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International Conference on Finite Difference Methods

FDM 2014: Finite Difference Methods,Theory and Applications pp 25–37Cite as

Well-Posedness in Hölder Spaces of Elliptic Differential and Difference Equations

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9045))

Abstract

In the present paper the well-posedness of the elliptic differential equation

$$\begin{aligned} -u^{\prime \prime }(t)+Au(t)=f(t)(-\infty <t<\infty ) \end{aligned}$$

in an arbitrary Banach space E with the general positive operator in Hö lder spaces \(C^{\beta }(\mathbb {R},E_{\alpha })\) is established. The exact estimates in Hölder norms for the solution of the problem for elliptic equations are obtained. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor’s decomposition on three points for the approximate solutions of this differential equation are studied. The well-posedness of the these difference schemes in the difference analogy of Hölder spaces \(C^{\beta }(\mathbb {R}_{\tau }, E_{\alpha })\) are obtained. The almost coercive inequality for solutions in \(C(\mathbb {R}_{\tau },E)\) of these difference schemes is established.

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References

  1. Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Elliptic Type. Nauka, Moscow (1973) (Russian)

    Google Scholar 

  2. Vishik, M.L., Myshkis, A.D., Oleinik, O.A.: Partial Differential Equations. Fizmatgiz, Moscow (1959) (Russian)

    Google Scholar 

  3. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman Advanced Publishing Program, London (1986)

    Google Scholar 

  4. Agmon, S.: Lectures on Elliptic Boundary Value Problems. D Van Nostrand, Princeton (1965)

    MATH  Google Scholar 

  5. Krein, S.G.: Linear Differential Equations in a Banach Space. Nauka, Moscow (1966) (Russian)

    Google Scholar 

  6. Skubachevskii, A.L.: Elliptic Functional Differential Equations and Applications. Birkhauser Verlag, Boston (1997)

    MATH  Google Scholar 

  7. Gorbachuk, V.L., Gorbachuk, M.L.: Boundary Value Problems for Differential-Operator Equations. Naukova Dumka, Kiev (1984) (Russian)

    Google Scholar 

  8. Sobolevskii, P.E.: On elliptic equations in a Banach space. Differential’nye Uravneninya 4(7), 1346–1348 (1969) (Russian)

    Google Scholar 

  9. Ashyralyev, A.: Method of positive operators of investigations of the high order of accuracy difference schemes for parabolic and elliptic equations. Doctor sciences thesis, Kiev (1992) (Russian)

    Google Scholar 

  10. Sobolevskii, P.E.: Well-posedness of difference elliptic equation. Discrete Dyn. Nat. Soc. 1(3), 219–231 (1997)

    Article  MATH  Google Scholar 

  11. Sobolevskii, P.E.: The theory of semigroups and the stability of difference schemes in operator theory in function spaces. Proc. School, Novosibirsk (1975). (pp. 304–307, “Nauka” Sibirsk. Otdel, Novosibirsk (1977)) (Russian)

    Google Scholar 

  12. Ashyralyev, A.: Well-posedness of the difference schemes for elliptic equations in \(C_{\tau }^{\beta,\gamma }(E)\) spaces. Appl. Math. Lett. 22, 390–395 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ashyralyev, A.: A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space. J. Math. Anal. Appl. 344(1), 557–573 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ashyralyev, A., Sobolevskii, P.E.: Well-posedness of the difference schemes of the high order of accuracy for elliptic equations. Discrete Dyn. Nat. Soc. 2006, 1–12 (2006)

    Article  MathSciNet  Google Scholar 

  15. Agarwal, R., Bohner, M., Shakhmurov, V.B.: Maximal regular boundary value problems in Banach-valued weighted spaces. Bound. Value Prob. 1, 9–42 (2005)

    MathSciNet  Google Scholar 

  16. Ashyralyev, A., Cuevas, C., Piskarev, S.: On well-posedness of difference schemes for abstract elliptic problems in \(L_{p}\left( \left[0,1\right], E\right)\) spaces. Numer. Funct. Anal. Optim. 29(1–2), 43–65 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ashyralyev, A.: On well-posedness of the nonlocal boundary value problem for elliptic equations. Numer. Funct. Anal. Optim. 24(1–2), 1–15 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ashyralyev, A., Sobolevskii, P.E.: Well-Posedness of Parabolic Difference Equations, vol. 69. Birkhäuser Verlag, Basel-Boston-Berlin (1994)

    Book  Google Scholar 

  19. Sobolevskii, P.E.: Imbedding theorems for elliptic and parabolic operators in C. Sov. Math., Dokl. 38(2), 262–265 (1989). Translation from Dokl. Akad. Nauk SSSR 302(1), 34–37 (1988)

    MathSciNet  Google Scholar 

  20. Ashyralyev, A., Sobolevskii, P.E.: New Difference Schemes for Partial Differential Equations, vol. 148. Birkhäuser Verlag, Basel (2004)

    Book  MATH  Google Scholar 

  21. Ashyralyev, A.: On the uniform difference schemes of a higher order of the approximation for elliptic equations with a small parameter. Appl. Anal. 36(3–4), 211–220 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ashyralyev, A., Fattorini, H.O.: On uniform difference schemes for second order singular perturbation problems in Banach spaces. SIAM J. Math. Anal. 23(1), 29–54 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Fatih University, Istanbul, Turkey

    Allaberen Ashyralyev

  2. ITTU, Ashgabat, Turkmenistan

    Allaberen Ashyralyev

Authors
  1. Allaberen Ashyralyev
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Correspondence to Allaberen Ashyralyev .

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Editors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Information and Communication Technologies, Sofia, Bulgaria

    Ivan Dimov

  2. Institue of Mathematics and MTA-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. University of Rousse, Rousse, Bulgaria

    Lubin Vulkov

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Ashyralyev, A. (2015). Well-Posedness in Hölder Spaces of Elliptic Differential and Difference Equations. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods,Theory and Applications. FDM 2014. Lecture Notes in Computer Science(), vol 9045. Springer, Cham. https://doi.org/10.1007/978-3-319-20239-6_3

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  • DOI: https://doi.org/10.1007/978-3-319-20239-6_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20238-9

  • Online ISBN: 978-3-319-20239-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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