Furthermore, if necessary, we choose smaller α such that We arbitrarily choose y$_{0}$ = ( y$_{1}$, y$_{2}$, . . . , y$_{m}$ ) ∈ R m satisfying We set Then (4.1), (4.2) and (4.4) yields, if x ∈ D and | y − y$_{0}$ | ≤ L, and if x ∈ D, | y − y$_{0}$ | ≤ L and | t | < T, r 2 > α $^{2}$L$^{2}$. (4.5) | y$_{0}$ | ≤ L − L ρ − ϵ. (4.6) ψ ( x, y, ∓ T ) = | x − x$_{0}$ | 2 − α | y − y$_{0}$ | 2 − βT 2 ≤ | x − x$_{0}$ | 2 − α | y − y$_{0}$ | 2 ≤ ˜ r 2 − αL 2 < 0 (4.7) ψ ( x, y, t ) = | x − x$_{0}$ | 2 − α | y − y$_{0}$ | 2 − β | t | 2 ≤ | x − x$_{0}$ | 2 − α | y − y$_{0}$ | 2 ≤ ˜ r 2 − αL 2 < 0 (4.8) G$_{1}$ = { y ∈ R $^{m}$; | y − y$_{0}$ | < L } , G$_{2}$ = { y ∈ R $^{m}$; | y | < 2 L } , Ω$_{0}$ = D × G$_{1}$ × { t = 0 } , Ω$_{1}$ = D × G$_{1}$ × ( − T, T ) , Ω$_{2}$ = D × G$_{2}$ × ( − T, T ) . 24