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1 Communicions inmhemicanysis Voume 11, Number 2, (211) ISSN GRONWALL-LIKE INEQUALITIES ON TIME SCALES WITH APPLICATIONS ELVAN AKIN-BOHNER Dermen of Mhemics nd Sisics Missouri Universiy of Science nd Technoogy Ro, Missouri 6549, USA MEHMET ÜNAL Dermen of Sofwre Engineering Bhçeşehir Universiy Beşikş-İsnbu, Turkey (Communiced by Mich Fečkn) Absrc Some new noniner dynmic inegr inequiies of Gronw ye for rerded funcions re esbished. These inequiies cn be used s bsic oos in he sudy of cerin csses of funcion dynmic equions s we s dynmic dey equions. AMS Subjec Cssificion: 39A1. Keywords: mesure chin, ime sces, gronw inequiies, dynmic equions, dey equions. 1 Inroducion Moived by Agrw e. ers 3, 4], our urose is o obin ime sces versions of some Gronw-ike inequiies used in he heory of differeni nd inegr equions. I is we known h Gronw ike inequiies in coninuous nd discree cses y cruci rue in sudying he quiive behvior of differeni nd difference equions. These inequiies hve been used o invesige he gob exisence, uniqueness, boundedness nd oher roeries of souions of vrious noniner differeni nd difference equions. For he bckground nd he summry on hese ricur subjecs, we refer he ineresed reder o he exceen monogrhs 9]-11] by Pche nd 1] by Agrw. Mny uhors E-mi ddress: kine@ms.edu E-mi ddress: mun@bhcesehir.edu.r

2 24 Evn Akın-Bohner, nd Mehme Ün hve sudied some fundmen inequiies used in nysis on ime sces, for exme, see 2], 5, 6], 13, 14]. In his er, we woud ike o sudy cerin csses of funcion dynmic equions s we s dynmic dey equions, see Secion 3. In Secion 1, we give brief inroducion o ccuus on ime sces s we s some imorn references. In Secion 2, we obin cerin ye of inequiies h re imorn o rove he min resus in his er. For comeeness, few hins concerning he bckground of ime sces, which hs receny received o of enion, migh be in order. In 1988, Sefn Higer 12] in his Ph.D. hesis dded new wrinke o he ccuus by inroducing he ccuus on ime sce, which is unificion nd exension of he heories of coninuous nd discree nysis. A ime sce is n rbirry nonemy cosed subse of he re numbers R, nd we usuy denoe i by he symbo T. The wo mos our exmes re T = R nd T = Z. Some oher ineresing ime sces exis, nd hey give rise o eny of icions such s he sudy of ouion dynmics modes (see 7], ges 15 nd 71). We define he forwrd nd bckwrd jum oerors σ, ρ : T T by σ() := inf{s T : s > } nd ρ() := su{s T : s < } (suemened by inf / = sut nd su / = inft). A oin T wih > inft is ced righ-scered, righ-dense, ef-scered nd ef-dense if σ() >, σ() =, ρ() < nd ρ() = hods, resecivey. Poins h re ef-dense nd righ-dense he sme ime re ced dense. The se T κ is derived from T s foows: If T hs ef-scered mximum m, hen T κ = T {m}. Oherwise, T κ = T. The grininess funcion µ : T, ) is defined by µ() := σ(). Hence he grininess funcion is if T = R whie i is 1 if T = Z. Le f be funcion defined on T, hen we define he de derivive of f T κ, denoed by f (), o be he number (rovided i exiss) wih he roery such h for every ε >, here exiss neighborhood U of wih f (σ()) f (s) f ()σ() s] ε σ() s for s U. Some eemenry fcs concerning he de derivive re: If f is differenibe, hen f σ () = f (σ()) = f () + µ() f (). If f nd g re differenibe, hen f g is differenibe wih ( f g) () = f σ ()g () + f ()g() = f ()g () + f ()g σ (). If f nd g re differenibe nd g()g(σ()), hen f g is differenibe wih ( ) f () = f ()g() f ()g (). g g()g(σ())

