Dynamic Sysems and Applicaions 2 (2003) 20-25 A SECOND-ORDER SELF-ADJOINT EQUATION ON A TIME SCALE KIRSTEN R. MESSER Universiy of Nebraska, Deparmen of Mahemaics and Saisics, Lincoln NE, 68588, USA. E-mail: kmesser@aol.com or kmesser@mah.unl.edu ABSTRACT. In his paper we examine he dynamic equaion [p()x ()] + q()x() = 0 on a ime scale. Lile work has been done on his equaion, which combines boh he dela and nabla derivaives. Several preliminary resuls are esablished, including Abel s formula and is converse. We hen proceed o invesigae oscillaion and disconjugacy of his dynamic equaion. AMS (MOS) Subjec Classificaion. 39A0.. NABLA DERIVATIVES In his paper, we are concerned wih he second-order self-adjoin dynamic equaion [p()x ] + q()x = 0. We begin our work by reviewing some properies of he nabla derivaive, and hen in Secion 2, we proceed o esablish several resuls concerning he ineracion of he wo ypes of derivaives. In he hird secion of he paper, we develop Abel s formula, and is converse, which we hen use o prove a reducion of order heorem. In he final secion, we urn our aenion o oscillaion and disconjugacy, esablishing firs an analogue of he Surm separaion heorem, and hen, via he Pólya and Trench facorizaions, we demonsrae he exisence of recessive and dominan soluions of he self-adjoin equaion. Here i is assumed ha he reader is already familiar wih he basic noions of calculus on a ime scale, using he dela-derivaive (or -derivaive). The reader may be less familiar, however, wih he nabla-derivaive (or -derivaive) on a ime scale, as developed by Aıcı and Guseinov [], and so we include here a brief inroducion o is properies, as previously esablished in oher works, saing hem wihou proof. Readers desiring more informaion are direced o [2] and []. Throughou, we assume ha T is a ime scale. The noaion [a, b] is undersood o mean he real inerval [a, b] inerseced wih T. Received Augus 23, 2002 056-276 $03.50 c Dynamic Publishers, Inc.
202 KIRSTEN R. MESSER Definiion.. Le T be a ime scale. For > inf(t), he backward jump operaor, ρ() is defined by ρ() := sups T : s < }, and he backward graininess funcion ν() is defined by ν() := ρ(). If f : T R, he noaion f ρ () is undersood o mean f(ρ()). Remark.2. Here, we reain he original definiion of ν(). This definiion is consisen wih he original lieraure published on -derivaives. I is inconsisen, however wih he curren work on α-derivaives. When working wih α-derivaives, he α-graininess, µ α is defined o be µ α := α(). When α() = ρ(), hen, we would have µ ρ := ρ() = ν(). This inconsisency is unforunae, bu we feel i is more imporan ha we remain consisen wih he way ν() was defined in previously published work. To minimize confusion, we recommend he noaion µ ρ () = ρ() be used in work ha is o be inerpreed in he more general α-derivaive seing. Definiion.3. Define he se T κ as follows: If T has a righ-scaered minimum m, se T κ := T \ m}; oherwise, se T κ = T. Definiion.4. Le T κ. Then he -derivaive of f a, denoed f (), is he number (provided i exiss) wih he propery ha given any ε > 0, here is a neighborhood U T of such ha for all s U. f(ρ()) f(s) f ()[ρ() s] ε ρ() s For T = R, he -derivaive is jus he usual derivaive. Tha is, f = f. For T = Z he -derivaive is he backward difference operaor, f () = f() := f() f( ). Definiion.5. A funcion f : T R is said o be lef-dense coninuous or ldconinous if i is coninuous a lef-dense poins, and if is righ-sided limi exiss (finie) a righ-dense poins. Definiion.6. I can be shown ha if f is ld-coninuous hen here is a funcion F, called a -aniderivaive, such ha F () = f() for all T. We hen define he -inegral ( -Cauchy inegral) of f by f(s) s = F () F ( ). Theorem.7. Assume f : T R is a funcion and le T κ. Then we have he following:
