmurova A., Nonlinear abstract boundary value problems atmospheric dispersion of pollutants, Nonlinear Anal. Real World Appl., (2010), 11(2), 932-951. 10.1016/j.nonrwa.2009.01.037Search in Google Scholar

[24] Triebel H., Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. Search in Google Scholar

[25] Triebel H., Spaces of distributions with weights. Multiplier on Lp spaces with weights. Math. Nachr., (1977)78, 339-356. 10.1002/mana.19770780131Search in Google Scholar

[26] Weis L, Operator-valued Fourier multiplier theorems and maximal Lp regularity, Math. Ann., (2001), 319, 735-758. 10.1007/PL00004457Search in Google Scholar

[27] Yakubov S. and Yakubov Ya., Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall /CRC, Boca Raton, 2000. Search in Google Scholar

Received: 2015-5-20
Accepted: 2015-7-16
Published Online: 2015-8-14

©2015 Veli B. Shakhmurov

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 27.2.2024 from https://www.degruyter.com/document/doi/10.1515/math-2015-0044/html
Scroll to top button