3 Gronw-ike Inequiies 25 We sy f : T R is righ-dense coninuous ( f C rd (T,R)) rovided f is coninuous righ-dense oins in T nd is ef-sided imi exiss (finie) ef-dense oins in T. The imornce of rd-coninuous funcions is h every rd-coninuous funcion ossesses n niderivive. A funcion F : T κ R is ced n niderivive of f : T R rovided F () = f () hods for T κ. In his cse we define he inegr of f by f (s) s = F() F() for T. Oher usefu formus re s foows: Z b Z b σ() Z Z c f () = f (s) s = µ() f (), Z b f () + c f (), Z b f ()g () = f (b)g(b) f ()g() f ()g σ (). Le f, g : T R be rd-coninuous nd, b T. If f () g() on,b), hen Z b Z b f () g(). The foowing resu is chin rue on T, see 7, Theorem 1.9]. Lemm 1.1. Le f : R R be coninuousy differenibe nd suose g : T R is de differenibe. Then f g : T R is de differenibe nd he formu { Z 1 ( f g) () = f ( g() + hµ()g () ) } dh g () hods. A comrehensive nd n exceen remen of ccuus on ime sces cn be found, for insnce, in 7, 8]. For convenience of noion, we e hroughou nd T, T =, ) T,,b T T,b] =,b] T. We e R =, ), R 1 = 1, ), nd C 1 (M,N) be he css of coninuousy differenibe funcions defined on he se M o he se N.

4 26 Evn Akın-Bohner, nd Mehme Ün 2 Gronw-Like Inequiies In his secion, we consider Gronw ye inequiies which wi be usefu o obin he gob exisence of souions for cerin dey dynmic equions. Theorem 2.1. Le b, f i, g i C rd (T,R ), i = 1,2,..., such h b is nondecresing nd e α : T T be nondecresing such h α() nd < = inf{α(s) : s T }. Suose h q is consn, ϕ C 1 (R,R ) is n incresing funcion wih ϕ( ) = on R, nd ψ is nondecresing coninuous funcion for u R wih ψ(u) > for u >. If u : T R nd for T, hen u() ϕ 1 ϕ(u()) b() + G 1 for T,β], where Ω 1 Ω G(b() + u q (α(s)) f i (s)ψ(u(α(s))) + g i (s)] s g i (s) s + Z r ds G(r) = (ϕ 1 (s)) q, r r >, r Z r ds Ω(r) = ψϕ 1 (G 1 (s))], r r >, r f i (s) s (2.1) G 1, nd Ω 1 denoe he inverse funcions of G, Ω, resecivey nd β is chosen such h Ω G(b()) + g i (s) s + f i (s) s Dom ( Ω 1) hods. Proof. Le ε > nd T T. Define funcion z : T,T ] R by z() = ε + b(t ) + u q (α(s)) f i (s)ψ(u(α(s))) + g i (s)] s. (2.2) Cery, z() is nondecresing, u() ϕ 1 (z()) for T,T ] nd z( ) = ε + b(t ). From (2.2), we obin z () = u q (α()) f i ()ψ(u(α())) + g i ()] ϕ 1 (z()) ] q fi ()ψ(ϕ 1 (z())) + g i () ], (2.3)

5 Gronw-ike Inequiies 27 for T κ,t ]. Here we use he fc h α() yieds z(α()) z(), ϕ is incresing, nd q > imies h u q (α()) ϕ 1 (z(α())) ] q ϕ 1 (z()) ] q, T,T ]. Using he monooniciy of ϕ 1 nd z, we obin ϕ 1 (z()) ] q ϕ 1 (z()) ] q = ϕ 1 (ε + b(t )) ] q >, T,T ]. Hence i foows from (2.3) we hve h z () ϕ 1 (z())] q fi ()ψ(ϕ 1 (z())) + g i () ], T κ,t ]. (2.4) On he oher hnd, king ino ccoun Lemm 1.1 nd he definiion of G, we hve (G(z())) = (G z) () Z 1 = z () G ( z() + µ()hz () ) dh, (2.5) for T κ,t ], where G 1 () = ϕ 1 ()] q. Since z() z() + µ()hz () for h 1, nd T κ,t ], we hve Hence from (2.5), we ge ϕ 1 (z()) ϕ 1 ( z() + µ()hz () ), T κ,t ]. Z 1 (G(z())) = z () G ( (z() + µ()hz () ) dh Z 1 = z 1 () ϕ 1 (z() + µ()hz ())] q dh Z 1 z 1 () ϕ 1 (z())] q dh for T κ,t ]. Combining (2.4) nd (2.6), we obin (G(z())) = z () ϕ 1 (z())] q, (2.6) fi ()ψ(ϕ 1 (z())) + g i () ], T κ,t ]. (2.7) Inegring (2.7) from o nd king ino ccoun h g i : T,T ] R for ech i, we deduce G(z()) G(ε + b(t )) + f i (s)ψ(ϕ 1 (z(s))) s + Z T g i (s) s. (2.8)

