The Solutions of Initial Value Problems for Nonlinear Fourth-Order Impulsive Integro-Differential Equations in Banach Spaces

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1 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo The Soluos of Ial Value Pobles fo Nolea Fouh-Ode Ipulsve Iego-Dffeeal Equaos Baach Spaces Zhag Lglg Y Jgy Lu Juguo Depae of aheacs of Ta Yua Uvesy of Techology Tayua 4 Shax PR Cha zllww@6coyjy-yue@6coqg88qg88@6co Absac:I hs pape we vesgae he axal ad al soluos fo al value poble of fouh ode pulsve dffeeal equaos by usg coe heoy ad he oooe eave ehod o soe exsece esuls of soluo ae obaed As a applcao we gve a exaple o llusae ou esuls Key-wo:Baach space Coe Ial value poble Ipulsve ego-dffeeal equaos Ioduco Ipulsve ego-dffeeal equaos have becoe oe poa ece yeas soe aheacal odels physcs checal echology populao dyacs boechology ad ecoocs Fo a oduco of he basc heoy of pulsve dffeeal equaos R see [] Ipulsve ego-dffeeal equaos boh fo fs ad secod ode have bee suded by ay auhos see [4-]Oly a few papes have pleeed he fouh ode pulsve equaos see [4-5]I [4]he auho use vaao eho ad a hee ccal pos heoe o vesgae pulsve equao whou pulsve dffeeal equales I hs pape by applyg a ew coespodg esul coeced wh fouh-ode pulsve dffeeal equales we apply coe heoy ad he oooe eave ehod o vesgae he axal ad al soluos Cosde he followg al value poble of fouh ode pulsve dffeeal equaos: (4) x () f( x() x () x () x () ( Tx)( )( Sx)( )) J x I ( x ( )) x I ( ( ) ( )) x x () x I ( ( )) x x I ( ( ) ( ) ( ) ( )) x x x x ( ) x( ) x x () x x () x x () x whee J [ a]( a > ) x x x x E θ s he zeo elee of E f C[ J E E E E E E < < < < < < ai C[ EE ] I CE [ EE ] I CEE [ ] I C[ E E E E ( ) ( Tx)( ) ( s) x( s) a ( Sx)( ) h( s) x( s) J CDR [ ] D {( s ) J J s} h CJ [ JR ] R [ ) x x ( ) ( ) x x x ( ) x ( ) x ( ) ad x ( ) deoe he gh ad lef ls of x a especvely Slaly x ( ) ad x ( ) deoe he gh ad lef ls of x a especvely x x ( ) ( ) x x x ( ) ( ) x x ( ) ad x ( ) deoe he gh ad lef ls of x a especvely Slaly x ( ) ad x ( ) deoe he gh ad lef ls of x a especvely Le [ J { x : J E x() s couous a x ( ) exs x ( ) x ( )} Ideed [ J E] s a Baach space wh he o x sup x ( ) J Le [ J E] { x [ J E] x ( ) s couous a x ( ) ad x ( ) exs } Fo x [ J we have E-ISSN: Issue Volue Mach

2 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo x ( ε ) x () x () s < < ε < ε > () Because x ( ) exss hee exss he l x ( ) of () as ε ad x ( ) x () x () s < < I he sae way we oba x ( ) x ( ) x ( ) ε Le x ( ) x ( ) x ( ) x ( ) x ( ) x ( ) Obvously x x x [ J Ideed [ J E ] s a Baach space wh he especve o: x ax{ x x x x } Le couous a Fo [ J E ] { x [ J x ( ) s x ( ) ad x ( ) exs } x [ J slaly x ( ) x ( ) exs Le x ( ) x ( ) x ( ) x ( ) Obvously x x [ J Ideed [ J E ] s a Baach space wh he especve o: x ax{ x x x } Le [ J { x [ J x ( ) s co- uous a x ( ) ad x ( ) exs} Fo x [ J le x ( ) x ( ) Obvously x [ J Ideed [ J E ] s a Baach space wh espec o he o: x ax{ x x } Le J J \{ } a J [ ] J ( ] J ( ] J ( a] τ ax{ } Deoe he o C [ J E ] he space C J [ E ] ad deoe he o ad he [ J [ J space [ J E ] ad J s he closue of [ J E ] especvely J If hee exss x such ha 4 x [ J C [ J ad IVP () he x s called he soluo of IVP () Pelaes Suppose ha E s a eal Baach space whch s paally odeed by a coe P E we say " x y" f ad oly f y x P Moeove P s called oal f hee exss a cosa N > such ha fo all xy E θ x y ples x N y I he case N s called he oaly cosa of P P s called egula f hee exss y E such ha x x x y ples x E such ha x x as Fuhe foao ca be foud [] Lea Assue ha p [ J C [ J sasfes p () M() p() M() p () J p Cp ( ) ( ) p Lp ( ) ( ) () Lp p () p() θ whee M () M () ae bouded wh M M o J ad M M L[ a] C L L ae all oegave cosas ad we have () () C) L) am am e am (( e ) M am M ( e ) ( C ( e )) am M ( e ) ( L (( e ) M M ( ) C( e )) L) am ( ( a C) M) ( L( M > (4) ( ) (5) whee M sup{ M( ) J} M sup{ M( ) J} C he p () θ p () θ J Poof Le P { g E gx ( ) x P} ay g P such ha v ( ) g( p ( )) he v [ J R] C [ J R] ad v () g( p ( )) v ( ) g( p ( )) J By () we have v () M() v () M() v () J v Cv ( ) v Lv ( ) ( )( ) (6) Lv v () v() Pu E-ISSN: Issue Volue Mach

