Analysis of Boundedness for Unknown Functions by a Delay Integral Inequality on Time Scales

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1 Inernaional Conference on Image, Viion and Comuing (ICIVC ) IPCSIT vol. 5 () () IACSIT Pre, Singaore DOI:.7763/IPCSIT..V5.45 Anali of Boundedne for Unknown Funcion b a Dela Inegral Ineuali on Time Scale Qinghua Feng School of Science, Shandong Univeri of Technolog, Zhangzhou Road, Zibo, Shandong, China, 5549 Abrac. In hi aer, new exlici bound for unknown funcion i derived b a new Volerra-Fredholm e dela inegral ineuali on ime cale, which can be ued a a hand ool in he inveigaion of ualiaive roerie a well a uaniaive roerie of dela dnamic euaion. Keword: Dela inegral ineuali; Time cale; Dnamic euaion; Bounded. Inroducion Inegral ineualiie la an imoran role in he reearch of ualiaive roerie of oluion of dnamic euaion, and man inegral ineualiie a well a difference ineualiie have been eablihed ince hen, for examle [-], and he reference herein. Our aim in hi aer i o eablih a new Volerra-Fredholm e dela inegral ineuali on ime cale, which rovide new bound for unknown funcion. In he re of he aer, R denoe he e of real number and R = [, ). T denoe an arbirar ime cale and T = [ x, ) T, T = [, ), where x, T. The e T κ i defined o be T if T doe no have a lef-caered maximum, oherwie i i T wihou he lef-caered maximum. On T we define he forward and backward jum oeraor σ ( TT, ) and ρ ( TT, ) uch ha σ () = inf{ T, > }, ρ ( ) = u{ T, < }. Definiion : A oin T wih > inft i aid o be lef-dene if ρ () = and righ-dene if σ () =, lef-caered if ρ () < and righ-caered if σ () >. Definiion : A funcion f ( T, R) i called rd-coninuou if i i coninuou in righ-dene oin and if he lef-ided limi exi in lef-dene oin, while f i called regreive if μ( ) f( ), where μ() = σ (). C denoe he e of rd-coninuou funcion, while R denoe he e of all regreive and rd-coninuou rd funcion, and R = { f f R, μ() f() >, T}. Definiion 3: For ome T κ Δ, and a funcion f ( T, R), he dela derivaive of f i denoed b f (), and aifie f ( σ()) f() f Δ ()( σ() ) ε σ() for ε >, where U, and U i a neighborhood of. The funcion f i called dela differenial on T κ Similarl, for ome T κ, and a funcion f ( T T, R), he arial dela derivaive of f wih reec o i denoed b ( f ( x, )) Δ, and aifie f ( x, σ( )) f( x, ) ( f( x, )) Δ ( σ( ) ) ε σ( ) for ε >, where U, and U i a neighborhood of. Definiion 4: For ome ab, T and a funcion f ( T, R), he Cauch inegral of f i defined b Correonding auhor. addre: fhua@ina.com

2 b a f () Δ = F( b) F( a), where F Δ () = f (), T κ. Similarl, for ome ab, T and a funcion f ( T T, R), he Cauch arial inegral of f wih reec o b i defined b f ( x, ) Δ = F( x, b) F( x, a), where ( F( x, )) Δ = f ( x, ), T κ.. Main Reul a We will give ome lemma for furher ue. Lemma. ([], Gronwall ineuali): Suoe X T i an arbiraril fixed number, and u( X, ), bx (, ) C, rd mx (, ) R wih reec o, mx (, ), hen imlie u( X, ) b( X, ) m( X, ) u( X, ) Δ, T u( X, ) b( X, ) e (, σ ( )) b( X, ) m( X, ) Δ, T m where e (, ) m i he uniue oluion of he following euaion, ( z( X, )) Δ = m( X, ) z( X, ), z( X, ) =. Lemma. []: Aume ha a,, and, hen for an K >, a K a K. Conider he following ineuali x u ( x, ) C L(,, u( τ ( ), τ ( ))) ΔΔ h( ξη, ) u ( τ( ξ), τ ( η)) ΔξΔηΔΔ x Lu (,, ( τ ( ), τ ( ))) ΔΔ h ( ξη, ) u ( τ( ξ), τ ( η)) ΔξΔηΔΔ wih he iniial condiion () ux (, ) = φ(, x), x[ α, x] Tor,, [ β, ], () φτ ( ( x), τ( )) C, τ( x) x, or, τ( ) where uh, i Crd( T T, R ), i=,,,, C m, C are conan, and,, C >, τ( TT, ), τ( x) x, < α= inf{ τ( x), xt} x, τ( T, T), τ( ), < β =, inf{ τ( ), T} φ Crd (([ α, x] [ β, ]), R ), M T, N T are wo fixed number. Theorem.: If for (, x ) ([ x, M] T ([, N] ), ux (, ) aifie (), and K > i an arbirar conan, hen he following ineuali hold C B ux B x B x 6 (, ) {[ ] 3(, ) 4(, )}, B5 (, x ) ([ x, M] T) ([, N] ) (3) rovided ha B 5 <, where

