Periodic solutions of functional dynamic equations with infinite delay

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1 Nonlinear Analysis 68 (28) Periodic soluions of funcional dynamic equaions wih infinie delay Li Bi a, Marin Bohner b, Meng Fan a, a School of Mahemaics and Saisics, KLAS and KLVE, Norheas Normal Universiy, Changchun, Jilin, 1324, PR China b Deparmen of Mahemaics and Saisics, Universiy of Missouri Rolla, Rolla, MO 6541, USA Received 2 Augus 26; acceped 6 December 26 Absrac In his paper, sufficien crieria are esablished for he exisence of periodic soluions of some funcional dynamic equaions wih infinie delays on ime scales, which generalize and incorporae as special cases many known resuls for differenial equaions and for difference equaions when he ime scale is he se of he real numbers or he inegers, respecively. The approach is mainly based on he Krasnosel skiĭ fixed poin heorem, which has been exensively applied in sudying exisence problems in differenial equaions and difference equaions bu rarely applied in sudying dynamic equaions on ime scales. This sudy shows ha one can unify such exisence sudies in he sense of dynamic equaions on general ime scales. c 27 Elsevier Ld. All righs reserved. Keywords: Periodic soluion; Time scale; Funcional dynamic equaion; Infinie delay 1. Inroducion In he real world, some processes vary coninuously while ohers vary discreely. These processes can be modeled by differenial equaions and difference equaions, respecively. There are also many processes ha vary boh coninuously and discreely. Thus an ineresing and challenging problem arises for mahemaicians: How can we model hese mixed processes? The erraic sop sar of he real world has long defeaed mahemaicians. Now he heory of ime scale calculus and dynamic equaions on ime scales provides us wih a powerful ool for aacking such mixed processes [2]. For example, ime scales are believed o provide a good way o undersand and conrol he Wes Nile virus since ime scale calculus and dynamic equaions on ime scales bridge he divide beween discree and coninuous aspecs of he Wes Nile virus [2]. In addiion, he choice of ime scale is very imporan in real world applicaions (see, e.g., [7 9]). The calculus on ime scales (see [4,5] and references cied herein) was iniiaed by Sefan Hilger in his 1988 Ph.D. disseraion [12] in order o unify coninuous and discree analysis, and i has a remendous poenial for applicaions and has recenly received much aenion since his foundaional work. The wo main feaures of he calculus on ime scales are unificaion and exension. The global exisence of periodic soluions of differenial equaions and difference equaions is a very basic and imporan problem, which plays a rôle similar o ha of globally sable equilibrium in an auonomous model. Thus, Corresponding auhor. addresses: bohner@umr.edu (M. Bohner), mfan@nenu.edu.cn (M. Fan) X/$ - see fron maer c 27 Elsevier Ld. All righs reserved. doi:1.116/j.na

2 L. Bi e al. / Nonlinear Analysis 68 (28) i is reasonable o seek condiions under which he resuling periodic nonauonomous sysem would have a periodic soluion. Much progress has been seen in his direcion and many crieria are esablished based on differen approaches (e.g., differenial equaions [6,15,17,19,22 24], difference equaions [1,14,16,18,25]). Careful invesigaion reveals ha i is similar o explore he exisence of periodic soluions for nonauonomous differenial equaions and heir discree analogue in he approaches, he mehods and he main resuls. For example, exensive research shows ha many resuls concerning he exisence of periodic soluions of differenial equaions can be carried over o heir discree analogues ([2,3] and references cied herein, [14 17,24,25]). I is naural o ask wheher we can explore such an exisence problem in a unified way and offer more general conclusions. Scholars in he fields of differenial equaions and difference equaions have long been aware of he sarling similariies of and inriguing differences beween he wo fields. Many resuls concerning differenial equaions carry over quie easily o corresponding resuls for difference equaions, while oher resuls seem o be compleely differen from heir coninuous counerpars. The sudy of dynamic equaions on general ime scales unifies and exends he fields of differenial and difference equaions, highlighing he similariies and providing insigh ino some of he differences. To prove a resul for a dynamic equaion on a ime scale is no only relaed o he se of real numbers or se of inegers bu also o hose peraining o more general ime scales. For example, exensive research reveals ha many resuls concerning he exisence of periodic soluions of predaor prey sysems modeled by differenial equaions can be carried over o heir discree analogues based on coincidence heory. On he basis of his fac, wih he help of he heory of dynamic equaions on ime scales and a coninuaion heorem in coincidence degree heory, Bohner, Fan and Zhang [2,3] sysemaically unified he exisence of periodic soluions of populaion models modeled by ordinary differenial equaions and heir discree analogues in he form of difference equaions and exended