3 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo v () v () ( J) he v [ J R] C [ J R] ad v( ) v() v ( s) v < < (7) < < v() v ( s) C v ( ) J So we have by (6) v ( ) M ( )( v() v ( s) Cv ( )) < < M () v () J v ( () ( ) (8) L v v s Cv ( )) Lv ( ) v () v() ( ) Nex we show v ( ) J (9) We suppose he equaly v ( ) J s o ue Ths eas ha we ca fd J such ha v ( ) > We have he ex wo cases: Case (a): Assue ha ( ]Le J j j j f v ( ) λ The λ () λ By (8) we have v ( ) v The v () s deceasg o [ ] so v ( ) v () Ths s a coadco wh v > ( ) () λ > Thee exss J { } such ha v ( ) λ o v ( ) λ Below we cuss oly he suao whe v ( ) λ (The poof s sla whe v ( ) λ )We oba by (8) v ( ) M( ac ) λ Mλ M λ [ ] () ( ) λ λ v L C L whee () M M ( a C ) M () The we have v ( ) v ( j) v ( ξ j)( j) j < ξ j < v ( j) v ( j ) v ( ξ j )( j j ) j < ξ j < j v ( ) v ( ) v ( ξ )( ) () < ξ < v ( ) v ( ) v ( ξ)( ) < ξ < By () we ow v ( ) v ( ) v v ( ) L ( C ) λ L λ (4) Cobg () () ad ()(4) hs yel j v ( ) v ( j) Lj( j C) λ Ljλ λm( j ) v v L C L j ( j) ( j ) j ( j ) λ j λm( j j ) v ( ) v ( ) L ( C ) λ L λ λm( ) v ( ) λ λm( ) Addg hose equaleswe have λ < v ( ) λ j λ L ( λm ( ) C ) λ λ L ( C ) j L λ (5) L λma (6) λ Ths eas ha < L( C) L Ma (7) Ths s a coadco wh (4) Case (b): whe () sasfes pug by (8) we have w () v() e M ( s ) E-ISSN: Issue Volue Mach

4 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo M ( s) ( ) M d s w ( ) M( )( v()( e ( e ) w( s) M ( s) ( C e ) w ( ) J < < w L v e M ( ()( ( s ) ) M ( ) d s ( e ) w() s M ( ) d C ( e ) w ( )) Lw ( ) w() v() ( ) I he sae way we have w ( ) Hece v ( ) I eas ha v ( ) J Ths yel < < v( ) v() v ( s) Cv( ) J Moeove fo ay g P we have p () θ p () θ J Ths e he poof Lea [] Le [ J R ] C [ D ] β ( ) s cosa ad () ( ) ( ) ss < < The ( ) Lea [] If H [ J R β ( ) J s a bouded ad couable se he we have ( H ( )) LJ [ R] ad α({ a a x () α d: x H}) α( H ( )) d Lea 4 [] Assue ha H [ J s bouded se ad he fucos belogg o H ae equcouy o J ( ) α ( H) ax{sup α( H ( ))sup α( H ( ))} J J whee α s a easue of ocopacess [ J E ] I ode o sudy he fouh-ode pulsve ego-dffeeal equaos we sudy he secodode pulsve dffeeal equaos fsly by ehod of he educo of ode Soe esuls of he secod ode pulsve dffeeal equaos We vesgae he followg secod ode pulsve dffeeal equaos: u ( ) f ( ( Bu)( )( Fu)( ) u( ) u ( ) ( TBu)( )( SBu)( )) J u I ( ( )) (8) u u I (( )( )( )( ) ( ) ( )) Bu Fu u u ( ) u() x u () x whee J [ a]( a > ) f C[ J E E E E E E < < < < < < a I C [ EE ] I C[ E E E E ( ) x x E ( Tu)( ) ( s) u( s) ( Su)( ) a h ( s ) u ( s ) J CDR [ ] D {( s ) J J s} h CJ [ JR ] R [ ) u u ( ) u ( ) u u ( ) u ( ) u ( ) ad u ( ) deoe he gh ad lef ls of u a especvely Slaly u ( ) ad u ( ) deoe he gh ad lef ls of u a especvely Defe wo opeaos B ad F B : [ J C [ J [ J C 4 [ J F : [ J C [ J [ J C [ J They ae couous ad ceasg opeaos Assue ha he followg codos ae sasfed: ( H) Thee exs u v [ J C [ J such ha u() v() u () v () J ad u ( ) f( ( Bu)( )( Fu)( ) u( ) u ( ) ( TBu)( )( SBu)( )) J u I ( ( )) (9) u u I (( )( )( )( ) ( ) ( )) Bu Fu u u ( ) u() x u () u() x x v ( ) f( ( Bv)( )( Fv)( ) v( ) v ( ) ( TBv)( )( SBv)( )) J v I ( ( )) () v v I (( )( )( )( ) ( ) ( )) Bv Fv v v ( ) v() x v () v() x x E-ISSN: Issue Volue Mach