6 28 Evn Akın-Bohner, nd Mehme Ün for T,T ]. Now define funcion v() by he righ hnd side of (2.8), h is, < v() = G(ε + b(t )) + Z T g i (s) s + f i (s)ψ(ϕ 1 (z(s))) s. Cery, v() is nondecresing, z() G 1 (v()) for T,T ] nd Therefore, for ny,t ] κ T, we hve Z T v( ) = G(ε + b(t )) + g i (s) s. v () = f i ()ψ(ϕ 1 (z())) ψ(ϕ 1 (G 1 (v()))) f i (). Using he monooniciy of ψ, ϕ 1, G 1, nd v yieds v () ψ(ϕ 1 (G 1 (v()))) f i (), T κ,t ]. (2.9) On he oher hnd, s before, king ino ccoun Lemm 1.1 nd he definiion of Ω, we hve (Ω(v())) = (Ω v) () Z 1 = v () Ω (v() + µ()hv ())dh, (2.1) for T κ,t ], where Ω 1 () = ψ(ϕ 1 (G 1 ())). Since v() v() + µ()hv () for h 1, T κ,t ], we hve nd so ψ ( ϕ 1 ( G 1 (v()) )) ψ ( ϕ 1 ( G 1 (v() + µ()v ()) )), T κ,t ] 1 ψ(ϕ 1 (G 1 (v() + µ()hv ()))) 1 ψ(ϕ 1 (G 1 (v()))), Tκ,T ]. Subsiuing his s inequiy ino (2.1) nd king ino ccoun (2.9) we obin (Ω(v())) v () ψ(ϕ 1 (G 1 (v()))) f i (), T κ,t ]. (2.11)

7 Inegring (2.11) from o yieds Ω(v()) Ω(v( )) + Gronw-ike Inequiies 29 = Ω G(ε + b(t )) + for T,T ] nd hence we obin v() Ω 1 Ω G(ε + b(t )) + ( ) f i (s) s Z T Z T g i (s) s + g i (s) s + for T,T ]. Since z() G 1 (v()) for T,T ], we ge Z T z() G 1 Ω G(ε + b(t )) + g i (s) s + Ω 1 ( ) f i (s) s, ( ) f i (s) s, ( ) f i (s) s for T,T ]. Leing ε nd king ino ccoun h u() ϕ 1 (z()) for T,T ] we obin Z T u() ϕ 1 G 1 ( ) Ω 1 Ω G(b(T )) + g i (s) s + f i (s) s. The s inequiy roduces he required inequiy (2.1) for T =, since T T rbirry, which comees he roof. ws The foowing corory foows from Theorem 2.1 when ϕ(u) = u, G(r) = r q, where > q re consns. Corory 2.2. Le b, f i, g i, α, nd ψ be s defined in Theorem 2.1. Suose h > q re consns. If u : T R nd for T, hen u() Ω 1 u () b() + for T,β], where Ω b()] q Z r Ω(r) = u q (α(s)) f i (s)ψ(u(α(s))) + g i (s)] s + q r ψ ds s 1 q g i (s) s + q ], r r >, f i (s) s 1 q

8 3 Evn Akın-Bohner, nd Mehme Ün nd Ω 1 denoes he inverse funcion of Ω nd β is chosen such h Ω b()] q + q g i (s) s + q f i (s) s Dom ( Ω 1) hods. Proof. The rgumen of he roof is sme s in he roof of Theorem 2.1 wih suibe modificion. Hence we omi he deis here. Remrk 2.3. Le T = Z. If = 2, q = 1, b() = c 2, α(s) = s, i = 1 in Corory 2.2, hen our resus deduces o he Pche inequiy in 9]. We finish his secion wih noher usefu noniner inegr inequiy wihou is roof since i is simir o he roof of Theorem 2.1. The roch is gin bsed on chin rue on ime sces. Theorem 2.4. Le b, f i, g i, i = 1,,, q, nd ϕ be s defined in Theorem 2.1. Suose h ψ j (u) ( j = 1,2) is nondecresing coninuous funcion for u R wih ψ j (u) > for u >. If u : T R 1 nd ϕ(u()) b() + for T, hen where for he cse ψ 1 (u) ψ 2 (og(u)), we hve u() ϕ 1 Ω 1 G(b())] + for T,β 1 ], nd G 1 u q (α(s)) f i (s)ψ 1 (u(α(s))) + g i (s)ψ 2 (ogu(α(s)))] s Ω 1 1 for he cse ψ 1 (u) < ψ 2 (og(u)), we hve u() ϕ 1 Ω 2 G(b())] + for T,β 2 ], G 1 Ω 1 2 Z r ds Ω m (r) = ψ m 1 (ϕ 1 (G 1 (s))), r r >, r f i (s) + g i (s)] s f i (s) + g i (s)] s G 1, nd Ω 1 m, m = 1,2 denoe he inverse funcions of G, Ω m, resecivey, G() is s defined in Theorem 2.1 for T, nd β m, m = 1,2 is chosen such h hods. Ω m G(b())] + f i (s) + g i (s)] s Dom ( Ω 1 ) m

