J. of Math. (PRC) u(t k ) = I k (u(t k )), k = 1, 2,, (1.6) , [3, 4] (1.1), (1.2), (1.3), [6 8]

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Vol 36 ( 216 ) No 3 J of Mah (PR) 1, 2, 3 (1, 4335) (2, 4365) (3, 431) :,,,, : ; ; ; MR(21) : 35A1; 35A2 : O17529 : A : 255-7797(216)3-591-7 1 d d [x() g(, x )] = f(, x ),, (11) x = ϕ(), [ r, ], (12) x( k ) = I k (x( k )), k = 1, 2, (13), [3, 4] Lyapunov (11), (12), [5] (11), (12), (13), [6 8],, : d d [u() g(, u )] = A[u() g(, u )] + f(, u ), (14) u = ϕ(), [ r, ], (15) u( k ) = I k (u( k )), k = 1, 2,, (16) Banach, A T (), u = u( + θ), r θ, = ([ r, ], ), ϕ ϕ = sup r θ ϕ : 215-3-11 : 215-8-2 : (1982 ),,,, :

592 Vol 36 : H 1 ) A Banach T (), K >, ω > > s, T ( s) Ke ω( s) H 2 ) f : R +, f(, ) =, F (R + R +, R + ) f(, ϕ F (, ϕ ), (, ϕ) R + ϕ 1, ϕ 2, : f(, ϕ 1 ) f(, ϕ 2 ) P (, ϕ 1, ϕ 2 ) ϕ 1 ϕ 2, P (, x, y) >, x, y, g(, ) =, L() [ r, ), g(, u 1 ) g(, u 2 ) L() u 1 u 2 H 3 ) I k : ([ r, ) ), I k (ϕ()) =, I k () =, q k > I k (x 1 ) I k (x 2 ) q k x 1 x 2, q k, Q = Kq k, < L() + Q < 1 k=1 k=1 H 4 ) h(, α) ([ r, ) R +, R + ) α, h() := h(, α), h() 1 δ Ke ω (1 + L())α + 1 Ke ω( s) F (s, h s )ds,, δ (17) h() α, [ r, ], h = sup r θ h( + θ), δ = inf [ r, ) (1 L() Q) 11 u( ) : [ r, ) (14), (15), u() u() = T ()[ϕ() g(, u )] + g(, u ) + u() = ϕ(), [ r, ] 2 T ( s)f(s, u s )ds,, (18) 21 u( ) : [ r, ) (14), (15), (16), u() : u() =T ()[ϕ() g(, u )] + g(, u ) + + T ( i )I i (u( i )),, < i< u() =ϕ(), [ r, ] T ( s)f(s, u s )ds (21) [ r, ], u() = ϕ(), (, 1 ],, u() = T ()[ϕ() g(, u )] + g(, u ) + T ( s)f(s, u s )ds,

No 3 : 593 ( 1, 2 ], 1 u( 1 ) = T ( 1 )[ϕ() g(, u )] + g( 1, u 1 ) + T ( 1 s)f(s, u s )ds + I 1 (u( 1 )), (22) ( 1, 2 ] (14), (22) u() = T ( 1 )[ϕ() g(, u )] + g(, u ) + T ( s)f(s, u s )ds + T ( 1 )I 1 (u( 1 )), (23) ( 2, 3 ], ( 3, 4 ], (21) 22 H 1 ) H 4 ), (14), (15), (16) u(, ϕ) : [ r, ) u() h(, ϕ ), [ r, ) (24) T, { u () () = T ()[ϕ() g(, u )] + g(, u () ), T, u () () = ϕ(), [ r, ], (25) [ r, ] u () () ϕ() K ϕ h(, ϕ ) T, I k (ϕ()) =, u () () = ϕ(), u () () T () ϕ() g(, u ) + g(, u () + < i< Ke ω ( ϕ + L() ϕ ) + L() u () + < i< T ( i ) Ik (u () ( i )) Ke ω( i) q k (u () ( i )) (26) η() = sup{ u () (s) : r s }, T (27) [ r, ] η() = u () ( ), [, ], < L() + Q < 1, δ = inf (1 L() Q) 1 > 1, (26), (27) [ r, ) δ H 3 ) η() Ke ω (1 + L()) ϕ + L()η() + Qη(), 1 η() (1 L() Q) Ke ω (1 + L()) ϕ 1 δ Ke ω (1 + L()) ϕ, (28) u () () 1 δ Ke ω (1 + L()) ϕ h(, ϕ ) (29) [ r, ], η() = ϕ, (28) (29) u () () h(, ϕ ), [ r, T ] u () h, T

594 Vol 36 u (k) () =T ()[ϕ() g(, ϕ())] + g(, u (k) ) + T ( s)f(s, u (k 1) s )ds + T ( i )I i (u (k) ( i )), T, < i< u (k) () = ϕ(), [ r, ], k = 1, 2, k = 1, [, T ], (21) (21) u (1) () T () ϕ() g(, u ) + g(, u (1) g(, )) + T ( s) f(s, () us ) ds + T ( i ) Ii (u (1) ( i )) < i< Ke ω ( ϕ + L() ϕ ) + L() (1) u + Ke ω( s) F (s, u () s )ds + Kq i e ω( i) u (1) ( i ) < i< (211) η() = sup{ u (1) (s) : r s }, T [ r, ] η() = u (1) ( ), [, ], η() Ke ω (1 + L()) ϕ + L()η() + + Kq i e ω( i) η(), < i< u (1) () 1 δ Ke ω (1 + L()) ϕ + 1 δ Ke ω( s) F (s, us () ) ds Ke ω( s) F (s, us () )ds h(, ϕ ) [ r, ], η() = ϕ, [ r, ], u (1) () ϕ() ϕ h(, ϕ ) u (1) () h(, ϕ ), [ r, T ] u (1) h, T k u (k) () h(, ϕ ), [ r, T ] u (k) h, T, (212), T, u (k+1) () T () ϕ() g(, ϕ()) + g(, (k+1) u ) g(, ) + T ( s) f(s, us (k) ) ds + < i< T ( i ) Ii (u (k+1) ( i )) Ke ω ( ϕ + L() ϕ ) + L() (k+1) u + Ke ω( s) F (s, u (k) s )ds + Kq i e ω( i) u (k+1) ( i ) < i< (213)