5 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo ( H ) Thee exs M ( ) M () ae bouded wh M M o J ad M M L[ a] C L L ( ) such ha fxyzuvw ( ) fxyzuvw ( ) M ( )( z z) M ( )( u u) J I ( u) I ( u) C ( u u) I ( x y z u) I ( x y z u) L ( z z) L ( u u) ( Bu )( ) x x ( Bv )( ) ( Fu)( ) y y ( Fv)( ) u() z z v() u () u u v () ( TBu )( ) v v ( TBv )( ) ( SBu )( ) w w ( SBv )( ) ( H ) Fo ay > hee exs d d ( ) ( ) ad b a ( ) such ha α ( f( J U U U U U U )) d α( U ) d α( U ) 4 U B ( 456) α( I ( V V V V )) b ax{ α( V ) α( V )} ( ) 4 4 V B ( j 4) ( ) j α( I ( V )) a ( V ) ( ) ( ) 4 α 4 V4 B whee B { u E u } α s he easue of ocopacess E wh he Kuaows popey Deoe [ u v] { u [ J u() u() v() u () u () v () J} Theoe Suppose E s a eal Baach space P s a oal coe B ad F ae bouded opeaos ad ( H) - ( H ) hold assue ha (4) o (5) s sasfed The hee exs oooe sequeces { u }{ v } [ J C [ J ae ufo covegece a u v [ J C [ J whee u s a al soluo ad v s a axal soluo of (8) o [ u v ] ad { u }{ v } ae covege a ( u ) ( v ) especvely ad u u u u u () () () () () () v () v () v () v () J u () u () u () ( u )() u () ( v )() v () v () v () J Poof Fo ay η [ u v] we cosde he soluo of lea pulsve dffeeal equao of ype u () M() u () M() u () σ () J u I ( ( )) ( ( ) ( )) () η Cu η u I (( )( )( )( ) ( ) ( )) Bη Fη η η L( u( ) η( )) L( u ( ) η ( )) ( ) u() x u () x whee σ( ) f( ( Bη)( ) ( Fη)( ) η( ) η ( )( TBη)( ) ( SBη )( )) M () η () M ( ) ( )) η Obvously u [ J C [ J s a soluo of () f ad oly f u [ J ad σ u() x x ( s)( () s M ()() s u s ( ( η ( )) M () s u ()) s I < < C ( u ( ) η ( ))) ( ) < < ( I (( Bη)( )( Fη)( ) η( ) η ( )) L( u ( ) η( )) L ( u ( ) η ( ))) () Nex we show ha u s a uque soluo of IVP ()Le f ( uu ) σ () M () u () M () u () J Fsly we cosde he followg lea dffeeal equao: u ( ) f ( uu ) J (4) u() x u () x I s easy o pove ha u C [ J s a soluo of (4) f ad oly f u C [ J σ u() x x ( s)( () s M ()() s u s M () s u ()) s Le ( Au )( ) x x ( s)( σ ( s) M()() sus The Fo ay M () s u ()) s (5) ( Au)() x ( σ () s M()() su s M () s u ()) s (6) [ ] uv C J E by (5) ad (6) we have E-ISSN: Issue Volue Mach

6 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo ( Au)( ) ( Av)( ) ( ) s ( M us ( ) vs ( ) M u ( s ) v ( s ) ) τ ( M us ( ) vs ( ) M u ( s ) v ( s ) ) ( τ )( M M ) u v J C [ J ( Au) ( ) ( Av) ( ) ( M u( s) v( s) M u ( s) v ( s) ) ( τ )( M M ) u v J C [ J ( Au)( ) ( Av)( ) τ ( M ( Au)( s) ( Av)( s) M ( A u) ( s) ( A v) ( s) ) ( τ ) ( ) ( ) ( Au) ( ) ( Av) ( ) M M u v C [ J J ( τ ) ( M M) ( ) u v C J [ J Hece ( Au)( ) ( Av)( ) (7) ( τ ) ( M M)( ) u v! C [ J E J ] ( Au) ( ) ( Av) ( ) (8) ( τ ) ( M M)( ) u v! C [ J E J ] ad ( Au) ( Av) (9) ( τ ) C [ J Thee exss τ ( M )( )! N such ha M u v C [ J E J ] τ ( τ ) ( M M) ( ) < ()! So by (9) () ad he Baach fxed po heoe he A has a uque fxed po w C [ J I eas ha w C [ J s a uque soluo of he (4) such ha w ( ) f ( w w ) J w() x w () x I he followg we cosde () u f ( uu ) J u( ) I( η ( )) C( w ( ) η ( )) w( ) u ( ) I(( Bη)( )( Fη)( ) η( ) η ( )) () L( w( ) η( )) L( w ( ) η ( )) w ( ) I s easy o pove ha u [ J C [( ) E ] s a soluo of () f ad oly f u [ J such ha u ( ) I( η ( )) C( w ( ) η ( )) w( ) ( ) ( I(( Bη )( ) ( Fη)( ) η( ) η ( )) L( w( ) η( )) L( w ( ) η ( )) w ( )) ( s)( σ () s M()() sus M () s u ()) s Pu ( Au)( ) I ( η ( )) C( w( ) w( ) ( )( I(( Bη )( ) ( Fη)( ) η( ) η ( )) L ( w ( ) η( )) η ( )) L ( w ( ) η ( )) w ( )) ( s)( σ () s M()() sus M () s u ()) s J () The fo ay J we have ( Au )( ) ( I (( Bη )( ) ( Fη)( ) η ( ) ( )) L ( w ( ) η( )) Obvously Fo ay η L ( w ( ) η ( )) w ( )) ( σ () s M()() sus M A : [ J [ J u v [ J ehod used (9) we oba ( Au) ( Av) [ J () s u ()) s usg he sla ( τ ) τ ( M M ) ( ) u v (4) [ J! By () (4) ad he Baach fxed po heoe A has a uque fxed po w [ J I eas ha soluo o () such ha w [ J s a uque E-ISSN: Issue Volue Mach