9 Gronw-ike Inequiies 31 The foowing corory foows from Theorem 2.4 when ϕ(u) = u, G(r) = r q, where > q re consns. Corory 2.5. Le b, f i, g i for i = 1,2,...,, ψ j, j = 1,2 nd α be s defined in Theorem 2.4. Suose h > q re consns. If u : T R 1 nd u () b() + for T, hen where for he cse ψ 1 (u) ψ 2 (og(u)), we hve u() for T,β 1 ], nd Ω 1 1 u q (α(s)) f i (s)ψ 1 (u(α(s))) + g i (s)ψ 2 (ogu(α(s)))] s ( ) Ω 1 b()] q for he cse ψ 1 (u) < ψ 2 (og(u)), we hve u() for T,β 2 ], Ω 1 2 ( ) Ω 2 b()] q + q + q f i (s) + g i (s)] s f i (s) + g i (s)] s Z r ds Ω m (r) = ψ m 1 (ϕ 1 (G 1 (s))), r r >, r G 1, nd Ω 1 m, m = 1,2 denoe he inverse funcions of G, Ω m, resecivey, G() is s defined in Theorem 2.1 for T, nd β m, m = 1,2 is chosen such h hods. ( ) Ω m b()] q + q f i (s) + g i (s)] s Dom ( Ω 1 ) m 1 q 1 q 3 Aicions Our resus re hefu in roving he gob exisence of souions o cerin dynmic equions wih ime dey. We firs consider he funcion dynmic equion φ (x()) = h() + F i,x(α()),w(x(α()))], (3.1) φ(x( )) = x

10 32 Evn Akın-Bohner, nd Mehme Ün where x is consn, φ C (R,R ) is n incresing funcion such h φ( x ) φ(x), h : T R is nondecresing, x : T R, α : T T is nondecresing such h α() nd < = inf{α(s) : s T }, w C (R,R) is nondecresing funcion, nd F i C ( T R 2,R ). The foowing heorem des wih he bound on he souion of (3.1). Theorem 3.1. Assume h F i : T R 2 R for i = 1,..., is coninuous funcion nd here exis coninuous funcions f i, g i C rd (T,R ), i = 1,..., such h nd F i,x(α()),w(x(α()))] x(α()) q f i ()ψ( x(α()) ) + g i (), (3.2) x + h(s) s b(), (3.3) where q is consn nd b(), ψ re s in Theorem 2.1. If x() is ny souion of (3.1) for T, hen x() φ 1 G 1 Ω 1 Ω G(b() + g i (s) s + f i (s) s (3.4) for T, where G nd Ω re defined s in Theorem 2.1. Proof. Le x() be souion of (3.1) for T. One cn show h x() sisfies he equiven equion φ(x()) = x + h(s) s + for T. I foows from (3.5) h φ(x()) x + h(s) s + F i s,x(α(s)),w(x(α(s)))] s, (3.5) F i s,x(α(s)),w(x(α(s)))] s (3.6) for T. Using he condiions (3.2), (3.3) on he righ hnd side of (3.6) we obin φ( x() ) b() + x(α(s)) q f i (s)ψ( x(α(s)) ) + g i (s)] s for T. Now n immedie icion of he inequiy esbished in Theorem 2.1 o he inequiy (3.4) yieds he resu. Remrk 3.2. We now consider he funcion dynmic equion wih he inii condiion (x ()) = h() + F i,x(α()),w(x(α()))], (3.7) x ( ) = x 1