3 C= C[ L(,, K ) (, ) m m h ξη K ΔξΔη ] ΔΔ x B (, x ) = [(,, L K ) (, ) h ξη K ΔξΔη ] ΔΔ x B (, x ) = [ A(,, K ) K x (, ) h ] ξη K ΔξΔη Δ. B (, x ) = e (, σ()) B (,) x Δ 3 B B (, x ) = B(, x ) e (, σ()) B (,) x B(,) x Δ 4 B x 5 3 B = A(,, K ) K B (,) ΔΔ h( ξη, ) K B3( ξη, ) ΔξΔη] ΔΔ. x 6 4 B = A(,, K ) K B (,) ΔΔ h( ξη, ) K B4( ξη, ) ΔξΔη] ΔΔ. Proof : Le he righ ide of () be vx. (, ) Then ux (, ) v ( x, ),( x, ) ([ x, M] T) ([, N] ) (4) From () we have u( τ ( x), τ ( )) v ( x, ),( x, ) ([ x, M] T) ([, N] ) (5) Given a fixed X [ x, M], and x[ x, X] T, [, N], hen vx (, ) vx (, ), x[ x, X] T, [, N] (6) Furhermore, conidering vx (, ) = C Lu (,, ( τ ( ), τ ( ))) ΔΔ h ( ξη, ) u ( τ( ξ), τ ( η)) ΔξΔηΔΔ (7) o we have X vx (, ) = C Lu (,, ( τ ( ), τ ( ))) ΔΔ X h( ξη, ) u ( τ( ξ), τ ( η)) ΔξΔηΔΔ Lu (,, ( τ ( ), τ ( ))) ΔΔ h ( ξη, ) u ( τ( ξ), τ ( η)) ΔξΔηΔΔ X C L(,, v (,)) ΔΔ X (, ) h (, ) ξηv ξηδξδηδδ

4 Lv (,, (,)) ΔΔ h ( ξη, ) u ( τ( ξ), τ ( η)) ΔξΔηΔΔ X = vx (, ) Lv (,, (, )) ΔΔ (, ) h (, ) ξηv ξηδξδηδδ From Lemma., we have X (8) v ( x, ) K v( x, ) K v ( x, ) K v( x, ) K (9) Combining (8), (9) we have X X vx (, ) = vx (, ) L (,,( K v (,) K)) ΔΔ (, )( (, ) h K v K ) X = vx (, ) [ L (,,( K v (,) K )) L (,, K ) L (,, K )] ΔΔ X (, )( (, ) h ) K v K X vx (, ) A (,, K ) K v (, ) ΔΔ X L (,, K ) ΔΔ X [ h( ξη, ) K ΔξΔηΔ] v( X, ) Δ (, ) h ξη K ΔξΔηΔΔ X X vx (, ) [ A (,, K ) K ΔvX ] (, ) Δ X L (,, K ) ΔΔ X [ h( ξη, ) K ΔξΔηΔ] v( X, ) Δ (, ) h ξη K ΔξΔηΔΔ X = vx (, ) B( X, ) B( XvX, ) (, ) Δ B Lemma. we obain vx (, ) vx (, ) B( X, ) e (, σ()) B( X,)(( v x, ) B( X,)) Δ B B σ B = vx (, )[ e (, ( )) B( X, ) Δ ] B( X, ) e (, σ ()) B( X,)(( vx, ) B( X,)) Δ, [, N] () Combining (6), (), i follow vx (, ) vx (, ) BX (, ) e(, σ( BXvx )) (, )( (, ) BX (, )) Δ B