7 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo w f ( w w ) J w( ) I( η ( )) C( w ( ) η ( )) w( ) w ( ) I(( Bη)( )( Fη)( ) η( ) η ( )) L( w( ) η( )) (5) L( w ( ) η ( )) w ( ) Aga we wa o pove ha lea dffeeal equao fo ay ( ) u f ( uu ) J u( ) I ( η ( )) C( w ( ) η ( )) w ( ) u ( ) I (( Bη)( )( Fη)( ) η( ) η ( )) L( w ( ) η( )) L( w ( ) η ( )) w ( ) has a uque soluo w [ J C [( ) such ha w f ( w w ) J w( ) I ( η ( )) C( w ( ) η ( )) w ( ) w ( ) I(( Bη)( )( Fη)( ) η( ) η ( )) (6) L( w ( ) η( )) L( w ( ) η ( )) w ( ) Le w( ) J w( ) J uη () (7) w( ) J Cobg () ad (5) (6) (7) we have u η [ J C [ J s a uque soluo of IVP () Pug u Aη The η A u v J E C J E :[ ] [ ] [ ] Nex we pove wo cases: Case (): u Au u ( Au) Av v ( Av ) v Case (): f η η [ u v] ad η η η η he Aη Aη( Aη) ( Aη) Fs cosde case () u Au p u u By () we have Pu u () M() u() M() u () M() u() M() u () f ( ( Bu)( )( Fu)( ) u u TBu SBu J ( ) ( )( )( )( )( )) u I ( u ( )) C( u ( ) u ( )) u I (( Bu )( )( Fu )( ) u ( ) u ( ) u ( )) L ( u ( ) u ( )) L ( u ( ) u ( )) ( ) u () x u () x Moeove by ( H ) we have p () u () u () M() p() M() p () J p u ( ) u Cp p u u Lp ( ) Lp ( )( ) p () u () u () u () x u() x p() θ Hece by Lea we oba p () θ p () θ J Ths eas ha u Au u ( Au ) I he sae way Av v( Av) v Nex cosde case (): Le η η [ u v] such ha η η η η ad pu p λ λ whee λ Aη λ Aη Cobg () ad ( H ) we have p () λ () λ () M() p() M() p () ( f( ( Bη)( )( Fη)( ) η( ) η ( ) ( TBη)( )( SBη)( )) f( ( Bη)( ) ( Fη)( ) η( ) η ( )( TBη)( )( SBη)( )) M( )( η() η( )) M( )( η ( ) η ( ))) M () p() M () p () J p λ ( ( )) λ I η C( λ ( ) η ( )) I ( η ( )) C( λ ( ) η ( )) Cp ( ) p λ λ I (( Bη)( )( Fη)( ) η( ) η ( )) L( λ( ) η( )) L( λ ( ) η ( )) I (( Bη)( )( Fη)( ) η( ) η ( )) L( λ( ) η( )) L( λ ( ) η ( )) Lp ( ) Lp ( )( ) p () p() θ ( 8) E-ISSN: 4-88 Issue Volue Mach

8 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo Moeove by Lea we oba p () θ p () θ J hs eas ha ( Aη )() ( Aη )()( Aη )() ( Aη )() Le u Au v Av ( ) (9) By Case () ad Case () we have u () u () u () v () v() v() J u () u () u () v () v () v () J (4) Le U { u } U { u } U () { u() } U () { u ( ) } J By oaly of P ad (4) he UU ae boh bouded ses [ J E ] Fo ay η [ u v] cobg ( H ) ad ( H ) we have u () M () u () M () u () f ( ( Bu )( )( Fu )( ) u ( ) u ( )( TBu )( ) SBu M() u() M() u () ( )( )) f( ( Bη)( )( Fη)( ) η( ) η ( )( TBη)( ) ( SBη)()) M () η() M () η () f ( ( Bv)( )( Fv)( ) v( ) v ( )( TBv)( ) ( SBv)()) M() v() M() v () v () M () v () M () v () (4) Moeove we oba { f ( Bη Fηηη TBη SBη) M () η M () η η [ u v ]} s a bouded se Hece hee exss a cosa γ > such ha f ( ( Bu )( )( Fu )( ) u ( ) u ( ) (4) ( TBu )( )( SBu )( )) M ( )( u ( ) u ( )) M()( u () u ()) γ J ( ) ad { σ } s a bouded se [ J E ] whee ( ) f ( ( Bu )( )( Fu )( ) u ( ) u ( ) σ ( TBu )( )( SBu )( )) M() u () M() u () By he defo of u () ad () we have u ( ) x x ( s)( f ( s( Bu )( s) ( Fu )( s) u ( s) u ( s)( TBu )( s) ( SBu )()) s M () s u () s M () su () s M() su() s M ( s) u ( s)) ( I( u ( )) < < C( u ( ) u ( ))) (4) ( )( I (( Bu )( ) < < ( Fu )( ) u ( ) u ( )) L ( u ( ) u ( )) L( u ( ) u ( ))) J ( ) The we have u ( ) x ( f ( s( Bu )( s)( Fu )( s) u ( s) u ( s)( TBu )( s)( SBu )( s)) M() su () s M () su () s M su() s < < M () s u ()) s ( I (( Bu )( )( Fu )( ) (44) u ( ) u ( )) L ( u ( ) u ( )) L( u ( ) u ( ))) J ( ) By (4) (4) ad (44) he fucos belogg o UU ae equv-couy o J ( ) So by Lea 4 we have J { } α ( U) ax sup α( U ( ))sup α( U( )) J J By ( H ) hee exs cosas d d ad ( ) ( ) b a ( ) such ha α ( f ( ( BU )( )( FU )( ) U ( ) U ( ) ( TBU )( )( SBU )( ))) dα( U ( )) dα ( U ( )) J (45) α ( I (( BU )( )( FU )( ) U ( ) U ( ))) ( b ) ax{ α( U ( )) α ( U ( ))} ( ) (46) ( ) α( I ( ( ))) U a α( U ( )) ( ) (47) Hece fo ay J cobg (4) (45) (46) (47) ad Lea we have α( U ( )) a ( α( f ( s( BU )( s)( FU )( s) U ( s) U ( s)( TBU )( s)( SBU )( s))) M α( U( s)) M α ( U ( s))) E-ISSN: 4-88 Issue Volue Mach