hese resuls o more general ime scales. Only a few papers have sudied he exisence of soluions of periodic boundary value problems of some dynamic equaions on ime scales (see, e.g., [1,21]). Among he known resuls on he exisence of periodic soluions of differenial equaions and difference equaions, many are achieved based on he Krasnosel skiĭ fixed poin heorem. The heoreical evidence suggess ha many resuls of he discree sysems are similar o hose of he corresponding coninuous sysems based on he Krasnosel skiĭ fixed poin heorem (e.g., [14 17,24,25]), which moivaes us o consider wheher we can unify hese resuls and exend hem o more general ime scales. Alhough he Krasnosel skiĭ fixed poin heorem has been proved o be powerful and effecive in dealing wih exisence problems and has been applied widely o sudy he exisence of periodic soluions of differenial equaions and difference equaions, i is rarely used o explore he exisence of periodic soluions of dynamic equaions on ime scales, especially hose wih infinie delays. In his paper, we will sysemaically invesigae he exisence of periodic soluions of some dynamic equaions wih infinie delay on ime scales, which will unify some relaed sudies of such problems for differenial equaions and difference equaions. The approach is based on he Krasnosel skiĭ fixed poin heorem. The se-up of his paper is as follows. In he coming secion, we presen some preliminary resuls on he calculus on ime scales. In Secion 3, we esablish a C h space for he funcional dynamic equaions wih infinie delay on ime scales. Then, we presen some preliminary resuls and he Krasnosel skiĭ fixed poin heorem. In he res of his paper, we sysemaically explore he exisence of periodic soluions of some dynamic equaions wih infinie delay on ime scales, which incorporae as special cases many well-known models in populaion dynamics, hemaopoiesis, ec. This sudy reveals ha, when we deal wih he exisence of posiive periodic soluions of differenial equaions and difference equaions, i is unnecessary o prove resuls for differenial equaions and separaely again for heir discree analogues (difference equaions). One can unify such problems in he framework of dynamic equaions on ime scales. 2. Preliminaries on ime scales In his secion, firs we will menion wihou proof several foundaional definiions and resuls from he calculus on ime scales so ha he paper is self-conained. For more deails, one can see [4,5]. A ime scale, which is a special case of a measure chain, is an arbirary nonempy closed subse of he real numbers R. Throughou his paper, we will denoe a ime scale by he symbol T, which has he opology inheried from he real numbers wih he sandard opology. Le R + =[, ), R = (, ] and a, b T, and define he inervals in T by T = T R, T + = T R +, (a, b) ={ T : a < < b}, [a, b] ={ T : a b}.

3 1228 L. Bi e al. / Nonlinear Analysis 68 (28) Oher inervals are defined accordingly. Definiion 2.1. Le T be a ime scale. For T, we define he forward and backward jump operaors σ, ρ : T T by σ() := inf{s T : s > } and ρ() := sup{s T : s < }. If < sup T and σ() =, hen is called righ-dense (oherwise: righ-scaered), and if > inf T and ρ() =, hen is called lef-dense (oherwise: lef-scaered). The graininess funcion µ : T R + is defined by µ() = σ(). Definiion 2.2. A funcion f : T R is called rd-coninuous provided i is coninuous a righ-dense poins in T and is lef-sided limis exis (finie) a lef-dense poins in T. The se of all rd-coninuous funcions f : T R is denoed by C rd = C rd (T) = C rd (T, R). The se of all bounded rd-coninuous funcions is denoed by BC rd.a funcion f : T R is called regulaed provided is righ-sided limis exis (finie) a all righ-dense poins in T and is lef-sided limis exis (finie) a all lef-dense poins in T. Lemma 2.1. Assume f : T R. (i) If f is coninuous, hen f is rd-coninuous. (ii) If f is rd-coninuous, hen f is regulaed. (iii) Assume f is coninuous. If g : T R is regulaed or rd-coninuous, hen f g has ha propery oo. (iv) Every regulaed funcion on a compac inerval is bounded. Definiion 2.3. Assume ha f : T R is a funcion and le T κ. Then we define f () o be he number (provided i exiss) wih he propery ha, for any given ε>, here is a neighborhood U of (i.e., U = ( δ, + δ) T for some δ>) such ha f (σ ()) f (s) f () σ() s ε σ() s for all s U. We call f () he dela (or Hilger) derivaive of f a. IfF = f, hen we define he Cauchy inegral by s r f (τ) τ = F(s) F(r) for r, s T. Lemma 2.2. If T κ and f : T R is dela differeniable a, hen f σ () = f () + µ() f (), where f σ = f σ. Lemma 2.3. If f C rd and T κ, hen σ() f (s) s = µ() f (). Lemma 2.4. If a, b, c T and f C rd, hen: (i) b a f () = c a f () + b c f () ; (ii) if f () g() for all [a, b), hen b a (iii) if f () for all a < b, hen b a f () b a g() ; f (). Definiion 2.4. If a T, sup T =, and f is rd-coninuous on [a, ), hen we define he improper inegral by b f () := lim f () a b a provided his limi exiss, and we say ha he improper inegral converges in his case. If his limi does no exis, hen we say ha he improper inegral diverges. Lemma 2.5. If f, g : T R are dela differeniable a T κ, hen ( fg) () = f ()g() + f σ ()g () = f ()g () + f ()g σ ().

