Some New Dynamic Inequalities for First Order Linear Dynamic Equations on Time Scales

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1 Applied Memicl Science, Vol. 1, 2007, no. 2, Some New Dynmic Inequliie for Fir Order Liner Dynmic Equion on Time Scle B. İ. Yşr, A. Tun, M. T. Djerdi nd S. Küükçü Deprmen of Memic, Fculy of Science nd Ar Univeriy of Gzi, Beşevler, 06500, Ankr, Turkey Abrc. We udy ome new dynmic inequliie for fir order liner dynmic equion on ime cle. Memic Subjec Clificion: 39A11, 26D10 Keyword: Time cle, Dynmic inequliy, Liner dynmic equion 1. Inroducion Hilger nd Aulbc [2, 5] generlized e definiion of derivive nd of n inegrl o ime cle in order o unify reul from e clculu of rel number wi reul from e difference clculu. So fer ime cle clculu creed. In i wy mny pper ime cle were wrien by Agrwl, Boner, Hilcer, Peeron nd join profeor. A ime cle i n rbirry nonempy cloed ube of e rel number. Te clculu of ime cle w iniied by B. Aulbc nd S. Hilger [2, 5] in order o cree eory cn unify dicree nd coninuou nlyi. For remen of e ingle vrible clculu of ime cle ee [3, 4, 8] nd e reference given erein. Afer mny eorie in rel number nd ineger number re exended o ime cle. In i pper, we udy ome new dynmic inequliie for fir order liner dynmic equion on ime cle. We conider nonomogeneou liner dynmicl equion of fir order (1.1) y ()+g()y σ ()+e g (σ(),)() 0, y() x We ume T [, b] i n rbirry inervl. We moreover g : T R, : T R, nd ϕ : T [0, ) re funcion uc for rbirry c

2 70 B. İ. Yşr e l T, g() nd e g (σ(),)() re inegrble on [, c], fuer uc ϕ()e g (, ) i inegrble on T, nd g() i regreive (i.e.,g R). We prove in Teorem.1 if rd-coninuouly differenible funcion y : T R ifie e differenil inequliy y ()+g()y σ ()+e g (σ(),)() (1.2) ϕ() for ll T k, en exi unique oluion y 0 () of e dynmicl equion (1.1) uc y() y 0 () e g (, ) ϕ(υ)e g (υ,) υ for ny T. Here, fir we menion everl foundionl definiion wiou proof nd reul from e clculu on ime cle in n excellen inroducory ex by Boner nd Peeron [3, 4]. 2. Generl Definiion Definiion 1. A ime cle T i nonempy cloed ube of R. We ume rougou T e opology i ineried from e ndrd opology on R. I lo umed rougou in T e inervl [, b] men e e { T : <} for e poin < b in T. Since ime cle my no be conneced, we need e following concep of jump operor. Definiion 2. Te mpping σ, ρ : T T defined by σ() inf { T : >} nd ρ() up { T : <} re clled e jump operor. Te jump operor σ nd ρ llow e clificion of poin in T in e following wy: Definiion 3. A nonmximl elemen T i id o be rig-dene if σ(), rig-cered if σ() >, lef-dene if ρ(),lef-cered if ρ() <. In e ce T R, we ve σ(), nd if T Z, >0, en σ() +. Definiion 4. Te mpping μ : T R + defined by μ() σ() i clled e grinine funcion. If T R, en μ() 0, nd wen T Z, we ve μ() 1. Definiion 5. Le f : T R. fi clled differenible T k, wi (del) derivive f () R if given ε>0 ere exi neigborood U of uc, for ll U, f σ () f() f ()[σ() ] ε σ(), were f σ f σ. If T R, en f df () (),nd if T Z, en f () d f( +1) f(). Some bic properie of del derivive re e following [3].

3 New dynmic inequliie on ime cle 71 Teorem 1. Aume f : T R nd le T k. (i) If f i differenible, en f i coninuou. (ii) If f i differenible nd i rig-cered, en f i differenible wi f () f σ () f(). σ() (iii) If f i differenible nd i rig-dene, en f f() f() () lim. (iv) If f i differenible, en f σ () f()+μ()f () Teorem 2. Aume f,g : T R re differenible T k. en: (i) Te um f + g : T R i differenible wi (f + g) () f ()+g (). (ii) For ny conn α, αf : T R i differenible wi (αf) () αf (). (iii) Te produc fg : T R i differenible wi (fg) () f ()g()+f(σ())g () f()g ()+f ()g(σ(). (iv) If f()f(σ()) 0, en 1 i differenible wi f ( ) 1 f f (). f()f (σ()) (v) If g()g(σ()) 0, en f i differenible nd g ( f g ) f ()g() f ()g () f()f (σ()). Definiion 6. Te funcion f : T R i id o be rd-coninuou (denoed by f C rd (T,R)) if, ll T, (i) f i coninuou every rig-dene poin T, (ii) lim f() exi nd i finie every lef-dene poin T. Definiion 7. Le f C rd (T,R). Ten g : T R i clled e niderivive of f on T if i i differenible on T nd ifie g () f() for ny T k. In i ce, we define f() g() g(), T.