9 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo ( ) ( a α( U ( )) Cα( U ( ))) < < ( ) ( ab ax{ α( U ( )) α U < < ( ( ))} al α( U ( )) al α ( U ( ))) a ( dα( U( s)) d α ( U ( s)) M α( U ( s)) M α ( U ( s))) ( ) ( a α( U ( )) Cα( U ( ))) < < ( ) ( ab ax{ α( U ( )) α( U ( ))} < < alα( U ( )) alα ( U ( ))) (48) ( U ( )) ( d ( U( s)) d ( U ( s)) α α α Mα( U( s)) Mα ( U ( s))) ( ) ( b ax{ α( U ( )) α( U ( ))} < < Lα( U( )) Lα ( U ( ))) (49) Le ( ) ax{ α( U( )) α ( U ( ))} Because he fucos belogg o UU ae equv-couy o J ( ) ad UU ae bouded we have ( ) [ J () Cobg (48) ad (49) we oba ( ) ( a )( d d M M ) ( s) (5) ( ) ( ) ( a C ( a )( b L L)) ( ) < < Moeove by Lea we have ( ) hs eas ( ) J oeove α( U ( )) α ( U ( )) J Ths yel ha U possesses he elavely copacess [ J E ] U possesses he elavely copacess [ J E ]Hece by (4) ad he oaly of u [ J { } P { u } u s covege a s covege a ( u ) ad u u u ( u ) (5) Because f s couous by he defo of σ ad (5) we have σ σ ( ) (5) whee σ ( ) f ( ( Bu )( )( Fu )( ) u ( ) ( u ) ( )( TBu )( )( SBu )( )) M() u() M()( u)() By (4) (5) (5) ad Lebesgue cool covege heoe we have l u u ( ) l u ( u ) '( ) Moeove u ( ) x x ( s) f ( s( Bu )( s) ( SBu )( s)) I u ( Fu )( s) u ( s)( u ) ( s)( TBu )( s) (( ) ( )) < < ( ) I (( Bu )( )( Fu )( ) < < u ( )( u )( )) J ( u ) ( ) x f ( s( Bu )( s)( Fu )( s) u ( s) ( u )( s)( TBu )( s)( SBu )( s)) I(( Bu )( )( Fu )( ) < < u ( )( u )( )) J I s easy o pove ha u [ J C [ J s a soluo of IVP (8)I he sae way hee exss v [ J C [ J such ha v v v ( v ) v s a soluo of IVP (8)ad by (4)we have u() u() u() u () v () v () v () v () J (5) () () () ( )() ( )() u u u u v v () v () v () J u [ J C [ J Fo s ay soluo of IVP (8) o [ u v ] he u() u () v() u () u () v () J Assue ha u () u () v () u () u () v ( ) J Le p () u() u () By () (9) ad ( H ) we have p () M() p() M() p () ( f ( ( Bu)( )( Fu)( ) u( ) u ( ) ( TBu)( )( SBu)( )) f ( ( Bu )( )( Fu )( ) u ( ) u ( )( TBu )( )( SBu )( )) M()( u () u ()) M ( )( u ( ) u ( ))) E-ISSN: 4-88 Issue Volue Mach

10 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo M() p() M() p () J p I ( u ( )) C ( u ( ) u ( )) I ( ( )) u Cp ( ) p I (( Bu )( )( Fu )( ) u ( ) u ( )) L ( u ( ) u ( )) L( u ( ) u ( )) I (( Bu)( )( Fu)( ) u( ) u ( )) Lp ( ) Lp ( ) ( ) p () p() θ By Lea we have p () θ p () θ J Moeove u () u() u () u () J I he sae way we ca show ha u() v () u () v () J Hece we oba u() u() v() u () u () v () J ( ) (54) Now f fo ay J u () u() v ()( u )() u () ( v )() By (5) he () hol Ths e he poof Theoe Suppose E s a eal Baach space P s a egula coe ad ( H)( H ) hold Assue (4) o (5) s sasfed he () hol Poof The poof s sla o he poof of Theoe he oly dffeece s ha we vefy elave copacess of UU ' ad he egulay of P by (4) sead of H Ths e he poof Coollay If E s a wea sequeally coplee Baach space P s a oal coe H Hhold ad (4) o (5) s sasfed he () hol Poof If E s a wea sequeally coplee Baach space he oaly of P s equvale o he egulay Hece () hol by Theoe Ths e he poof Rea f s elave o opeaos BF To y owledge all papes coeced wh he secod ode pulsve ego-dffeeal equao has bee o vesgaed hs suao so IVP (8) s a ew poble Rea BFelave o Theoe ae bouded ad couous opeaos howeve BF elave o Theoe ae couous ad ceasg 4 Soe esuls of he fou ode pulsve dffeeal equaos Le us ls he followg assupos fo coveece: ( G ) Thee exs 4 y z [ J C [ J such ha y() z() y () z () y () z () y () z () J (4) y () f( y() y () y () y () ( Ty)( )( Sy)( )) J y I ( ( )) y y I ( y ( ) ( )) y (55) y I ( ( ))( ) y y I ( ( ) ( ) ( y y y ) y ( )) y() x y () x y () x y () y () x x (4) z ( ) f( z( ) z ( ) z ( ) z ( )( Tz)( ) ( Sz)( )) J z I ( ( )) z z I ( z ( ) ( )) z (56) z I ( ( ))( ) z z I ( ( ) ( ) z z z ( ) z ( )) z() x z () x z () x z () z () x x ( G ) Thee exs 4 y z [ J C [ J such ha y() z() y () z () y () z () y () z () J (4) y () f( y() y () y () y () ( Ty)( )( Sy)( )) J y I ( ( )) y y I ( y ( ) ( )) y (57) y I ( ( )) y y I ( ( ) ( ) ( ) ( y y y y )) y() x y () x y () x y () y () x x E-ISSN: 4-88 Issue Volue Mach