4 L. Bi e al. / Nonlinear Analysis 68 (28) Lemma 2.6. Every rd-coninuous funcion has an aniderivaive. In paricular, if F() := f (τ) τ for all T is an aniderivaive of f, i.e., F = f. T, hen F defined by Definiion 2.5. A funcion r : T R is called regressive provided 1 + µ()r() for all T κ. The se of all regressive and rd-coninuous funcions will be denoed by R. Definiion 2.6. We define he se R + of all posiively regressive elemens of R by R + ={p R : 1 + µ()p() > for all T}. Definiion 2.7. If p R, hen he dela exponenial funcion e p (, s) is defined as he unique soluion of he iniial value problem y = p()y, y(s) = 1, where s T. Furhermore, for p, q R, we also define p q = p + q + µpq and p q = p q 1 + µq. Lemma 2.7. If p, q R, hen e p (, ) = 1, e p (, s) = 1/e p (s, ), e p (, u)e p (u, s) = e p (, s), e p (σ (), s) = (1 + µ()p())e p (, s), e p (s,σ()) = e p(s, ) 1 + µ()p() e p (, s) = pe p(, s), e p (s, ) = ( p)e p(s, ), e p q = e p e q, e p q = e p e q. Lemma 2.8. If p R + and T, hen e p (, )> for all T. Definiion 2.8. Le ω>. A ime scale T is called ω-periodic if +ω T whenever T. We also wrie T+ω T, i.e., { + ω : T} T. A funcion p is said o be ω-periodic on T if p( + ω) = p() for all T. Theorem 2.1. Suppose T is ω-periodic, p C rd (T) is ω-periodic, and a, b T. Then σ( + ω) = σ() + ω, ρ( + ω) = ρ() + ω, µ( + ω) = µ(), b+ω a+ω p() = b a p(), e p (b, a) = e p (b + ω,a + ω) if p R, k p := e p ( + ω,) 1 is independen of T whenever p R. Proof. The firs wo saemens follow from T + ω = T. Nex, µ( + ω) = σ( + ω) ( + ω) = σ() + ω ω = σ() = µ(). If p C rd is ω-periodic on T, hen we use [4, Theorem 1.98] wih ν() = + ω so ha ν is sricly increasing and saisfies ν () = 1 and ν(t) = T + ω = T =: T. Hence, using [4, Theorem 1.98], b a p() = = b a b+ω a+ω p()ν () = p(s ω) s = ν(b) ν(a) b+ω a+ω (p ν 1 )(s) s p(s) s.

5 123 L. Bi e al. / Nonlinear Analysis 68 (28) Nex, if p R, we use he represenaion of he exponenial funcion in erms of he cylinder ransform ξ [4, Definiion 2.3] o conclude { b } { b+ω } e p (b, a) = exp ξ µ() (p()) = exp ξ µ() (p()) = e p (b + ω,a + ω). a a+ω Finally, le p R and define f () := e p ( + ω,) for T. Le T. Then f () = e p ( + ω, )e p (, ) = e p (, ω)e p (, ) so ha f = e p (, ω)e p (, ) and hence, using Lemma 2.7, f = e p (, ω)e p (, ) + e σ p (, ω)e p (, ) = pe p (, ω)e p (, ) + (1 + µp)e p (, ω)( p)e p (, ) p = pf (1 + µp) f = pf pf =. 1 + µp Therefore k p = e p ( + ω,) 1 does indeed no depend on T. Lemma 2.9. Le T be ω-periodic and suppose f : T T R saisfies he assumpions of [4, Theorem 1.117]. Define g() = +ω f (, s) s. If f (, s) denoes he derivaive of f wih respec o, hen g () = +ω f (, s) s + f (σ (), + ω) f (σ (), ). Proof. Le T and use a proof similar o ha of he fourh par of Theorem 2.1 o obain g() = +ω f (, s) s + +ω +ω Now we apply [4, Theorem 1.117] o arrive a g () = = +ω +ω This concludes he proof. 