4 72 B. İ. Yşr e l 2.1. Te Hilger complex plne. For >0, define e Hilger complex number, e Hilger rel xi, e Hilger lerning xi, nd e Hilger imginry circle by C { z C : z } 1, R { } z R : z> 1 A { } z R : z< 1, I { z C : } z repecively. For 0, le C 0 : C, R 0 : R, A 0 :, nd I 0 : ir. Le >0nd z C. Te Hilger rel pr of z i defined by Re (z) : z+1, nd e Hilger imginry pr of z i defined by Im (z) : Arg(x+1), were Arg(z) denoe e principle rgumen of z (i.e., π < Argz π). For >0, define e rip Z : { z C : π <Argz π }, nd for 0, e Z 0 : C. Ten we cn define e cylinder rnformion ξ C Z by ξ (z) 1 Log(1 + z), > 0 were Log i e principle logrim funcion. Wen 0, we define ξ 0 (z) z, for ll z C. I en follow e invere cylinder rnformion : Z C i given by ξ 1 ξ 1 (z) 1 ez. Since e grinine my no be conn for given ime cle, we will inercngebly ubcrip vriou quniie (uc ξ nd ξ 1 ) wi μ μ() ined of o reflec i Generlized exponenil Funcion. Te funcion p : T R i regreive if 1 + μ()p() 0 for ll T k, nd i concep moive e definiion of e following e: R { p : T R : p C rd (T) nd 1 + μ()p() 0 T k}, R + { p R:1+μ()p() > 0 for ll T k}. Te funcion p : T R i uniformly regreive on T ere exi poiive conn δ uc 0 <δ 1 1+μ()p(), T k. If p R, en we define e generlized ime cle exponenil funcion by e p (, ) exp ξ μ(τ ) (p(τ)) τ for ll, T Te following eorem i compilion of properie of e p (, ) (ome of wic re counerinuiive) we need in e min body of e pper. Teorem 3. Te funcion e p (, ) e following properie: (i) If p R, en e p (, r)e p (r, ) e p (, ) for ll r,, T. (ii) e p (σ(),) (1 + μ()p())e p (, ). (iii) If p R +, en e p (, 0 ) > 0 for ll T. (iv) If 1+μ()p() < 0 for ome T k, en e p (, 0 )e p (σ(), 0 ) < 0.

5 New dynmic inequliie on ime cle 73 R p(τ )dτ (v) If T R, en e p (, ) e. Morever, If p i conn, en ep (, ) e p( ). (vi) If T Z, en e p (, ) Π 1 τ (1 + p(τ)). Morever, If T Z, wi >0 nd p i conn, en e p (, ) (1 + p) ( ) Definiion 8. If p Rnd f : T R i rd-coninuou, en e dynmic equion. (2.1) i clled regreive. y () p()y()+f() Teorem 4. If p, q R, en (i) e 0 (, ) 1 nd e p (, ) 1; (ii) e p (σ(),) (1 + μ()p())e p (, ); 1 (iii) e p(, e p(,) ); (iv) e p (, ) 1 e p(, e p(,) ); (v) e p (, )e p (, τ) e p (, τ); (vi) e p (, )e q (, ) e p q (, ); (vii) ep(,) e e q(,) p q(, ); Teorem 5. (Vriion of conn). Le 0 T nd y( 0 )y 0 R.Ten e regreive IVP (2.1) unique oluion y : T R given by y() y 0 e p (, 0 )+ 0 e p (, σ(τ))f(τ) τ. Teorem 6. (Vriion of conn). Suppoe (2.1) i regreive. Le 0 T nd x( 0 )x 0 R. Te unique oluion of e iniil vlue problem i given by x () p()x σ + f(), x( 0 )x 0 x() e p (, 0 )x Min Reul 0 e p (, τ)f(τ) τ. Teorem 7. Le T [, b] i n rbirry inervl, were, b R {± } re rbirrily given wi <b.aume g : T R wi g() R, : T R, re rd-coninuou funcion uc g() nd e g (σ(),)() re inegrble [, c] for ec c T. Moreover, uppoe ϕ : T [0, ) i funcion uc ϕ()e g (, ) i inegrble on T. If rd-coninuouly differenible funcion

6 74 B. İ. Yşr e l y : T R ifie e differenil inequliy (1.2) for ll T, en ere exi unique x R uc y() e g(, (3.1) )(x e g (σ(υ),)(υ) υ) e g (, ) ϕ(υ)e g (υ,) υ for every T. Proof. For impliciy, we ue e following noion: z() :e g (, )y()+ e g (σ(υ),)(υ) υ for ec T. By mking ue of i noion nd by (1.2), we ge z() z() e g(, )y() e g (, )y()+ e g (σ(υ),)(υ) υ [e g (v, )y(v)] υ + e g (σ(υ),)(υ) υ ( ) [e g (v, )y(v)] + e g (σ(υ),)(υ) υ [ eg (v, )y (v)+g(v)e g (v, )y σ (v)+e g (σ(υ),)(υ) ] υ e g (v, ) [ y (v)+g(v)y σ (v)+e g (σ(υ),υ)(υ) ] υ (3.2) ϕ(υ)e g (v, ) υ for ny, T. Finlly, i follow from (3.2) nd e bove rgumen for ny T, y() e g(, )(x e g (σ(υ),)(υ) υ) e g (, )(z() x) e g (, )(z() z()) + e g (, )(z() x)