11 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo (4) z ( ) f( z( ) z ( ) z ( ) z ( )( Tz)( ) ( Sz)( )) J z I ( ( )) z z I ( z ( ) ( )) z z I ( ( )) z (58) z I ( ( ) ( ) ( ) ( z z z z )) ( ) z() x z () x z () x z () z () x x ( G ) Thee exs M() M() ae bouded wh M M o J ad M M L[ a] L L ( ) ae all C oegave cosas such ha fxyzuvw ( ) fxyzuvw ( ) M ( )( z z) M ( )( u u) J I ( z) I ( z) I ( y z) I ( y z) I ( u) I ( u) C ( u u) I ( x y z u) I ( x y z u) L( z z) L( u u) ( ) whee y() x x z() y () y y z () y () z z z () y () u u z () ( Ty )( ) v v ( Tz )( ) ( Sy )( ) w w ( Sz )( ) J ( G ) Thee exs b a b ( ) such ha I ( z) I ( z) b z z I ( y z) I ( y z) a y y b z z yzyz E ( ) Deoe [ y z] { y [ J y() y() z() y () y () z () y () y () z () y () y () z () J} Theoe 4 Suppose E s a eal Baach space P s oal coe ad ( G)( G)( G)( H ) hold Assue (4) o (5) s sasfed he IVP () has he axal ad al soluos 4 y z [ J C [ J o [ y z ] Poof Cosde IVP () Le x () u () J The we have x () u () J u ( ) f( x( ) x ( ) u( ) u ( )( Tx)( )( Sx)( )) J x I ( ( )) u x I ( ( ) ( )) (59) x u u I ( ( )) u u I ( ( ) ( ) ( ) ( )) x x u u ( ) x() x x () x u() x u () x Fo ay u [ J we have x () u () J x ( ( )) I u (6) x I ( x ( ) ( ))( ) u x() x x () x Obvously f x [ J C [ J s a soluo of (6) f ad oly f x( ) x x ( s) u( s) I( u( )) < < ( ) I ( x ( ) u ( )) (6) < < ad x ( ) x u( s) I ( x ( ) u( )) (6) < < Le x( ) ( Bu)( ) J (6) x ( ) ( Fu)( ) J (64) The defe wo opeaos BF B J E J E C J E : [ ] [ ] [ ] F: [ J [ J C [ J Nex we show ha () B s bouded ad couous Whe 4 fo ay y y [ J by (6)(6) we have ( By )( ) ( By )( ) ( s) y ( s) y ( s) << I ( y ( )) I ( y ( )) < < ( ) I (( Fy )( ) y ( )) E-ISSN: Issue Volue Mach

12 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo I (( Fy )( ) y ( )) a y y b y y (65) a a ( Fy )( ) ( Fy )( ) a b y y a y y b y y a b y y a( a ( Fy )( ) ( Fy )( ) a ( Fy )( ) ( Fy )( ) ) a y y b y y a b y y a(( a ) a ( Fy )( ) ( Fy )( ) a y y a b y y ) a y y b y y (66) a b y y aa ( b ( a ( ( a )) b) j j a a ( a )) y y j j Hece By By N y y whee a N b ab aa ( b ( a ( ( a )) b ) j j a a ( a )) j j I he sae way (67) whee So ( By ) ( By ) N y y ( (68) N a b a b ( a ( ( a )) b j j a a ( a )) j j By By N y y ax{ } whee N N N Hece B s bouded ad couous Whe he poof s sla () B s ceasg Fo ay y y [ J y y by ( G ) ad (6) we have ( By )( ) ( By )( ) ( )( ( ) ( )) s y s y s θ J The ( By )( ) ( By )( ) J I pacula ( By)( ) ( By)( ) Moeove fo ay J we have < < < < ( I ( y ( )) I ( y ( ))) ( )( I (( By )( ) y ( )) I (( By )( ) y ( ))) (69) I ( y ( )) I ( y ( )) ( )( I (( By )( ) y ( )) I (( By )( ) y ( ))) θ The ( By )( ) ( By )( ) J I pacula ( By)( ) ( By)( ) I he sae way we have ( By)( ) ( By)( ) J ( By )( ) ( By )( )( ) Hece ( By )( ) ( By )( ) J he By By I he sae way F s a bouded couous opeao wh ceasg Cobg (6) ad (6) s easy o show f y [ J C [ J E-ISSN: Issue Volue Mach

13 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo 4 he By [ J C [ J ad f y [ J C [ J he Fy [ J C [ J I he sae way we ca show 4 B: [ J C [ J [ J C [ J F: [ J C [ J [ J C [ J They ae all bouded couous opeaos wh ceasg Hece by (59)-(64) IVP () s equvale o IVP (8) Obvously f u [ J C [ J s a soluo of IVP (8) he x() [ J C 4 [ J s a soluo of IVP () by(6) Pug u() y () v() z () J we have u v By ( G ) we oba y () x x ( s) u ( s) I ( u ( )) (7) < < < < ( ) I ( y ( ) u ( )) J z () x x ( s) v ( s) I ( v ( )) (7) < < < < ( ) I ( z ( ) v ( )) J y () x u ( s) I ( y ( ) u( )) J (7) < < I ( z ( ) v( )) J (7) < < z () x v ( s) he y ( ) ( Bu )( ) z ( ) ( Bv )( ) y ( ) ( Fu )( ) z ( ) ( Fv )( ) J whee u v sasfy ( H ) By ( G ) s easy o ow ha ( H ) hol Hece applyg Theoe hee exs he axal ad al soluos u v [ J C [ J of IVP (8) o [ u v ] Le y Bu z Bv The ad [ ] [ ] 4 y z J E C J E y () x x ( s) u () s I (74) ( u ( )) < < ( ) I (( y )( ) u ( )) < < J By (74) we have ( y)() u() J y ( ( )) (75) I u ( y ) I (( ) ( ) ( ))( ) y u y () x( y )() x If hee exs u such ha (8) ad y such ha (75) he y s a soluo of IVP ()I he sae way z s a soluo of IVP ()I s easy o vefy 4 y z [ J C [ J ae he axal ad al soluos of IVP of () o [ y z ] especvely Ths e he poof Theoe 4 Suppose E s a eal Baach space P s a egula coe ad ( G)( G)( G ) hold Assue (4) o (5) s sasfed he hee exs he axal ad al soluos y z [ J C 4 [ J of IVP () o [ y z ] especvely Poof The poof s sla o he poof of Theoe 4If Theoe sasfes he hee exs u v [ J C [ J he axal ad al soluos of IVP () especvely Ths e he poof Coollay 4 If E s a wea sequeally coplee Baach space P s a oal coe ( G )( G )( G ) hold ad (4) o (5) s sasfed he IVP () has he axal ad al soluos 4 y z [ J C [ J o [ y z ] Poof If E s a wea sequeally coplee Baach space he oaly of P s equvale o he egulay of P Hece he cocluso of Coollay4 hol by Theoe 4Ths e he poof Theoe 4 Suppose E s a eal Baach space P s egula coe ad ( G )( G)( G)( H) hold Assue (4) o (5) s sasfed If fo ay zu E f( xyzuvw ) f( x yzuv w) x x y yv vw w (76) he IVP () has he axal ad al soluos E-ISSN: Issue Volue Mach

