LAUSR

We consider a difference-operator approximation Ahx$$ {A}_h^x $$ of the differential operatorAxux=−a11xux1x1x−a22xux2x2x+σux,x=x1x2,$$ {A}^xu(x)=-{a}_{11}(x){u}_{x_1{x}_1}(x)-{a}_{22}(x){u}_{x_2{x}_2}(x)+\sigma u(x),\kern1em x=\left({x}_1,{x}_2\right), $$defined in the region ℝ+ × ℝ with the boundary condition u0x2=0,x2∈ℝ.$$ u\left(0,{x}_2\right)=0,\kern1em {x}_2\in \mathbb{R}. $$ Here, the coefficients aii(x), i = 1, 2, are continuously differentiable, satisfy the condition of uniform ellipticity a112x+a222x≥δ>0$$ {a}_{11}^2(x)+{a}_{22}^2(x)\ge \delta >0 $$, 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.