3. The C h space f (, s) s = f (, s) s f (σ (), ) + +ω f (, s) s + f (σ (), + ω) f (σ (), ). f (, s) s + f (, s + ω) s. f (, s + ω) s + f (σ (), + ω) I is well known ha he developmen of he heory of funcional differenial equaions wih infinie delay primarily depends on he choice of a phase space. Many auhors have acively devoed heir research o his opic, and various phase spaces have been proposed (see [13,23] and references cied herein). The phase space for iniial funcions plays a very imporan rôle in he sudy of funcional dynamic equaions wih infinie delay on ime scales. However, no aemps have been made o consruc a phase space for funcional dynamic equaions wih infinie delay on ime scales. In his secion, we esablish a phase space for funcional dynamic equaions wih infinie delay on ime scales. Suppose inf T =and 1, 2 T imply T. Le h C rd (T, R + ), h(s) > for all s T, and h(s) s = 1. Define { } C h = ϕ C rd (T, R n ) : h(s) ϕ [s,] s <, where ϕ [s,] = sup ϕ(θ). s θ

6 L. Bi e al. / Nonlinear Analysis 68 (28) One can easily verify ha C h is a linear subspace of C rd and BC rd is a linear subspace of C h.forϕ C h, define ϕ h = h(s) ϕ [s,] s < ; hen (C h, h ) is a normed space. For simpliciy, we denoe i by C h. Lemma 3.1. For any ε> and K >, here exiss δ = δ(ε, K )> such ha, for any ϕ 1,ϕ 2 C h,if ϕ 1 ϕ 2 h δ, hen ϕ 1 ϕ 2 [ K,] ε. Proof. We complee he proof by conradicion. Suppose ha here exis ε > and K > such ha, for any δ>, here exis ϕ1 δ,ϕδ 2 C h, such ha ϕ1 δ ϕδ 2 h δ bu ϕ1 δ ϕδ 2 [ K,] >ε. Le δ = ε K 2 h(s) s > and pu ϕ 1 = ϕδ 1 and ϕ 2 = ϕδ 2. Then ϕ 1 ϕ 2 h δ bu ϕ1 ϕ 2 [ K,] >ε. Hence δ ϕ 1 ϕ 2 h = = K K K h(s) ϕ 1 ϕ 2 [s,] s h(s) ϕ 1 ϕ 2 [s,] s + h(s) ϕ 1 ϕ 2 [s,] s K h(s) ϕ 1 ϕ 2 [s,] s K h(s) ϕ1 ϕ 2 [ K,] s >ε h(s) s = 2δ, a conradicion. The proof is complee. Lemma 3.2. Suppose ha {ϕ n } C rd (T, R n ) is uniformly bounded. Then lim n ϕ n ϕ h = if and only if for any K N, we have lim n ϕ n ϕ [ K,] =. Proof. We prove necessiy firs. By Lemma 3.1, for any ε>and K >, here exiss δ = δ(ε, K ) such ha, for any ϕ 1,ϕ 2 C h wih ϕ 1 ϕ 2 h δ, wehave ϕ 1 ϕ 2 [ K,] <ε. Since lim n ϕ n ϕ h =, here exiss N N such ha, for any n > N,wehave ϕ n ϕ h δ for all n N. Hence ϕ n ϕ [ K,] <ε, i.e., lim ϕ n ϕ [ K,] =. n Now we show sufficiency. Suppose ha {ϕ n } is uniformly bounded, i.e., here exiss H > such ha ϕ n H for all n N. Le ε>. Since h(s) s = 1 <, here exiss K N such ha K h(s) s <ε. Moreover, for all k N, here exiss N k N such ha if n N k, hen ϕ n ϕ [ k,] ε, whence ϕ [ k,] H +ε,so ϕ H +ε. Therefore, for n N K,wehave ϕ n ϕ h = = K h(s) ϕ n ϕ [s,] s h(s) ϕ n ϕ [s,] s + h(s) ϕ n ϕ [s,] s K (2H + ε)ε + ε = (2H + ε + 1)ε. This complees he proof. Lemma 3.3. The space (C rd [a, b], R k ) is complee when endowed wih he supremum norm. Proof. Clearly C rd [a, b] is a linear space. By Lemma 2.1, every rd-coninuous funcion is bounded. Le { f n } C rd [a, b] be a Cauchy sequence. Le ε>. There exiss N ε N such ha for any m, n N ε,we have f m f n [a,b] = sup [a,b] f m () f n () <ε. Now le m, n N ε and [a, b]. Then f m () f n () sup [a,b] f m () f n () <ε. Thus { f n ()} R k is a Cauchy sequence and hence convergen o, say, f (). Le n N ε. Then f n () f () = lim f n() f m () ε m 3. We will show ha f C rd [a, b].

