exist and Lemma 3 shows that (taking into account Remark 2) for fixed (0 $_{, ξ}$′$_{)}$ ∈ D × G ′ , the functions U = ∂$_{x}$ 1 q and V = ϝ 2 satisfy the conditions of Lemma 6 for y$_{0}$ = 0, in where the variable y is replaced by η . Therefore, by (6.2) and Lemma 6, we conclude that where $^{U}$± $_{q}$(0 $_{, ξ}$′ $_{i}$) = ∂$_{x}$ 1 q ( 0 , ± 0 $_{, ξ}$′ i ) . On the other hand, from (6.2) by Lemmata 2 and 3, it is not difficult to obtain Consequently, we have By Lemma 5, and Lemmata 2, 3 and 5 yield Hence, for x = 0 , the mean value theorem on interval [0 , ε ] (e.g., [13], p. 186) implies that where θ$_{1}$ ∈ (0 , 1) is a constant depending on $_{ξ}$′ $_{i}$. By using (6.5) in (6.4), we obtain 15 U$_{+}$$_{q}$ (0 $_{, ξ}$′ $_{i}$) − $^{U}$− $_{q}$(0 $_{, ξ}$′ $_{i}$) = q$_{i}$ (0 $_{, ξ}$′ $_{i}$) ε $^{2}$, (6.6) ∞ ∫ −∞$^{∂$_{x}$}$$^{i}$$^{p}$$^{(0}$ $_{, η, ξ}$′ $_{i}$) dη = ε ∞ ∫ −∞$^{∂$_{ξ}$}$$^{i}$$^{∂$_{x}$}$$^{i}$$^{p}$$^{(0}$ $_{, η, ξ}$′ $_{i}$θ$_{1}$ ) dη, (6.5) ∂$_{ξ}$ i ∂$_{x}$ i p ∈ L$_{1}$ ( R 1 η ) ∩ L$_{2}$ ( R 1 η ) ∩ C ( Δ ρ η × $_{G}$′ ) , ρ = − 1 , 1. ∞ ∫ −∞$^{∂$_{x}$}$$^{s}$$^{p}$$^{(0}$$^{, η,}$$^{0)}$$^{dη}$ = 0, 1 ≤ s ≤ n , U$_{+}$$_{q}$ (0 $_{, ξ}$′ $_{i}$) − $^{U}$− $_{q}$(0 $_{, ξ}$′ $_{i}$) = − ε ∞ ∫ −∞$^{∂$_{x}$}$$^{i}$$^{p}$$^{(0}$ $_{, η, ξ}$′ $_{i}$) dη . (6.4) U$_{+}$$_{q}$ (0 $_{, ξ}$′ $_{i}$) = −$^{∞}$ ∫ 0 F$_{2}$$_{i}$ (0 $_{, η, ξ}$′ $_{i}$) dη, $^{U}$− $_{q}$(0 $_{, ξ}$′ $_{i}$) = 0 ∫ −∞$^{F$_{2}$$_{i}$}$$^{(0}$ $_{, η, ξ}$′ $_{i}$) dη . U$_{+}$$_{q}$ (0 $_{, ξ}$′ $_{i}$) − $^{U}$− $_{q}$(0 $_{, ξ}$′ $_{i}$) = − 2 πa$_{ii}$ (0) ε $^{2}$, (6.3) dU ( y ) dy = V ( y ) + ( U ( y$_{0}$ + 0) − U ( y$_{0}$ − 0)) δ ( y$_{0}$ ) .