A SECOND-ORDER SELF-ADJOINT DYNAMIC EQUATION 203. If f is nabla-differeniable a, hen f is coninuous a. 2. If f is coninuous a and is lef-scaered, hen f is nabla-differeniable a wih f () = f() f(ρ()). ν() 3. If is lef-dense, hen f is nabla-differeniable a iff he limi f() f(s) lim s s exiss as a finie number. In his case 4. If f is nabla-differeniable a, hen f f() f(s) () = lim. s s f ρ () = f() ν()f (). Oher properies of boh he -derivaive and he -inegral are analogous o he properies of he -derivaive and -inegral. For example, boh differeniaion and inegraion are linear operaions, and here are produc and quoien rules for differeniaion, as well as inegraion by pars formulas. Readers ineresed in he specifics can find more deails in [2]. 2. PRELIMINARY RESULTS We are ineresed in he second-order self-adjoin dynamic equaion (2.) Lx = 0 where Lx = [p()x ] + q()x. Here we assume ha p is coninuous, q is ld-coninuous and ha p() > 0 for all T. Define he se D o be he se of all funcions x : T R such ha x : T κ R is coninuous and such ha [p()x ] : T κ κ R is ld-coninuous. A funcion x D is said o be a soluion of Lx = 0 on T provided Lx() = 0 for all T κ κ. Since he equaion we are ineresed in, equaion (2.), conains boh - and - derivaives, we esablish here some resuls regarding he relaionship beween hese wo ypes of derivaives on ime scales. One of he following resuls relies on L Hôpial s rule. A version of L Hôpial s rule involving -derivaives is conained in [2]. We sae is analog for -derivaives here. As we may wish o use L Hôpial s rule o evaluae a limi as ±, we make he following definiion.
204 KIRSTEN R. MESSER Definiion 2.. Le ε > 0. If T is unbounded below, we define a righ neighborhood of, denoed R ε ( ) by R ε ( ) = T : < }. ε We nex define a righ neighborhood for poins in T. Definiion 2.2. Le ε > 0. For any righ-dense T, define a righ neighborhood of, denoed R ε ( ), by R ε ( ) := T : 0 < < ε}. Theorem 2.3 (L Hôpial s Rule). Assume f and g are -differeniable on T and le T }. If T, assume is righ-dense. Furhermore, assume lim f() = lim g() = 0, + 0 + 0 and suppose here exiss ε > 0 such ha g()g () > 0 for all R ε ( ). Then lim inf + 0 f () g () lim inf + 0 f() g() lim sup + 0 f() g() lim sup + 0 f () g (). Proof. Wihou loss of generaliy, assume g() and g () are boh sricly posiive on R ε ( ). Le δ (0, ε], and le a := inf τ Rδ ( ) f (τ), b := sup f (τ) g (τ) τ R δ ( ). To complee g (τ) he proof, i suffices o show a f(τ) inf τ R δ ( ) g(τ) sup f(τ) τ R δ ( ) g(τ) b, as we may hen le δ 0 o obain he desired resul. We mus be careful here, as eiher a or b could possibly be infinie. Noe, however, ha since g (τ) > 0 on R δ ( ), we have a <. Similarly, b >. So our only concern is if a = or b =. Bu, if a =, we have immediaely ha a f(τ) inf τ R δ ( ) g(τ), as desired, and if b = we have immediaely ha f(τ) sup τ R δ ( ) g(τ) b, as desired. Therefore, we may assume ha boh a and b are finie. Then ag (τ) f (τ) bg (τ) for all τ R δ ( ), and by a heorem of Guseinov and Kaymakçalan [3], s ag (τ) τ s f (τ) τ s bg (τ) τ for all s, R δ ( ), < s.
Inegraing, we see ha A SECOND-ORDER SELF-ADJOINT DYNAMIC EQUATION 205 ag(s) ag() f(s) f() bg(s) bg() for all s, R δ ( ), < s. Leing + 0, we ge ag(s) f(s) bg(s) for all s R δ ( ), and hus f(s) a inf s R δ ( ) g(s) sup f(s) s R δ ( ) g(s) b. Then, by he discussion above, he proof is complee. Remark 2.4. Alhough he preceding heorem is only saed in erms of one-sided limis, an analogous resul can be esablished if he limi is aken from he oher direcion. Lef neighborhoods of or of poins in T are defined in similar manner o righ neighborhoods. To apply L Hôpial s rule using a lef-sided limi, mus be lef-dense (or if T is unbounded above), and gg mus be sricly negaive on some lef neighborhood of. In order o deermine when he wo ypes of derivaives may be inerchanged, we need o consider some of he poins in our ime scale separaely, so le Addiionally, le A := T is lef-dense and righ-scaered}, T A := T \ A. B := T is righ-dense and lef-scaered}, T B := T \ B. The following lemma is very easy o prove, and we omi he proof here. Lemma 2.5. If T A hen σ(ρ()) =. If T B hen ρ(σ()) =. Theorem 2.6. If f : T R is -differeniable on T κ and f is rd-coninuous on T κ hen f is -differeniable on T κ, and f f (ρ()) T A () = lim s f (s) A. If g : T R is -differeniable on T κ and g is ld-coninuous on T κ hen g is -differeniable on T κ, and g () = g (σ()) T B lim s + g (s) B. Proof. We will only prove he firs saemen. The proof of he second saemen is similar. Firs, assume T A. Then here are wo cases: Eiher. is lef-scaered, or 2. is boh lef-dense and righ-dense.