7 1232 L. Bi e al. / Nonlinear Analysis 68 (28) Suppose [a, b] is righ-dense. Le ε>. Since f Nε/3 C rd [a, b], here exiss a neighborhood U 1 of such ha for any U 1,wehave f Nε/3 ( ) f Nε/3 () < 3 ε. Le U 1. Then f ( ) f () f () f Nε/3 ( ) + f Nε/3 ( ) f Nε/3 () + f Nε/3 () f () <ε. I follows ha f is coninuous a he righ-dense poin. Now suppose [a, b] is lef-dense. Le ε>. Since f Nε/2 C rd [a, b], we can conclude ha here exis δ> and α R such ha f Nε/2 () α < ε for all U 2 = ( δ, ) T. 2 Le U 2. Then f () α f () f Nε/2 () + f Nε/2 () α <ε. Thus f has he finie limi α a he lef-dense poin. Therefore, f C rd [a, b]. This implies ha C rd [a, b] is complee. Theorem 3.1. (C h, h ) is a Banach space. Proof. Le {ϕ n } C h be a Cauchy sequence. Thus {ϕ n } is bounded, i.e., here exiss M > such ha ϕ n h M for all n N. Le ε>. Le K > be such ha K ε h(s) s < 4M + ε. Since lim m,n ϕ n ϕ m h =, by Lemma 3.2, wehavelim m,n ϕ n ϕ m [ K,] =, so {ϕ n } is a Cauchy sequence in C rd ([ K, ]). Hence, by Lemma 3.3, here exiss ϕ C rd ([ K, ]) such ha ϕ n ϕ wih respec o he supremum norm on [ K, ]. Therefore here exiss N N such ha ϕ n ϕ [ K,] < ε 2 for all n N. Hence for K, ϕ() ϕ n () ϕ() + ϕ n () ϕ n ϕ [ K,] + M < ε 2 + M. Now define ϕ() = ϕ( K ) for all < K. Then ϕ C rd (T ) and for n N we now have ϕ n ϕ h = < < Hence ϕ n ϕ in C h. K K ( 2M + ε 2 h(s) ϕ n ϕ [s,] s ( 2M + ε ) 2 ( h(s) ϕ n [s,] + ϕ [s,]) s + h(s) (M + ε ) 2 + M s + ) K h(s) s + ε 2 ε 4M + ε + ε 2 = ε. K K h(s) ε 2 s h(s) s h(s) ϕ n ϕ [ K,] s Theorem 3.2. Suppose ha ϕ C h and x (θ) = x( + θ) for θ (, ]. (i) Le A (, ). Suppose ha x : (, A) R n is rd-coninuous on [, A] and saisfies x = ϕ. Then for any [, A], we have x C h and x is rd-coninuous wih respec o. (ii) There exiss K 1 such ha ϕ() K 1 ϕ h.

8 Proof. We firs show (i). In fac, for any [, A], h(s) x [s,] s = = L. Bi e al. / Nonlinear Analysis 68 (28) h(s) x [s,] s + h(s) x [s,] s ( h(s) max x [s+,], x [,]) s + h(s) x [s+,] s + h(s) ϕ [s,] s + 2 h(s) x [,] s + h(s) x [,A] s h(s) ϕ [s,] s + 2 x [,A] < h(s) x [,] s h(s) x [,] s so ha x C h for [, A]. Nex we show ha x is rd-coninuous wih respec o. We prove ha x is coninuous a righ-dense poins (in a similar way i can be shown ha x has finie lef-sided limis a lef-dense poins, so he deails are omied here). Le [, A] be righ-dense. If is also lef-dense (again, in a similar way, we can show he corresponding conclusion for he case ha is lef-scaered, and hence he deails are omied here), le [, ). Since x C h, for any ε>, here exiss M(,ε)> such ha M M h(s) x [s,] s < ε and h(s) s < ε 4 4L. By Lemma 2.1(iv), we know ha x is bounded on [, A], say x [,A] L. Since C h is a subspace of C rd and x C h, we have x C rd. Assume M θ. If θ is righ-dense, since x C rd, we can choose sufficienly small δ 1 such ha for any ( δ 1, + δ 1 ) T,wehave x (θ) x (θ) = x (θ) x (θ + ) < ε 4. If θ is righ-scaered and lef-scaered, i is obvious ha here exiss δ 2 > such ha if θ (θ δ 2,θ + δ 2 ) T, hen θ = θ. So, when <δ 2,wehaveθ δ 2 <θ+ <θ+ δ 2, i.e., θ + (θ δ 2,θ + δ 2 ). Hence θ + = θ. Therefore x (θ) x (θ) = x (θ) x (θ + ) = < ε 4. Assume θ is righ-scaered and lef-dense. Noe ha since x C rd, by definiion, x has a finie lef-sided limi a θ. Hence here exis δ 3 > and α R such ha x (s) α < ε 8 for any s (θ δ 3,θ + δ 3 ) T. Then x (s) x (θ) < x (s) α + x (θ) α < ε 8 + ε 8 = ε 4. Le <δ 3. Then θ + (θ δ 3,θ + δ 3 ) T. Thus x (θ) x (θ) = x (θ + ) x (θ) < ε 4. So, for any θ [ M, ], here is δ>such ha, if ( δ, + δ) T, hen max M θ x (θ) x (θ) < 4 ε. Thus x x h = M h(s) x x [s,] s h(s)( x [s,] + x [s,] ) s + h(s) x x [s,] s M