5 = vx (, )[ e (, σ( )) B( X, ) Δ ] B( X, ) B e (, σ ()) B( X,)(( v x, ) B( X,)) Δ B, x [ x, X] T, [, N] () Seing x = X in (), conidering X i eleced from [ x, M] arbiraril, ubiuing X wih x, ield v(, x ) v( x, ) B(, x ) e (, σ()) B(,)(( x v x, ) B(,)) x Δ B = vx (, )[ e (, σ( )) B( x, ) Δ ] B( X, ) B e (, σ ()) B(,)(( x vx, ) B(,)) x Δ B, x [ x, X] T, [, N] () ha i, vx (, ) vx (, ) B(, x) B(, x) x [ x, X] T, [, N] (3) 3 4 On he oher hand, from (5), (7) (9) we have vx (, ) = C Lv (,, (, )) ΔΔ (, ) h (, ) ξηv ξηδξδηδδ C L (,, K v (,) K ) ΔΔ (, )( (, ) h ) K v K C [ L (,, K v (, ) K ) L(,, K ) L(,, K )] ΔΔ (, )( (, ) h ) K v K ξ Δ η Δ Δ C A(,, K ) K v(,) ΔΔ L (,, K ) ΔΔ (, )( (, ) h K v K ) = C A(,, K ) K v(,) ΔΔ (, )( (, ) h K v K ) (4) Then uing (3) in (4) ield 3 4 vx (, ) C A (,, K ) K [ vx (, ) B(, ) B(, )] ΔΔ 3 4 h(,)( ξη K [( v x, ) B(,) ξη B(,)] ξη K ) ΔξΔη ΔΔ x 3 = C v( x, ){ A (,, K ) K B(,) ΔΔ h( ξη, ) K B3( ξη, ) ΔξΔη] ΔΔ}

6 x A(,, K ) K B4 (,) ΔΔ h( ξη, ) K B4( ξη, ) ξ η] Δ Δ Δ Δ = C v( x, ) B B (5) 5 6 which i followed b C B6 vx (, ) B 5 (6) Combining (4), (3) and (6) we can obain he deired ineuali (3). 3. Concluion In hi aer, new exlici bound for unknown funcion i derived b ue of a new Volerra- Fredholm e inegral ineuali on ime cale, which rovide a hand ool in he ualiaive anali of oluion of dnamic euaion on ime cale. 4. Reference [] W.N. Li, Some dela inegral ineualiie on ime cale, Comu. Mah. Al. 59 () [] F.H. Wong, C.C. Yeh, S.L. Yu, C.H. Hong, Young ineuali and relaed reul on ime cale, Al. Mah. Le. 8 (5) [3] F.H. Wong, C.C. Yeh, W.C. Lian, An exenion of Jenen, ineuali on ime cale, Adv. Dnam. S. Al. () (6) 3-. [4] M.Z. Sarikaa, On weighed Iengar e ineualiie on ime cale, Al. Mah. Le. (9) [5] W.J. Liu, C.C. Li, Y.M. Hao, Furher generalizaion of ome double inegral ineualiie and alicaion, Aca. Mah. Univ. Comenian. 77 () (8) [6] X.L. Cheng, Imrovemen of ome Orowki-Gru e ineualiie, Comu. Mah. Al. 4 () 9-4. [7] M. Bohner, T. Mahew, The Gru ineuali on ime cale, Commun. Mah. Anal. 3 () (7) -8. [8] Q.A. Ngo, Some mean value heorem for inegral on ime cale, Al. Mah. Comu. 3 (9) [9] W.j. Liu, Q.A. Ngo, Some Iengar-e ineualiie on ime cale for funcion whoe econd derivaive are bounded, Al. Mah. Comu. 6 () [] W.j. Liu, Q.A. Ngo, A generalizaion of Orowki ineuali on imecale for k oin, Al. Mah. Comu. 3 (8) [] R. Agarwal, M. Bohner, A. Peeron, Ineualiie on ime cale: a urve, Mah. Ineual. Al. 4 (4) () [] F.C. Jiang, F.W. Meng, Exlici bound on ome new nonlinearinegral ineuali wih dela, J. Comu. Al. Mah. 5 (7)