206 KIRSTEN R. MESSER Case : Suppose is lef-scaered and f is -differeniable on T κ. Then f is coninuous a, and is herefore -differeniable a. Nex, noe ha ρ() is righ-scaered, and f (ρ()) = = f(σ(ρ())) f(ρ()) σ(ρ()) ρ() f() f(ρ()) ρ() = f (). Case 2: Now, suppose is boh lef-dense and righ-dense, and f : T R is coninuous on T and -differeniable a. Since is righ-dense and f is -differeniable a, we have ha f() f(s) lim s s exiss. Bu is lef-dense as well, so his expression also defines f (), and we see ha f () = f() f(s) lim s s = f () = f (ρ()). So, we have esablished he desired resul in he case where T A. exiss. Now suppose A. Then is lef-dense. Hence f () exiss provided f() f(s) lim s s As is righ-scaered, we need only consider he limi as s from he lef. Then we apply L Hôpial s rule [2], differeniaing wih respec o s o ge f() f(s) lim s s = lim s f (s) = lim s f (s). Since we have assumed ha f is rd-coninuous, his limi exiss. Hence f is -differeniable, and f () = lim s f (), as desired. Corollary 2.7. If T, and f : T R is rd-coninuous on T hen f(τ) τ is -differeniable on T and [ ] f(ρ()) if T A f(τ) τ = lim s f(s) A. If T, and g : T R is ld-coninuous on T hen g(τ) τ is -differeniable on T and [ ] g(σ()) if T B g(τ) τ = lim s + g(s) B.
A SECOND-ORDER SELF-ADJOINT DYNAMIC EQUATION 207 The following corollary was previously esablished by Aıcı and Guseinov in heir work []. Corollary 2.8. If f : T R is -differeniable on T κ and if f is coninuous on T κ, hen f is -differeniable on T κ and f () = f ρ () for T κ. If g : T R is -differeniable on T κ and if g is coninuous on T κ, hen g is -differeniable on T κ and g () = g σ () for T κ. 3. ABEL S FORMULA AND REDUCTION OF ORDER We begin his secion by looking a he Lagrange Ideniy for he dynamic equaion (2.). We esablish several corollaries and relaed resuls, including Abel s formula and is converse. We conclude he secion wih a reducion of order heorem. Some of he resuls in his secion are due o Aıcı and Guseinov. Specifically, Theorem 3. and Corollary 3.5 were previously esablished in heir work []. Our condiions on p and q are less resricive han Aıcı and Guseinov s, and our domain of ineres, D, is defined more broadly. In spie of his, however, many of he proofs conained in [] remain valid. As his is he case, we have omied he proofs of some of he following heorems, and refer he reader o Aıcı and Guseinov s work. Theorem 3.. If T, and x 0 and x are given consans, hen he iniial value problem Lx = 0, x( ) = x 0, x ( ) = x has a unique soluion, and his soluion exiss on all of T. Definiion 3.2. If x, y are -differeniable on T κ, hen he Wronskian of x and y, denoed W (x, y)() is defined by x() y() W (x, y)() = for T κ. x () y () Definiion 3.3. If x, y are -differeniable on T κ, hen he Lagrange bracke of x and y is defined by x; y}() = p()w (x, y)() for T κ. Theorem 3.4 (Lagrange Ideniy). If x, y D, hen x()ly() y()lx() = x; y} () for T κ κ.