9 1234 L. Bi e al. / Nonlinear Analysis 68 (28) M M [ { h(s) max x [,], x [s,]} + x [s,]] s + h(s) x x [s,] s M h(s)(l + 2 x [s,] ) s + x x [ M,] L ε 4L + 2 ε 4 + ε 4 = ε. To conclude, we have proved ha x is rd-coninuous wih respec o on [, A]. For (ii), noe ha ϕ h = h(s) ϕ [s,] s h(s) ϕ() s = ϕ(). Remark 3.1. If T = R, hen he C h space is he phase space esablished in [23] for funcional differenial equaions wih infinie delay. 4. Krasnosel skiĭ s fixed poin heorem and preliminary resuls Le ω T be posiive. Assume ha T is an ω-periodic ime scale (see Definiion 2.8). In he nex secion, we will firs explore he exisence and nonexisence of posiive ω-periodic soluions of he funcional dynamic equaion wih infinie delay on T of he form x () = a(, x())x(σ ()) + f (, x ), T, (4.1) where x C h and x (θ) = x( + θ) for θ (, ]. Throughou his paper, we assume: (H 1 ) a(, x) is rd-coninuous and ω-periodic in and is coninuous in x. There exis ω-periodic α, β C rd such ha α() a(, x) β() and k α > (where k α is defined as in Theorem 2.1). (H 2 ) f (,ϕ)is rd-coninuous in and is coninuous in ϕ wih f ( + ω,ϕ) = f (,ϕ)for ϕ C h, and f (,ϕ)maps bounded ses ino bounded ses and f (,ϕ) for ϕ C h wih ϕ(θ), θ T. In order o explore he periodiciy of (4.1), we firs inroduce he essenial ool o be applied hroughou his paper and prove some preliminary resuls. Definiion 4.1. Le (X, ) be a Banach space. A nonempy closed subse K X is called a cone if i saisfies he following wo condiions: (i) For any u,v K,α,β >, we have αu + βv K. (ii) u, u K implies u =. Now le us inroduce he famous Krasnosel skiĭ fixed poin heorem [11], which will come ino play soon. Lemma 4.1. Le X be a Banach space and le K X be a cone. Assume ha Ω 1, Ω 2 are bounded open subses of X wih Ω 1, Ω 1 Ω 2, and le F : K (Ω 2 \ Ω 1 ) K be a compleely coninuous operaor such ha one of (i) Fu u for any u K Ω 1, and Fu u for any u K Ω 2 ; (ii) Fu u for any u K Ω 1, and Fu u for any u K Ω 2 holds. Then F has a fixed poin in K (Ω 2 \ Ω 1 ). Le and define P w ={u C rd (T, R) : u( + ω) = u()} u = max [,ω] u() for u P w. Lemma 4.2. Suppose ha {u n } P ω,u P ω, and u n uasn. Then {u n } converges uniformly o u C h wih respec o.

10 L. Bi e al. / Nonlinear Analysis 68 (28) Proof. Le ε>. Since u n u asn, here exiss N N such ha for n N, wehave u n u <ε. Le n N. Then u n u h = h(s) u n u [s,] s = h(s) u n u [s+,] s u n u h(s) s <ε. This complees he proof. Lemma 4.3. Le b, p P ω. Then x () = b()x(σ ()) + p() (4.2) has a unique ω-periodic soluion x given by x() = 1 +ω p(s)e b (s, ) s. (4.3) k b Proof. Firs we show ha x defined by (4.3) is an ω-periodic soluion of (4.2). Using Theorem 2.1, wefind x( + ω) = 1 +2ω p(s)e b (s, + ω) s = 1 +ω p(s + ω)e b (s + ω, + ω) s k b k b +ω = 1 +ω p(s)e b (s, ) s = x() k b so ha x is ω-periodic. Nex, we use Lemma 2.9, Theorem 2.1, Lemmas 2.7 and 2.3 o calculae +ω { k b x () + b()x(σ ()) } = p(s)( b)()e b (s, ) s + p( + ω)e b ( + ω,σ()) p()e b (,σ()) { +ω σ() } σ(+ω) + b() p(s)e b (s,σ()) s p(s)e b (s,σ()) s + p(s)e b (s,σ()) s +ω = b() + b() +ω { +ω p(s)e b (s,σ()) s + p()e b ( + ω,σ()) p()e b (,σ()) } p(s)e b (s,σ()) s µ()p()e b (,σ()) + µ( + ω)p( + ω)e b ( + ω,σ()) = p() {e b ( + ω,σ()) e b (,σ()) µ()b()e b (,σ()) + µ()b()e b ( + ω,σ())} = p() {(1 + µ()b())e b ( + ω,σ()) (1 + µ()b())e b (,σ())} = p() {e b ( + ω,) e b (, )} = k b p() so ha x solves (4.2). Now we assume ha x is any ω-periodic soluion of (4.2). Le T. ByLemmas 2.5 and 2.7, wehave [xe b (, )] = x e b (, ) + x σ e b (, ) = e b (, )(x + bx σ ) = e b (, )p. Inegraing boh sides of his equaion from o + ω produces +ω e b (s, )p(s) s = x( + ω)e b ( + ω, ) x()e b (, ) = x()[e b ( + ω, ) e b (, )]=x()e b (, )k b so ha indeed x is equal o he righ-hand side of (4.3). For any u P ω, consider he dynamic equaion x () = a(, u())x(σ ()) + f (, u ). (4.4)