No 3 : 595 u (k+1) () 1 δ Ke ω (1 + L()) ϕ + 1 δ Ke ω( s) F (s, h s )ds h(, ϕ ) [ r, ], u (k+1) () ϕ() ϕ h(, ϕ ), (212) {u (k) ()} [, T ] T, (25) (21), u (1) () u () () g(, (1) u ) g(, u () ) + + < i< T ( i ) Ii (u (1) ( i )) I i (u () ( i )) T ( s) f(s, us () ) f(s, ) ds L() (1) u u () + Ke ω( s) P (s, () us, ) () us ds + Kq i e ω( i) u (1) ( i ) u () ( i ) < i< (214) ξ() = sup{ u (1) (s) u () (s) : r s }, T, [ r, ] ξ() = u (1) ( ) u () ( ), [, ], [, T ] ξ() L()ξ() + u (1) () u () () 1 δ Ke ω( s) P (s, () us, ) () us ds + Qξ(), 1 δ Ke ω( s) P (s, us (), ) us () ds Ke ω( s) P (s, h s, h s )h s ds (215) [ r, ], η() =, (215) h, P (, h, h ), [, T 1 ], M >, N > h N, KP (, h δ, h ) M (215) u (1) () u () MN, u (1) u () MN, u (2) () u (1) 1 δ u (2) u (1) N M 2 2 2 N M 2 T 2 2! Ke ω( s) P ( s, u (1) s N M 2 T 2 2!, u () s ) u (1) s u () s ds (216)

596 Vol 36, u (k) () u (k 1) () N M k T k, k! u (k) u (k 1) N M k T k k!, { u (k) () } [, T ], u(), lim k u (k) () = u(), u() (14), (15), (16) u() v() (14), (15), (16) v h, v() =T ()[ϕ() g(, v )] + g(, v ) + + T ( i )I i (v( i )),, < i< v() =ϕ(), [ r, ] T ( s)f(s, v s )ds (217) T, (25) (21) u() v() g(, u ) g(, v ) + T ( s) f(s, u s ) f(s, v s ) ds + T ( i ) I i (u( i )) I i (v( i )) < i< L() u v + Ke ω( s) P (s, u s, v s ) u s v s ds + Kq i e ω( i) u( i ) v( i ) < i< (218) ζ() = sup{ u(s) v(s) : r s }, T, ε >, ζ() 1 δ Ke ω( s) P (s, u s, v s )ζ(s)ds M ζ(s)ds + ε Gronwall ζ() εe M εe MT, u() v() εe MT, u() v() 22 3,,,,,,

No 3 : 597 [1] [M] :, 21 [2] [M] :, 1994 [3] Arino O, Benkhali R, Ezzinbi K Exisence resuls for iniial value problems for neural funcional differenial equaions[j] J Diff Equ, 1997, 138(2): 188 193 [4] Wu Huachen, Zhi Hongguan Uniform asympoic sabiliy for perurbed neural delay differenial equaions[j] J Mah Anal Appl, 24, 291(3): 578 595 [5], [J] ( ), 21, 23(4): 488 491 [6] He Mengxing, Liu Anping, Ou Zhuoling Sabiliy for large sysems of parial funcional differenial equaions: ieraive analysis mehod[j] Appl Mah ompu, 22, 132(2): 489 53 [7] He Mengxing Global exisence and sabiliy of soluions for reacion diffusion funcional differenial equaions[j] J Mah Anal Appl, 1996, 199(2): 842 858 [8] He Mengxing, Luo Ronggui Asympoic behavior and convergence of Soluions of a semilinear ranspor equaion wih delay[j] J Mah Anal Appl, 21, 254(1): 464 483 [9] He Lianhua, Liu Anping Periodic soluions of firs-order impulsive differenial equaion[j] J Mah, 212, 32(5): 825 831 [1] Tang iaoping, Li Jingyun, Gao Wenjie Exisence of posiive periodic soluions of an impulsive holling-ii predaor-prey sysem wih ime delay[j] J Mah, 29, 29(6): 761 768 THE EISTENE AND UNIQUENESS OF THE SOLUTION FOR NEUTRAL IMPULSIVE EVOLUTION EQUATIONS WANG iao-mei 1, ZHANG Zhi-qiang 2, ZHU Hua 3 (1Deparmen of Mahemaics and Physics, Miliary Economics Academy, Wuhan 4335, hina) (2ollege of Informaion Engineering, Wuchang Insiue of Technology, Wuhan 4365, hina) (3Deparmen of Basic Sciences, Zhixing ollege, HuBei Universiy, Wuhan 431, hina) Absrac: In his paper, he exisence and uniqueness of soluions are obained By sudying he srucure of ieraive sequence of soluions o a class of nonlinear impulsive evoluion equaions wih ieraive analysis mehod and semigroup heorem, we show ha he exisence and uniqueness of soluions are inseparable linked wih impulsive delay condiion I has cerain superioriy o solve such problems wih ieraive analysis Keywords: impulsive; ieraive; exisence; uniqueness 21 MR Subjec lassificaion: 35A1; 35A2

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