208 KIRSTEN R. MESSER Proof. Le x, y D. We have x; y} = [pw (x, y)] = [xpy ypx ] = x p ρ y ρ + x[py ] y p ρ x ρ y[px ] = x p ρ y + x[py ] y p ρ x y[px ] = x[py ] y[px ] = x([py ] + qy) y([px ] + qx) = xly ylx, where we have made use of he fac ha x and y are coninuous and applied Corollary 2.8. Corollary 3.5 (Abel s Formula). If x, y are soluions of (2.) hen where C is a consan. W (x, y)() = C p() for T κ, Definiion 3.6. Define he inner produc of x and y on [a, b] by x, y := b a x()y(). Corollary 3.7 (Green s Formula). If x, y D hen x, Ly Lx, y = [p()w (x, y)] b a. Theorem 3.8 (Converse of Abel s Formula). Assume u is a soluion of (2.) wih u() 0 for T. If v D saisfies hen v is also a soluion of (2.). W (u, v)() = C p(), Proof. Suppose ha u is a soluion of (2.) wih u() 0 for any, and assume ha v D saisfies W (u, v)() = C. Then by Theorem 3.4, we have p() u()lv() v()lu() = u; v} () u()lv() = [p()w (u, v)()] = [p() C p() ] = C = 0.
A SECOND-ORDER SELF-ADJOINT DYNAMIC EQUATION 209 As u() 0 for any, we can divide hrough by i o ge Hence v is a soluion of (2.) on T. Lv() = 0 for T κ κ. Theorem 3.9 (Reducion of Order). Le T κ, and assume u is a soluion of (2.) wih u() 0 for any. Then a second, linearly independen soluion, v, of (2.) is given by for T. v() = u() p(s)u(s)u σ (s) s Proof. By Theorem 3.8, we need only show ha v D and ha W (u, v)() = C p() for some consan C. Consider firs W (u, v)() = u()v () v()u () [ = u() u () p(s)u(s)u σ (s) s + u ()u() = u()u () = p(). u()u () p(s)u(s)u σ (s) s p(s)u(s)u σ (s) s + p(s)u(s)u σ (s) s ] u σ () p()u()u σ () u()uσ () p()u()u σ () Here we have C =. I remains o show ha v D. We have ha v () = u () = u () p(s)u(s)u σ (s) s + u σ () p()u()u σ () p(s)u(s)u σ (s) s + p()u(). Since u D, u() 0 and p is coninuous, we have ha v is coninuous. Nex, consider [p()v ()] = [ ] [ ] p()u () p(s)u(s)u σ (s) s + u() = [p()u ()] p(s)u(s)u σ (s) s [ ] +p ρ ()u ρ () p(s)u(s)u σ (s) s u () u()u ρ ().
20 KIRSTEN R. MESSER Now, he firs and las erms are ld-coninuous. I is no as clear ha he cener erm is ld-coninuous. Specifically, we are concerned abou wheher or no he expression [ ] p(s)u(s)u σ (s) s is ld-coninuous. Noe ha he inegrand is rd-coninuous. Hence Corollary 2.7 applies and yields [ p(τ)u(τ)u σ (τ) τ ] = Simplificaion of his expression gives [ ] p(τ)u(τ)u σ (τ) τ = p ρ ()u ρ ()u σρ () lim s if T A A. p(s)u(s)u σ (s) p ρ ()u ρ ()u() for T. This funcion is ld-coninuous, and so we have ha v D. Hence by Theorem 3.8, v is also a soluion of (2.). Finally, noe ha as W (u, v)() = p() v are linearly independen. 4. OSCILLATION AND DISCONJUGACY 0 for any, u and In his secion, we esablish resuls concerning generalized zeros of soluions of (2.), and examine disconjugacy and oscillaion of soluions. Definiion 4.. We say ha a soluion, x, of (2.) has a generalized zero a if x() = 0 or, if is lef-scaered and x(ρ())x() < 0. Definiion 4.2. We say ha (2.) is disconjugae on an inerval [a, b] if he following hold.. If x is a nonrivial soluion of (2.) wih x(a) = 0, hen x has no generalized zeros in (a, b]. 2. If x is a nonrivial soluion of (2.) wih x(a) 0, hen x has a mos one generalized zero in (a, b]. We will invesigae oscillaion of (2.) as approaches he supremum of he ime scale. Le ω = sup T. If ω <, we assume ρ(ω) = ω. Furhermore, if ω <, we allow he possibiliy ha ω is a singular poin for p or q. Definiion 4.3. Le ω = sup T be as described above, and le a T. We say ha (2.) is oscillaory on [a, ω) if every nonrivial real-valued soluion has infiniely many generalized zeros in [a, ω). We say (2.) is nonoscillaory if i is no oscillaory.