11 1236 L. Bi e al. / Nonlinear Analysis 68 (28) From (H 2 ), Lemma 2.1 and Theorem 3.2, i follows ha f (, u ) P ω. Lemma 4.3 ells us ha he unique ω-periodic soluion of (4.4) is given by where x u () = +ω G(, s) f (s, u s ) s, G(, s) = e a(s, ) wih a() = a(, u()) for u P ω. k a Define he operaor F : P ω P ω by (Fu)() = +ω G(, s) f (s, u s ) s for u P ω and T. (4.5) One can easily show ha x is an ω-periodic soluion of (4.1) if and only if x is a fixed poin of F in P ω. Define γ 1 = inf e α(s, ), γ 2 = sup e β (s, ), s ω s ω δ = γ 1k α γ 2 k β. From (H 1 ), Definiion 2.7, and he definiion of G(, s), we can conclude he following. Lemma 4.4. The funcion G(, s) saisfies: (i) G(, s) = G( + ω,s + ω) for any s [, + ω]; (ii) A := γ 1 k β G(, s) γ 2 k α =: B for any s [, + ω]. I is clear ha δ = A/B and <δ 1. Define K δ ={x P ω : x() δ x for all T}. I is rivial o show ha K δ is a cone. Lemma 4.5. F(K δ ) K δ. Proof. For any u K δ, we show ha Fu K δ. I is easy o see ha Fu P ω by he definiion of F. Since u K δ, we have u s (θ) = u(s + θ) for θ T.By(H 2 ) and Lemma 4.4, wehave f (s, u s ), (Fu)(), and (Fu)() = +ω G(, s) f (s, u s ) s B Hence Fu B f (s, u s ) s. Moreover (Fu)() = +ω which implies Fu K δ. G(, s) f (s, u s ) s A f (s, u s ) s. f (s, u s ) s A Fu = δ Fu, B Lemma 4.6. Le η> and Ω ={x P ω : x <η}. Then F : K δ Ω K δ is compleely coninuous. Proof. Firs we show ha F is coninuous. Suppose ha where u n, u K δ Ω, G n (, s) = e a n (s, ) k an u n u, (Fu n )() = wih a n () := a(, u n ()). +ω G n (, s) f (s, u n s ) s,

12 Now, using Lemma 2.9,wefind +ω L. Bi e al. / Nonlinear Analysis 68 (28) (Fu n ) () = ( a n )()G(, s) f (s, u n s ) s + G(σ (), + ω) f (, un ) G(σ (), ) f (, un ) = ( a n )()(Fu n )() + f (, un ) { } 1 k a 1 + µ()a n () e 1 a n ( + ω,) 1 + µ()a n () = ( a n )()(Fu n f (, u n )() + ) 1 + µ()a n (). Hence for v n = Fu n Fu,wehave where (v n ) () = (Fu n ) () (Fu) () = ( a n )()(Fu n f (, u n )() + ) 1 + µ()a n () ( a)()(fu)() f (, u ) 1 + µ()a() = ( a n )()v n () + f n (), { f n () = {( a n )() ( a)()} (Fu)() + f (, u n ) 1 + a n ()µ() f (, u ) 1 + a()µ() By Lemma 4.2 and (H 2 ), one has u n u and f (, u n ) f (, u ) as n. Moreover, since a(, x) is coninuous wih respec o x on {(, x) : ω, x η} and a(, u n ()) a(, u()) (i.e., a n a in brief), we have f n () asn. Since f maps bounded ses ino bounded ses by (H 2 ), here exiss M 1 > such ha f (s, x s ) M 1 for any x Ω and s [, + ω]. In addiion, here exiss M 2 > such ha a(, x) M 2. Hence, f n () 2M 2 (Fu)() +2M 1 2M 2 BωM 1 + 2M 1. Consequenly, v n B f n (s) s, where B = sup,s ω G n (, s), G n (, s) = e an (s, ), an k = a n. a n By he dominaed convergence heorem, we have v n = Fu n Fu asn, which shows ha F is coninuous. Nex, we prove ha F(K δ Ω) is compac. In fac, for any u K δ Ω, wehave (Fu)() B +ω f (s, u s ) s BωM 1 and (Fu) () = ( a)()(fu)() + f (, u ) 1 + a()µ() BωM 1 + M 1. So F(K δ Ω) is uniformly bounded and equiconinuous, and hence, by he Arzelà Ascoli heorem, F(K δ Ω) is compac. Therefore F : K δ Ω K δ is compleely coninuous. 5. Exisence and nonexisence of posiive periodic soluions For convenience in he subsequen discussion, we inroduce he following noaion: f = f = lim max ϕ h [,ω] lim max ϕ h [,ω] f (,ϕ), f = lim ϕ h f (,ϕ), f = lim ϕ h min ϕ h [,ω] ϕ h [,ω] D δ ={ϕ C h : ϕ(θ) δ ϕ h for all θ T }. min f (,ϕ) ϕ h, f (,ϕ) ϕ h, }.

