By Remark 2 and taking into account Lemma 2, we see that the functions ̂ u , $^{∂$_{η}$}$̂ u are continuously differentiable in the region D × G ′ for both cases η > 0 and η < 0. Then using (4.3), we obtain that for both cases η > 0 and η < 0 the function ̂ u satisfies the equation √ − 1 ∂$_{η}$ ∂$_{x}$ 1 ̂ u − $_{2}$√ − 1 n ∑ j,k =2 Γ j 1 $_{k}$ξ$^{k}$∂$_{η}$ ∂$_{ξ}$ j ̂ u + n ∑ j =2 ξ $^{j}$∂$_{x}$ j ̂ u − √ − 1 n ∑ j,k =2 Γ 1 $_{jk}$ξ$^{k}$ξ $^{$^{j}$η}$̂ u − n ∑ j,k,s =2 Γ s $_{jk}$ξ$^{k}$ξ$^{j}$∂$_{ξ}$ s ̂ u = 0 (4.4) ∂$_{η}$∂$_{x}$ 1 p − 2 n ∑ j,k =2 Γ j 1 $_{k}$ξ$^{k}$∂$_{η}$ ∂$_{ξ}$ j p = ϝ $_{1}$, (4.5) ∂$_{η}$∂$_{x}$ 1 q − 2 n ∑ j,k =2 Γ j 1 $_{k}$ξ$^{k}$∂$_{η}$ ∂$_{ξ}$ j q = ϝ $_{2}$, (4.6) ϝ 1 = n ∑ j,k =2 Γ 1 $_{jk}$ξ$^{k}$ξ$^{j}$ηp − n ∑ j =2 ξ $^{j}$∂$_{x}$ j q + n ∑ s,j,k =2 Γ s $_{jk}$ξ$^{k}$ξ$^{j}$∂$_{ξ}$ s q, ϝ 2 = n ∑ j,k =2 Γ 1 $_{jk}$ξ$^{k}$ξ$^{j}$ηq + n ∑ j =2 ξ $^{j}$∂$_{x}$ j p − n ∑ s,j,k =2 Γ s $_{jk}$ξ$^{k}$ξ$^{j}$∂$_{ξ}$ s p. u ( x, ξ ) = 0 , ( x, ξ ) ∈ ∂D × G, ( ̂ u ( $_{x, η, ξ}$′$_{) = 0}$$_{,}$ ( $_{x, ξ}$′$_{)}$ ∈ ∂D × $_{G}$′ ) . (4.7) in the classical sense. Putting ̂ u = p + √ − 1 q and separating the real and the imaginary parts of the left-hand side of equation (4.4) for both η > 0 and η < 0, we have the following equations with respect to the functions ∂$_{η}$p and ∂$_{η}$ q respectively: where In this work, the uniqueness of the solution of the problem is investigated under the assumption of the existence of the solution . Hence it is assumed that there exists a solution u ( x, ξ ) to equation (3.5), i.e., there exists a solution ̂ u = p + √ − 1 q to equation (4.3) which satisfies the properties indicated in Remark 2 and the condition Lemma 3. Let the conditions of Lemma 1 and condition (3.6) be satisfied. 8