A SECOND-ORDER SELF-ADJOINT DYNAMIC EQUATION 2 The following Lemma is a direc consequence of he definiion of nonoscillaory. Lemma 4.4. Le ω = sup T be as described above, and le a T. Then if (2.) is nonoscillaory on [a, ω), here is some T, a, such ha (2.) has a posiive soluion on [, ω). Theorem 4.5 (Surm Separaion Theorem). Le u and v be linearly independen soluion of (2.). Then u and v have no common zeros in T κ. If u has a zero a T, and a generalized zero a 2 > T, hen v has a generalized zero in (, 2 ]. If u has generalized zeros a T and 2 > T, hen v has a generalized zero in [, 2 ]. Proof. If u and v have a common zero a T κ, hen u( ) v( ) W (u, v)( ) = u ( ) v ( ) = 0. Hence u and v are linearly dependen. Now suppose u has a zero a T, and a generalized zero a 2 > T. Wihou loss of generaliy, we may assume 2 > σ( ) is he firs generalized zero o he righ of, u() > 0 on (, 2 ), and u( 2 ) 0. Assume v is a linearly independen soluion of (2.) wih no generalized zero in (, 2 ]. Wihou loss of generaliy, v() > 0 on [, 2 ]. Then on [, 2 ], ( u v ) () = v()u () u()v () v()v σ () = C p()v()v σ (), which is of one sign on [, 2 ). Thus u v is monoone on [, 2 ]. Fix 3 (, 2 ). Noe ha Bu u( ) v( ) = 0, and u( 3) v( 3 ) > 0. u( 2 ) v( 2 ) 0, which conradics he fac ha u v is monoone on [, 2 ]. Hence v mus have a generalized zero in (, 2 ]. Finally, suppose u has generalized zeros a T and 2 > T. Assume 2 > σ( ) is he firs generalized zero o he righ of. If u( ) = 0, we are in he previous case, so assume u( ) 0. Then, as u has a generalized zero a, we have ha is lef-scaered. Wihou loss of generaliy, we may assume u() > 0 on [, 2 ), u(ρ( )) < 0 and u( 2 ) 0. Assume v is a linearly independen soluion of (2.) wih no generalized zero in [, 2 ). Wihou loss of generaliy, v() > 0 on
22 KIRSTEN R. MESSER [, 2 ], and v(ρ( )) > 0. In a similar fashion o he previous case, we apply Abel s formula o ge ha u v is monoone on [ρ( ), 2 ]. Bu u(ρ( )) v(ρ( )) < 0, u( ) v( ) > 0, and u( 2) v( 2 ) 0, which is a conradicion. Hence v mus have a generalized zero in [, 2 ]. Theorem 4.6. If (2.) has a posiive soluion on an inerval I T hen (2.) is disconjugae on I. Conversely, if a, b T κ κ and (2.) is disconjugae on [ρ(a), σ(b)] T, hen (2.) has a posiive soluion on [ρ(a), σ(b)]. Proof. If (2.) has a posiive soluion, u on I T, hen disconjugacy follows from he Surm separaion heorem. Conversely, if (2.) is disconjugae on he compac inerval [ρ(a), σ(b)], hen le u, v be he soluions of (2.) saisfying u(ρ(a)) = 0, u (ρ(a)) = and v(σ(b)) = 0, v (b) =. Since (2.) is disconjugae on [ρ(a), σ(b)], we have ha u() > 0 on (ρ(a), σ(b)], and v() > 0 on [ρ(a), σ(b)). Then is he desired posiive soluion. x() = u() + v() Theorem 4.7 (Pólya Facorizaion). If (2.) has a posiive soluion, u, on an inerval I T, hen for any x D, we ge he Pólya facorizaion Lx = α ()α 2 [α x] } () for I, where α := u > 0 on I, and α 2 := puu σ > 0 on I. Proof. Assume ha u is a posiive soluion of (2.) on I, and le x D. Then by he Lagrange Ideniy (Theorem 3.4), u()lx() x()lu() = u; x} () Lx() = u() pw (u, x)} () [ } x = puu σ () u() u] = α ()α 2 [α x] } (), for I, where α and α 2 are as described in he heorem.