14 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo 4 y z [ J C [ J o [ y z ] Poof Sla o he poof of Theoe 4we cosde IVP ()Le x () u () J he x( ) ( Bu)( ) x ( ) ( Fu)( ) J Hece IVP () s equvale o IVP (8)Le u () y () v () z () J (77) The u v Cobg (77) ad ( G ) fo ay J we have y ( ) y () y () ( s) u ( s) y ( ) ( ) y ( ) < < < < y ( ) y () u ( s) y ( ) J < < z ( ) z () z () ( s) v ( s) z ( ) ( ) z ( ) < < < < z ( ) z () v ( s) z ( ) < < I s easy o vefy y ( ) ( Bu )( ) y ( ) ( Fu )( ) ( Bv )( ) z ( )( Fv )( ) z ( ) J I pacula y ( ) ( Bu )( ) y ( ) ( Fu )( ) ( Bv)( ) z( )( Fv)( ) z ( ) Moeove we have fo ay ( ) y ( ) ( Bu )( ) y ( ) ( Fu )( ) ( Bv)( ) z( )( Fv)( ) z ( ) J y ( ) ( Bu )( ) y ( ) ( Fu )( ) ( Bv )( ) z ( )( Fv )( ) z ( ) So we have y Bu y Fu Bv z Fv z Hece by ( G ) we ow ( H ) hol Sla o he poof Theoe 4we oba he cocluso Ths e he poof Theoe 44 Suppose E s a eal Baach space P s egula coe ( G )( G)( G) hold Assue (4) o (5) s sasfed If fo ay zu E (76) hol he IVP () has he axal ad al soluos 4 y z [ J C [ J [ y z ] o Poof Sla o Theoe 4 s easy o ow ( H ) hol The he es of he poof s sla o he poof of Theoe 4Ths e of he poof Coollay 4 If E s a wea sequeally coplee Baach space P s a oal coe ( G ) ( G ) ( G ) hold Assue (4) o (5) s sasfed If fo ay zu E (76) hol he IVP () has he axal ad al soluos 4 y z [ J C [ J [ y z ] o Poof If E s a wea sequeally coplee Baach space he oaly of P s equvale o he egulay of P Hece he cocluso of Coollay4 hol by Theoe 44Ths e he poof Rea 4 I Theoe ad Theoe 4 Theoe 44 he codo ( H ) s oe easy o use ad vefy 5 Applcao Exaple 5 Cosde he followg al value poble fo fouh-ode pulsve egodffeeal equaos: (4) x () ( x ()) ( x ()) 4 ( x ( )) ( x ( )) 8 9 ( s e x () ) 6 s x ( s) s x x ( ) 5( ) x x ( ) x ( ) (78) x x ( ) 4 x x( ) x ( ) x ( ) 4 5 x ( ) ( ) 8 x() x () x () x () Cocluso IVP (78) has he axal ad al 4 soluos belogg o C o [ ) ( )] such ha E-ISSN: Issue Volue Mach

15 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo 4 [ ] 4 x () 4 ( ] ( ) [ ] 6 x () ( ] ( ) [ ] x () ( ] ( ) [ ] x () ( ] ( ) Poof Le E c { x ( x x x ): x } wh he o x sup x P { x ( x x x ) c : x } The P s a oal coe E ad (78) s a al value poble E whee s a s ( ) e hs ( ) s x x x x ( ) x ( x x x ) y ( y y y ) z ( z z z ) u ( u u u ) v ( v v v ) w ( w w w ) f ( f f f ) ad f( xyzuvw ) ( x ) ( y) 4 ( z) ( u) 8 9 ( v) w 6 (79) I ( I I I ) I ( I I I ) I ( I I I ) I ( I I I ) whee I( z) z 5( ) I ( yz ) y z I ( u) u 4 I( xyzu ) x y z u Le J [] obvously f C[ J E E E E E E Le y ( ) ( ) [] ( ) [ ] z() ( ) ( ] 4 We have y ( ) ( ) [] y ( ) ( ) [] y ( ) ( ) [] (4) y ( ) ( ) [] ( ) [ ] 6 6 z () ( ) ( ] 6 ( ) [ ] 4 z () ( ) ( ] ( ) [ ] z () ( ) ( ] ( ) [ ] (4) z () ( ) ( ] 4 Hece we have y z [ J C [ J y () z () y () z () y () z () J ad y () z () ( ) x y () z () ( ) x y () z () ( ) x y () z () ( ) x E-ISSN: Issue Volue Mach

16 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo f ( y ( ) y ( ) y ( ) y ( )( Ty )( )( Sy )( )) [] whe f ( z ( ) z ( ) z ( ) z ( )( Tz )( )( Sz )( )) 4 ( ) ( ) s s ( ) whe < f ( z ( ) z ( ) z ( ) z ( )( Tz )( )( Sz )( )) 4 ( ) ( ) ( ) s s ( ) ( ) y ( ) I( y ( )) y ( ) I( y ( ) y ( )) y ( ) I( y ( )) y ( ) I( y( ) y ( ) y ( ) y ( )) z ( ) I( z ( )) z ( ) I( z ( ) z ( )) z ( ) I( z ( )) z ( ) 4 I( z( ) z ( ) z ( ) z ( )) so ( G ) s sasfed O he ohe had fo ay J y() x x z() y () y y z () y () z z z () y () u u z () Ty() v v Tz() Sy() w w Sz() we have fxyzuvw ( ) fxyzuvw ( ) ( x x ) ( y y) ( z z) 4 8 ( u ) ( ) ( ) 9 u v v w w 6 ( z z) ( u u)( ) 8 9 I ( z) I ( z) I ( y z) I ( y z) I( u) I( u) ( u u) 4 I ( x y z u) I ( x y z u) ( z z) ( u u) 5 8 so ( G ) s sasfed whee M() M() C L L I s easy o vefy (4) hol Obvously fo ay yzyz E we have I( z) I( z) z z I( y z) I( y z) y y z z so ( G ) s sasfed By (79) he () () () () () () f f f f ( f f f ) () () () () f ( f f f ) whee f () ( xyzuvw ) ( x ) ( y) 4 ( z) ( v) w (8) 8 6 f () ( xyzuvw ) ( u) (8) 9 ( b) Fo ay > assue { } J b ( b) { b) ( b) ( b) ( b) { x } { y } { z } { u } { v } b b b b b { } ( b) w B b whee ( ) { ) ( ) ( ) x b ( x b x b x b ) ( ) ( ) ( ) ( ) y b ( y b y b y b ) E-ISSN: Issue Volue Mach

