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
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.


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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

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