content_image/324642.jpg content_image/324640.jpg content_image/324641.jpg content_image/324639.jpg content_image/324637.jpg content_image/324636.jpg content_image/324638.jpg content_image/324635.jpg
exist and 

<img src='content_image/324642.jpg'>

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 

<img src='content_image/324640.jpg'>

<img src='content_image/324641.jpg'>

<img src='content_image/324639.jpg'>

<img src='content_image/324637.jpg'>

<img src='content_image/324636.jpg'>

<img src='content_image/324638.jpg'>

where θ$_{1}$ ∈ (0 , 1) is a constant depending on $_{ξ}$′ $_{i}$. By using (6.5) in (6.4), we obtain 

<img src='content_image/324635.jpg'>