11 Gronw-ike Inequiies 33 where >, x 1 re consns. Assume h F i : T R 2 R for i = 1,..., is coninuous funcion nd here exis coninuous funcions f i, g i : T R, i = 1,..., such h he inequiies (3.2) nd (3.3) hod, where q is consn such h > q nd b(), ψ re defined s in Corory 2.2. If x() is ny souion of he robem (3.7) for T, hen i sisfies he equiven equion x () = x 1 + h(s) s + for T. I foows from (3.8) h x() x 1 + h(s) s + F i s,x(α(s)),w(x(α(s)))] s, (3.8) F i s,x(α(s)),w(x(α(s)))] s (3.9) for T. Using he condiions (3.2), (3.3) on he righ hnd side of (3.9) yieds x() b() + x(α(s)) q f i (s)ψ( x(α(s)) ) + g i ()] s, (3.1) where T. Now n immedie icion of he inequiy esbished in Corory 2.2 o (3.1) yieds x() Ω 1 Ω b()] q + q for T, where Ω is s in Corory 2.2. g i (s) s + q f i (s) s In he foowing heorem we give necessry condiions o obin unique souion of (3.7). Theorem 3.3. Assume h F i : T R 3 R for i = 1,..., is coninuous funcion nd here exiss coninuous nonnegive funcion f i () for i = 1,..., for T such h F (,x,w(x)) F (,x,w(x)) f i () x x, (3.11) where > 1 is consn, hen he robem (3.7) hs unique souion on T. Proof. Le x() nd x() be wo souions of (3.7) for T. Then we hve x () x () = F i s,x(α(s)),w(x(α(s)))] F i s,x(α(s)),w(x(α(s)))]] s, (3.12) for T. From (3.11) nd (3.12), we ge x () x () 1 q f i (s) x (α(s)) x (α(s)) s (3.13)

12 34 Evn Akın-Bohner, nd Mehme Ün for T. Rerrnging equion (3.13) yieds ( ) x () x () 1 ] x (α(s)) x (α(s)) 1 1 ] fi (s) x (α(s)) x (α(s)) 1 s, (3.14) where T when ψ(u) = u, q = 1, suibe icion of he inequiy in Corory 2.2 o he funcion x () x () 1 nd he inequiy (3.14) ed us o he inequiy x () x () 1 for T. Hence we obin x() = x() for T. Acknowedgmens The uhors hnk he referees for heir crefu reding of he mnuscri nd insighfu commens. References 1] R. Agrw, Difference Equions nd Inequiies, Mrce Dekker, New York, ] R. Agrw, M. Bohner nd A. Peerson, Inequiies on ime sces: survey. Mh. Inequ. A. 4 (21), ] R. Agrw, Y. H. Kim nd S. K. Sen, New rerded discree inequiies wih icions, In. J. Difference Equ., 4 (29), ] R. Agrw, Y. H. Kim nd S. K. Sen, New noniner inegr inequiies wih icions, Funcion Differeni Equions, 16 (29), ] E. Akın-Bohner, M. Bohner nd F. Akın, Pche inequiies on ime sces, J. Inequ. Pure. A. Mh., 6 (25), Ar. 6. ONLINE: h://jim.vu.edu.u/rice.h?sid=475]. 6] D. R. Anderson, Noniner Dynmic Inegr Inequiies in wo Indeenden Vribes on Time Sce Pirs, Advnces in Dynmic Sysems nd Aicions, 3 (28), ] M. Bohner nd A. Peerson, Dynmic Equions on Time Sces, An Inroducion wih Aicions, Birkhäuser, Boson, 21. 8] M. Bohner nd A. Peerson, ediors, Advnces in Dynmic Equions on Time Sces, Birkhäuser, Boson, 23. 9] B.G. Pche, Inequiies for Finie Difference Equions, Mrce Dekker, New York, 22.

13 Gronw-ike Inequiies 35 1] B.G. Pche, On some new inequiies reed o cerin inequiies in he heory of differeni equions, J. Mh. An. A. 189 (1995), ] B.G. Pche, Inequiies for Differeni nd Inegr Equions, Acdemic Press, New York, ] S. Higer, Ein Mβkeenkkü mi Anwendung uf Zenrumsmnnigfigkeien, PhD hesis, Universiä Würzburg, ] P. Rehk, Hrdy inequiy on ime sces nd is icion o hf iner dynmic equions, J. Inequ. A., 5 (25), ] F. Wong, C. C. Yeh nd C. H. Hong, Gronw inequiies on ime sces, Mh. Inequ. A., 1 (26),