A SECOND-ORDER SELF-ADJOINT DYNAMIC EQUATION 23 Theorem 4.8 (Trench Facorizaion). Le a T, and le ω := sup T. If ω <, assume ρ(ω) = ω. If (2.) is nonoscillaory on [a, ω), hen here is T such ha for any x D, we ge he Trench facorizaion Lx() = β ()β 2 [β x] } () for [, ω), where β, β 2 > 0 on [, ω), and =. β 2 () Proof. Since (2.) is nonoscillaory on [a, ω), (2.) has a posiive soluion, u on [, ω) for some T. Then by Theorem 4.7, Lx has a Pólya facorizaion on [, ω). Thus here are funcions α and α 2 such ha Lx() = α ()α 2 [α x] } () for [, ω), defined as described in he preceding heorem. Now, if =, α 2 () hen ake β () = α (), and β 2 () = α 2 (), and we are done. Therefore, assume ha In his case, le β () = <. α 2 () α () s and β 2() = α 2 () α 2 (s) ω α 2 (s) s σ() α 2 (s) s for [, ω). Noe ha as α, α 2 > 0, we have β, β 2 > 0 as well. Also, = lim β 2 () b ω b b = lim b ω = lim b ω =. Now le x D. Then [ ] [β x] α ()x() () = = for [, ω). So we ge s α 2 (s) [ α 2 () b [ s α 2 (s) s α 2 (s) β 2 ()[β ()x] = α 2 ()[α ()x()] s α 2 (s) ] ] σ() s α 2 (s) s[α α 2 (s) ()x()] + α ()x() α 2 () s s α 2 (s) σ() α 2 (s) α 2 (s) s + α ()x()
24 KIRSTEN R. MESSER for [, ω). Taking he -derivaive of boh sides gives β2 ()[β ()x()] } = α2 ()[α ()x()] } α 2 (s) s + α 2 ()[α ()x()] } [ ρ α 2 (s) s +[α ()x()] for [, ω). We now claim ha he las wo erms in his expression cancel. To see his, pu he expression back in erms of our posiive soluion u, and consider A and T A separaely. Careful applicaion of Theorem 2.6 hen shows ha hese erms do, in fac cancel, and we ge β2 ()[β ()x()] } = α2 ()[α ()x()] } α 2 (s) s. I hen follows ha β () β 2 ()[β ()x()] } = α () α 2 ()[α ()x()] } = Lx(), for [, ω) and he proof is complee. Theorem 4.9 (Recessive and Dominan Soluions). Le a T, and le ω := sup T. If ω < he we assume ρ(ω) = ω. If (2.) is nonoscillaory on [a, ω), hen here is a soluion, u, called a recessive soluion a ω, such ha u is posiive on [, ω) for some T, and if v is any second, linearly independen soluion, called a dominan soluion a ω, he following hold.. lim ω u() v() = 0 2. 3. b 4. p()v () v() = p()u()u σ () < for b < ω, sufficienly close, and p()v()v σ () > p()u () u() for < ω, sufficienly close. The recessive soluion, u, is unique, up o muliplicaion by a nonzero consan. Proof. The proof of his heorem is direcly analogous o he sandard proof used in he differenial equaions case. See, for example, [5]. Research suppored by NSF Gran072505. The views expressed in his aricle are hose of he auhor and do no reflec he official policy or posiion of he Unied Saes Air Force, Deparmen of Defense, or he U.S. Governmen. ] REFERENCES [] F. Merdivenci Aıcı and G. Sh. Guseinov. On Green s funcions and posiive soluions for boundary value problems. J. Compu. Appl. Mah., 4:75 99, 2002. [2] M. Bohner and A. Peerson. Dynamic Equaions on Time Scales: An Inroducion wih Applicaions. Birkhäuser, 200.
A SECOND-ORDER SELF-ADJOINT DYNAMIC EQUATION 25 [3] G. Sh. Guseinov and B. Kaymakçalan. On he Riemann inegraion on ime scales. In B. Aulbach, S. Elaydi, and G. Ladas, ediors, Conference Proceedings of he Sixh Inernaional Conference on Difference Equaions and Applicaions, Augsburg, 200. Taylor and Francis. To appear. [4] W. Kelley and A. Peerson. Difference Equaions: An Inroducion wih Applicaions. Academic Press, 2nd ediion, 200. [5] W. Kelley and A. Peerson The Theory of Differenial Equaions: Classical and Qualiaive. Prenice Hall. To appear.