A New Generalized Gronwall-Bellman Type Inequality

Transcription

1 22 Inernaonal Conference on Image, Vson and Comung (ICIVC 22) IPCSIT vol. 5 (22) (22) IACSIT Press, Sngaore DOI:.7763/IPCSIT.22.V5.46 A New Generalzed Gronwall-Bellman Tye Ineualy Qnghua Feng School of Scence, Shandong Unversy of Technology, Zhangzhou Road 2, Zbo, Shandong, Chna, Absrac. In hs aer, a new nonlnear negral neualy s esablshed, whch rovde a handy ool for analyzng he global exsence of soluons of dfferenal and negral euaons. Keywords: Inegral neualy; Global exsence; Inegral euaon; Dfferenal euaon. Inroducon Durng he as decades, wh he develomen of he heory of dfferenal and negral euaons, a lo of negral neuales, for examle [-3], have been dscovered, whch lay an moran role n he research of boundedness, global exsence, sably of soluons of dfferenal and negral euaons. In [2], Jang roved he followng heorem: Theorem A: R = [, ). Suose ha x(), f (), h() CR (, R ). he followng form of delay negral neualy: x ( ) C [ f ( s) x ( s) h( s) x ( ( s))] ds, R σ wh he nal condon x ( ) = φ( ), [ α,], φ( σ( )) ( ρ( )), R, σ( ) where, are consans, >, >,. σ () C( R, R), σ( ), < α = nf{ σ( ), R }, φ C([ α,], R ), mles ha f () s x( ) ex( ds) [ C h( s)ex( f ( τ) dτ) ds] s for [, ], where s a osve number sasfyng nf { C h( s)ex( f ( τ) dτ) ds} >. [, ] s In hs aer, movaed by he above work, we wll rove more general heorem and esablsh a new negral neualy. Also we wll gve one examle so as o llusrae he valdy of he resen negral neualy. Corresondng auhor. E-mal address: fhua@sna.com

2 2. Man Resuls Theorem 2.: Assume ha x, a C( R, R ), a () s non-decreasng. f, g, f, g C( R R, R ). ω CR (, R ) be nondecreasng wh ω ( u) > on (, ). If x() sasfes he followng delay negral neualy: σ σ2 ω σ3 x ( ) a( ) [ f( s, ) x ( ( s)) g( s, ) x ( ( s)) ( x( ( s)))] ds, R () wh he nal condon x ( ) = φ( ), [ α,], φ( σ ( )) ( a ( )), R, σ ( ), =, 2,3 (2) where, are consans, > >, σ CR (, R), σ( ), < α= nf{mn{ σ( ), =,2,3}, R }, φ C([ α,], R ), hen for R, x F a () { Ω [ Ω(ex( ()) ()) ex( F( )) F '( s)ex( F( s)) ds]} 2 (3) where 2 F () = f (,) s ds, F () = g(,) s ds, (4) r Ω () r = ds, Ω s he nverse of Ω. ω( s ) Proof: We noce (3) holds for = obvously. Le he rgh sde of () be ϕ (). x() ϕ() (5) When σ (), we have x( σ ( )) ϕ( σ ( )) ϕ( ) (6) When σ (), we have x( σ ( )) = φ( σ ( )) a ( ) ϕ( ) (7) So from (6), (7) we always have x( σ ( )) ϕ( ), =,2,3. Fx T >. for (, T], we have ϕ () at ( ) [ f(,) sϕ () s gs (,) ϕ () sω( ϕ())] s ds (8) Le he rgh sde of (8) be u (). ϕ() u(), (, T] (9) and