17 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo ( ) ( ) ( ) ( ) z b ( z b z b z b ) ( ) ( ) ( ) ( ) u b ( u b u b u b ) ( b) ( b) ( b) ( b) v ( v v v ) ( ) ( ) ( ) ( ) w b ( w b w b w b ) By (8) we have () ( b) ( b) ( b) ( b) ( b) ( b) ( b) f ( x y z u v w ) ( b) ( b) ( x ) ( y ) 4 ( b) ( b) ( b) ( z ) ( v ) w 8 6 ( ) ( ) ( ) 4 8 ( ) b ( ) 6 (8) So () ( ) ( ) ( ) ( ) ( ) ( ) ( ) { f ( b x b y b z b u b v b w b )} s bouded oeove we choose subsequece { b } {} b such ha () ( ) ( ) ( ) ( ) ( ) ( ) ( ) f ( b b b b b b b x y z u v w ) ζ ( ) (8) Cobg (8) ad (8) we have ζ ( ) ( ) ( ) 4 8 ( ) ( ) 6 (84) ζ ( ζ ζ ζ ) c E Fo ay ε > so by (8) ad (84)hee exss a posve ege such ha () ( ) ( ) ( ) ( ) ( ) ( ) ( ) f ( b x b y b z b u b v b w b ) < ε ζ < ε > ( ) (85) By (8) hee exss a posve ege such ha () ( ) ( ) ( ) ( ) ( ) ( ) ( ) f ( b x b y b z b u b v b w b ) ζ < ε > ( ) (86) The cobg (85) ad (86)we have () ( ) ( ) ( ) ( ) ( ) ( ) ( ) f ( b x b y b z b u b v b w b ) ζ () ( b ) ( ) ( ) ( ) ( ) ( ) ( ) sup ( b b b j b b b f x y z u v w ) ζ ε > Hece () ( ) ( ) ( ) ( ) ( ) ( ) ( ) f ( b x b y b z b u b v b w b ) ζ Thus () α ( f ( J U U U U U U )) U B ( 456) (87) O he ohe had applyg (8) () α( f ( J U U U U4 U5 U6)) α( U4) 9 U B ( 456) (88) By (87) ad (88) we have α( f( J U U U U4 U5 U6)) α( U4) 9 U B ( 456) (89) I he sae way α( I( V V V V4)) α( V) 5 V B ( j 4) (9) j α( I( V4 )) α( V4 ) V4 B (9) 4 Hece ( H ) hol whee () () d d b a Fally s easy o pove (76) hol The we have he cocluso by Theoe 4Ths e he poof Refeeces: [] V Lashaha DD Baov P S Seoov Theoy of Ipulsve Dffeeal Equaos Wold Scefc Sgapoe 989 [] D Guo V Lashaha Nolea Pobles Absac Coes Acadec Pess Ic Boso 988 [] D GuoV Lashaha XZ Lu Nolea Iegal Equaos Absac Spaces Kluwe Acadec Publshes Dodech 996 [4] D J Guo Ial value pobles fo olea secod-ode pulsve ego-dffeeal equaos Baach spaces J Mah Aal Appl (996) - [5] D J Guo Secod-ode pulsve ego -dffeeal equaos o ubouded doas Baach spaces Nolea Aal 5 (999)4-4 [6] L S Lu C X Wu F Guo A uque soluo of al value pobles fo fs-ode pulsve ego-dffeeal equaos of xed ype Baach Spaces J Mah Aal Appl 75 () [7] J L SuY H Ma Ial value pobles fo he secod-ode xed oooe ype of pulsve dffeeal equaos Baach Spaces J Mah Aal Appl 47 () [8] Y X L Z Lu Moooe eave echque fo addessg pulsve ego-dffeeal equaos E-ISSN: 4-88 Issue Volue Mach

18 WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo Baach spaces Nolea Aal 66 (7) 8-9 [9] X G Zhag L S Lu Ial value pobles fo olea secod-ode pulsve ego-dffeeal equaos of xed ype Baach spaces Nolea Aal 64 (6) [] L S Lu Y H Wu X G Zhag O wellposedess of a al value poble fo olea secod-ode pulsve ego-dffeeal equaos of Volea ype Baach spaces J Mah Aal Appl 7 (6) [] W X Wag L L Zhag ZD Lag Ial value pobles fo olea pulsve ego- -dffeeal equaos Baach space J Mah Aal Appl (6) 5-57 [] M Q Feg H H Pag A class of hee-po bouday-value pobles fo secod-ode pulsve ego-dffeeal equaos Baach spaces Nolea Aalyss 7 (9) 64-8 [] X M Zhag M Q Feg W G Ge Exsece of soluos of bouday value pobles wh egal bouday codos fo secod-ode pulsve ego-dffeeal equaos Baach spaces Joual of Copuaoal ad Appled Maheacs () [4] JT Su HB Che TJ Zhou Mulplcy of soluos fo a fouh-ode pulsve dffeeal equao va vaaoal eho Bulle of he Ausala Maheacal Socy 8 () [5] J T Su H B Che L Yag Vaaoal eho o fouh-ode pulsve dffeeal equaos Joual of Appled Maheacs ad Copug 5 () - 4 E-ISSN: 4-88 Issue Volue Mach