Advertisement

Ukrainian Mathematical Journal

, Volume 70, Issue 8, pp 1176–1191 | Cite as

The Structure of Fractional Spaces Generated by a Two-dimensional Difference Operator in a Half Plane

  • A. Ashyralyev
  • S. Akturk
Article
  • 1 Downloads
We consider a difference-operator approximation \( {A}_h^x \) of the differential operator
$$ {A}^xu(x)=-{a}_{11}(x){u}_{x_1{x}_1}(x)-{a}_{22}(x){u}_{x_2{x}_2}(x)+\sigma u(x),\kern1em x=\left({x}_1,{x}_2\right), $$

defined in the region ℝ+ × ℝ with the boundary condition

$$ u\left(0,{x}_2\right)=0,\kern1em {x}_2\in \mathbb{R}. $$

Here, the coefficients aii(x), i = 1, 2, are continuously differentiable, satisfy the condition of uniform ellipticity \( {a}_{11}^2(x)+{a}_{22}^2(x)\ge \delta >0 \), and σ > 0. We study the structure of the fractional spaces generated by the analyzed difference operator. The theorems on well-posedness of difference elliptic problems in a Hölder space are obtained as applications.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. O. Fattorini, “Second order linear differential equations in Banach spaces,” in: North-Holland Mathematics Studies, North Holland (1985), 108.Google Scholar
  2. 2.
    P. Grisvard, “Elliptic problems in nonsmooth domains,” in: Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA (1985).Google Scholar
  3. 3.
    M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustyl’nik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff, Leiden (1976).Google Scholar
  4. 4.
    S. G. Krein, “Linear differential equations in a Banach space,” Transl. Math. Monogr., Amer. Math. Soc., Providence, RI (1968).Google Scholar
  5. 5.
    A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel, etc. (1997).Google Scholar
  6. 6.
    V. V. Vlasov and N. A. Rautian, Spectral Analysis of Functional Differential Equations [in Russian], MAKS Press, Moscow (2016).Google Scholar
  7. 7.
    T. S. Kalmenov and D. Suragan, “Initial boundary-value problems for the wave equation,” Electron. J. Different. Equat., 48, 1–6 (2014).Google Scholar
  8. 8.
    A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, etc. (1995)Google Scholar
  9. 9.
    M. Z. Solomyak, “Estimation of the norm of resolvent of an elliptic operator in the spaces L p,” Uspekhi Mat. Nauk, 15, No. 6, 141–148 (1960).Google Scholar
  10. 10.
    H. B. Stewart, “Generation of analytic semigroups by strongly elliptic operators under general boundary conditions,” Trans. Amer. Math. Soc., 259, 299–310 (1980).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kh. A. Alibekov and P. E. Sobolevskii, “Stability and convergence of difference schemes of a high order for parabolic differential equations,” Ukr. Math. Zh., 31, No. 6, 627–634 (1979); English translation: Ukr. Math. Zh., 31, No. 6, 483–489 (1979).Google Scholar
  12. 12.
    S. I. Danelich, Fractional Powers of Positive Difference Operators, Dissertation, Voronezh State Univ., Voronezh (1989).Google Scholar
  13. 13.
    Yu. A. Simirnitskii and P. E. Sobolevskii, “Positivity of multidimensional difference operators in the C-norm,” Uspekhi Mat. Nauk, 36, No. 4, 202–203 (1981).Google Scholar
  14. 14.
    A. Ashyralyev and S. Akturk, “Positivity of a one-dimensional difference operator in the half-line and its applications,” Appl. Comput. Math., 14, No. 2, 204–220 (2015).MathSciNetzbMATHGoogle Scholar
  15. 15.
    G. H. Hardy, J. E. Littlewood, and G. P´olya, Inequalities, Cambridge Univ. Press, Cambridge (1988).Google Scholar
  16. 16.
    A. Ashyralyev, “A survey of results in the theory of fractional spaces generated by positive operators,” TWMS J. Pure Appl. Math., 6, No. 2, 129–157 (2015).MathSciNetzbMATHGoogle Scholar
  17. 17.
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam–New York (1978).Google Scholar
  18. 18.
    A. Ashyralyev and F. S. Tetikoglu, “A note on fractional spaces generated by the positive operator with periodic conditions and applications,” Bound. Value Probl., 31 (2015).Google Scholar
  19. 19.
    A. Ashyralyev, N. Nalbant, and Y. Sozen, “Structure of fractional spaces generated by second order difference operators,” J. Franklin Inst., 351, No. 2, 713–731 (2014).MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. Ashyralyev and S. Akturk, “A note on positivity of two-dimensional differential operators,” Filomat, 31, No. 14, 4651–4663 (2017).MathSciNetCrossRefGoogle Scholar
  21. 21.
    A. Ashyralyev, S. Akturk, and Y. Sozen, “The structure of fractional spaces generated by two-dimensional elliptic differential operator and its applications,” Bound. Value Probl., 3 (2014).Google Scholar
  22. 22.
    A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Birkhäuser, Basel, etc. (2004).Google Scholar
  23. 23.
    A. Ashyralyev, “On well-posedness of the nonlocal boundary-value problems for elliptic equations,” Numer. Funct. Anal. Optim., 24, 1–15 (2003).MathSciNetCrossRefGoogle Scholar
  24. 24.
    V. Shakhmurov and H. Musaev, “Maximal regular convolution-differential equations in weighted Besov spaces,” Appl. Comput. Math., 16, No. 2, 190–200 (2017).MathSciNetzbMATHGoogle Scholar
  25. 25.
    S. Akturk and Y. Sozen, “The structure of fractional spaces generated by the difference operator on the half plane,” AIP Conf. Proc., 1479, 611–614 (2012).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • A. Ashyralyev
    • 1
  • S. Akturk
    • 2
  1. 1.Near East University, North Nicosia, Mersin, TRNC; Institute of Mathematics and Mathematical ModelingAlmatyKazakhstan
  2. 2.Yakuplu the Neighborhood Street KubilayIstanbulTurkey

Personalised recommendations