3 f( s, ) gs (,) u () u'() = [ ϕ ( s) ds f(,) ϕ () ϕ ( s) ω( ϕ( s)) ds g(, ) ϕ ( ) ω( ϕ( ))] d f(,) sds d gs (,) ω(()) us ds ϕ () ϕ () d d d f(,) sds d gs (,) ω(()) us ds u () u () d d () d f(,) sds d gs (,) ω(()) us ds u u u d d '( ) ( ) ( ) () Le v () = u (). d f(,) sds d gs (,) ω(()) us ds d f(,) sds d gsds (,) v'( ) v( ) v () ( v ()) d d d d ω ha s, v'() F '() v() F '() ( v ()) 2 ω ( 2) Mullyng ex( F ( )) on boh sdes of (2), we have d[ex( F ( )) v ( )] F2'( ) ω( v ( ))ex( F( )) (3) d Inegrang (3) from o, follows ex( F ( )) v ( ) v () F2'( s) ω( v ( s))ex( F( s)) ds (4) Snce v() = u () = a ( T), we have and v () { a ( T) F2'( s) ω( v ( s))ex( F( s)) ds}ex( F( )) (5) u () ex( F ()) { a ( T) F2'( s) ω( u( s))ex( F( s)) ds} (6)

4 Le k () = a ( T) F2'() sω( v ())ex( s F()) s ds. u () ex( F ()) k () (7) and k'() = F2'() ω( u())ex( F()) F2'( ) ω(ex( F( T)) k ( ))ex( F( )) k'( ) F 2'( )ex( F( )) (8) ω(ex( FT ( )) k ( )) Inegrang (8) from o, follows Ω[ex( FT ( )) k ( )] Ω[ex( FT ( )) a ( T)] 2 ex( F( T)) F '()ex( s F()) s ds (9) k ( ) ex( FT ( )) [ (ex( FT ( )) a ( T)) Ω Ω 2 ex( F( T)) F '( s)ex( F( s)) ds] (2) From (7) and (2), follows u ( ) Ω { [ Ω (ex( FT ( )) a ( T)) ex( F( T)) F2'( s)ex( F( s)) ds]}, (, T] (2) Combnng (5), (9), (2) we have x ( ) Ω { [ Ω (ex( FT ( )) a ( T)) ex( F( T)) F2'( s)ex( F( s)) ds]}, (, T] (22) Seng = T and consderng T R s arbrary, we have comleed he roof. Corallary 2.2: Assume ha x, afg,, CR (, R ), m C( R, R ). If xa,,,, σ( ), α, φ, ω are he same as n Theorem 2., and x() sasfes he followng delay negral neualy: σ σ2 ω σ3 x ( ) a ( ) m ( ) [ f( sx ) ( ( s)) gsx ( ) ( ( s)) ( x( ( s)))] ds, R (23) wh he nal condon (2), hen for R, x( ) { Ω [ Ω(ex( F ( )) a ( )) ex( F( )) F2'( s)ex( F( s)) ds]} (24)

5 where F () = m() f () s ds, F () = m() g() s ds 2 (25) 3. Alcaon Examle: We consder he followng delay dfferenal euaon ( x ( )) ' = F(, x( σ ( )), x( σ ( )), x( σ ( ))) (26) 2 3 wh he nal condon x () = φ(), [ α,], φ( σ ()) A, R, σ (), =,2,3 3 where A > s a consans and A= x (), F CR ( R, R),. () C( R, R), σ (), < α σ = nf{mn{ σ ( ), =,2,3}, R }, φ C([ α,], R ), Assume (,,, ) ( ) F xyz f x g ( ) x v( z ) where f, gv, CR (, R), v s nondecreasng, and Inegrang (26) from o, follows vsds ( ) =. > > x ( ) x () F( s, x( σ ( s)), x( σ ( s)), x( σ ( s))) ds = o 2 3 So ( ) () ( ) x x x x () F ( s, x( σ( s)), x( σ2( s)), x( σ3( s))) ds o [ f ( s) x( σ ( s)) g( s) x( σ ( s)) v( x( σ ( s)) )] ds o 2 3 Takng ω ( u) = v( u), from Theorem 2., we can reach he esmae x ( ) { Ω [ Ω (ex( F ( )) A ) ex( F( )) F '( s)ex( F( s)) ds]} 2 where F () = f () s ds, F () = g() s ds, 2 whch shows x() does no blow u n fne me. So he soluon of (26) s global. 4. Conclusons In hs aer, we esablsh a new negral neualy, whch rovdes a handy ool n he ualave analyss of soluons of negral euaons and dfferenal euaons. Our resul generalzes he resul n [2].

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