∫ ∫ ∫ ∫

The proof of Theorem 3.3 can see (Geni etal. 2001, Zheng & Qiao 2009, Yuan & Xu 2011).

Theorem 3.4 { | Rγ γe > 0, or γ ρ +i R a A≠ E} ( )0 . Proof. For any G g x gg g(g , , (g ), ( )x ) ,X0 1 2, 3 4 5 considering the equation [ ( )]γ A E P G+ = ,

∀ ∈G X , namely,

( ) ( ) ( ) ,γ μ μ μ ∞

∫∑P P P− P dx g=j j j

(13)

( ,γ λ) P g=1 1 0 1 (14)

( ,γ λ) P g=s2 2 0 2 (15)

( ) ,γ λ λ λP − P g=s3 2 1 32 (16)

P x P g x jj j j j′( )x ( ( )) ( )x ( ) , .+ = 4 5 (17)

And we can assume

P P4 3( )0 ,λ (18)

P P5 1( )0 .λ (19)

Solving the equation (17) with the help of (18)– (19) gets

P P d

jd

( )x ( )e p[ ( ( )) ]

, , .exp ( ( ))

=

=+ −

∫ 4 5 0

γ μ η η

γ μ η η τ⎣⎢

⎤ ⎦⎥∫0x jg d( ) τ

(20)

Noting that g Lj ( )x [ , )∈ ∞ 1 0 , j = 4 5, , together

with Theorem 3.1, we know P Lj ( )x [ , )∈ ∞ 1 0 ,

j = 4 5, . This implies that [ ( )]γ A E+ is an onto mapping.