7 New dynmic inequliie on ime cle 75 e g (, ) z() z() + e g (, ) z() x e g (, ) ϕ(υ)e g (υ,) υ + e g(, ) z() x e g (, ) ϕ(υ)e g (υ,) υ b, ince z() x b. I now remin o prove e uniquen of x R. Aume x 1 R lo ifie e inequliy (3.1) in plce of x. Ten, we ve e g (, )(x 1 x) 2e g (, ) ϕ(υ)e g (υ,) υ for ny T. I follow from e iegrbiliy ypoee x 1 x 2 ϕ(υ)e g (υ,) υ 0 b. Ti implie e uniquene of x R. Remrk 1. we my now remrk y() e g (, )(x e g (σ(υ),)(υ) υ) i e generl oluion of e differenil equion (1.1), were x R i n rbirry elemen Exmple. In i ecion, we will inroduce ome exmple for liner differenil equion of fir order wenever T R follow. Exmple 1. If we ke T [, b] R i n rbirry inervl in R, nd we e () 0, ϕ() ε in Teorem.1, we obin e following reul: Le T [, b] i n rbirry inervl in R, were, b R {± } re rbirrily given wi <b.i i cler wen T R, en σ(), y σ () y() nd μ 0. Alo, wen T R, en from (1.1) equion we ge y ()+g()y()+() 0,...y() x for ll T, nd (1.2) inequliy y ()+g()y()+() ϕ() for ny T. Aume g : T R i { coninuou nd inegrble } funcion on [, c] for ec c T uc exp g(u)du i inegrble on T. If coninuouly differenible funcion y : T R ifie e differenil inequliy y ()+g()y() ε for ll T, en ere } exi unique x R uc g(u)du x y() exp { ε exp for ec T. { g(u)du } b { v } exp g(u)du υ

8 76 B. İ. Yşr e l Exmple 2. Le g < 0 nd be fixed rel number, le T [, ) be n inervl wi R, nd ϕ : T R ifie e differenil inequliy y ()+g()y()+() ϕ() for ll T. We cn eily verify e coice of g,, ϕ nd T re conien wi e ypoee of Teorm.1. Hence, ere exi unique c 0 R uc y() c 0 e g + (1 g e g( ) ) e g ϕ(υ)e gυ dυ for ny T. Furer, we know y 0 () c 0 e g g (1 e g( ) ) i (priculr) oluion of e differenil equion y ()+g()y()+() 0. If we e ϕ() ε nd T [, ) wi 0 in e bove emen, en ere exi unique oluion y 0 () of e differenil equion y ()+g()y()+ () 0uc y() y 0 () ε g for ll T. (We my compre i wi [1] or [10].) Exmple 3. If we ke T [, b] in rel inervl in R, i i cler we cn ve e me Teorem 1. in ([7]). Reference [1] C. Alin nd R. Ger, On ome inequliie nd biliy reul reled o e exponenil funcion, J. Inequl. Appl., 2 (1998) [2] B. Aulbc, S. Hilger, Liner dynmic proce wi inomogeneou ime cle, in: Nonliner Dynmic nd Qunum Dynmicl yem, Pperfrom e In. Seminr ISAM-90, Guig, 1990, G.A. Leonov, V. Reimnn, W. Timmermnn (Ed), Memicl Reerc, Vol. 59, Akdemie Verlg, Berlin, 1990, pp [3] M. Boner, A. Peeron, Dynmic equion on ime cle, An Inroducion wi Applicion, Birkuer, Boon, [4] M. Boner, A. Peeron, Advnce in Dynmic Equion on Time Scle, Birkäuer, Boon, [5] S. Hilger, Anlyi on meure cin- unified pproc o coninuou nd dicree clculu, Reul M. 18 (1990) [6] S.M. Jung., Hyer-Ulm biliy of liner differenil equion of fir order, Applied Memic Leer, 17 (2004) [7] S.M. Jung., Hyer-Ulm biliy of liner differenil equion of fir order II, Applied Memic Leer, inpre. [8] B. Kymkçln, V. Lkmiknm. nd S. Sivundrm, Dynmic Syyem on Meure Cin, Volume 370 of Memic nd i Applicion. Kluwer Acdemic Publier Group, Dordrec, [9] T.Miur, S. E. Tki nd H. Cod, On e Hyer-Ulm biliy of rel coninuou funcion vlued differenible mp, Tokyo J. M., 24 (2001) [10] S. E. Tki, T. Miur, S. Miyjim. On e Hyer-Ulm biliy of e Bnc pce-vlued differenil equion y λy, Bull. Koren M. Soc. 39 (2002) Received: July 1, 2006