13 1238 L. Bi e al. / Nonlinear Analysis 68 (28) Posiive periodic soluions of he sysem (4.1) Assume ha: (H 3 ) There exiss K 1 > such ha for any ϕ D δ wih ϕ h [δk 1, K 1 ],wehave f (,ϕ)> K 1 /(Aω). (H 4 ) There exiss K 2 > such ha for any ϕ D δ wih ϕ h K 2,wehave f (,ϕ)< K 2 /(Bω). Theorem 5.1. (i) If (H 3 ) holds and f = f =, hen (4.1) has a leas wo posiive ω-periodic soluions u 1 and u 2 wih < u 1 < K 1 < u 2. (ii) If (H 4 ) holds and f = f =, hen (4.1) has a leas wo posiive ω-periodic soluions u 1 and u 2 wih < u 1 < K 2 < u 2. Proof. We only prove (i) since he proof of (ii) is very similar. From f =, i follows ha, for any ε wih <ε 1/(Bω), here exiss r < K 1 such ha f (,ϕ) ε ϕ h for ϕ D δ wih < ϕ h r. (5.1) Le Ω r ={u P ω : u < r }. Then for any u K δ Ω r,wehave u = r and δ u h δ u u (θ). Hence u D δ and δr u h r.by(4.5) and (5.1), wehave Fu f (s, u s ) s Bε u s h s Bεω u u, ha is, Fu u for u K δ Ω r. On he oher hand, we know from f = ha, for any ε wih <ε<1/(2bω), here is N 1 > K 1 such ha f (,ϕ) ε ϕ h for ϕ D δ wih ϕ h N 1. (5.2) Le Ω r1 ={u P ω : u < r 1 }, where r 1 is chosen such ha r 1 > N Bω sup f (,ϕ). (5.3) ϕ h N 1,ϕ D δ [,ω] Then, for any u K δ Ω r1,wehaveδ u h δ u u (θ). Hence u D δ and δr 1 u h r 1 = u. I follows from (4.5), (5.2) and (5.3) ha +ω Fu B f (s, u s ) s = B f (s, u s ) s + B f (s, u s ) s I 1 I 2 B r 1 2Bω s + B ε u s h s r Br 1εω < r 1 = u, where I 1 ={s [,ω]: u s h N 1 } and I 2 ={s [,ω]: u s > N 1 }. This means Fu u for u K δ Ω r1. Le Ω K1 ={u P ω : u < K 1 } wih K 1 > r. Then, for any u K δ Ω K1, we obain K 1 u h δk 1.By (H 3 ), f (, x )>K 1 /(Aω).ByLemma 4.4, +ω Fu A f (s, u s ) s > AK 1ω Aω = K 1 = u. This shows ha Fu u for u K δ Ω K1. I follows from Lemmas 4.5 and 4.6 ha F : K δ (Ω r1 \ Ω K1 ) K δ and F : K δ (Ω K1 \ Ω r ) K δ are compleely coninuous.

14 L. Bi e al. / Nonlinear Analysis 68 (28) Thus, by Lemma 4.1, F has fixed poins u 1 and u 2 in K δ (Ω K1 \ Ω r ) and K δ (Ω r1 \ Ω K1 ), respecively. Tha is o say, (4.1) has a leas wo posiive ω-periodic soluions u 1 and u 2 wih < u 1 < K 1 < u 2. The following heorem is crucial in our subsequen argumens. Lemma 5.1. If (H 3 ) and (H 4 ) hold, hen (4.1) has a leas one posiive ω-periodic soluion u wih u lying beween K 1 and K 2, where K 1 and K 2 are defined in (H 3 ) and (H 4 ), respecively. Proof. Wihou any loss of generaliy, we can assume ha K 2 < K 1. Oherwise, we can employ Lemma 4.1(ii) o prove his lemma. Le Ω K2 ={u P ω : u < K 2 }. Then for any u K δ Ω K2, i follows from (4.5) and (H 4 ) ha +ω Fu B f (s, u s ) s < BωK 2 Bω = K 2 = u. This means ha Fu u for any u K δ Ω K2. Le Ω K1 ={u P ω : u < K 1 }. Then for any u K δ Ω K1,wehave u h K 1. I follows from (4.5) and (H 3 ) ha +ω Fu A f (s, u s ) s > AωK 1 Aω = K 1 = u. This means ha Fu > u for u K δ Ω K1. Hence he proof is complee by Lemma 4.1. Theorem 5.2. (i) If f [, 1/(Bω)) and f (1/(Aδω), ), hen (4.1) has a leas one posiive ω-periodic soluion. (ii) If f [, 1/(Bω)) and f (1/(Aδω), ), hen (4.1) has a leas one posiive ω-periodic soluion. Proof. We firs show (i). Assume ha f = α 1 [, 1/(Bω)) and f = β 1 (1/(Aδω), ). Forε = 1/(Bω) α 1 >, here exiss a sufficienly small R 1 > such ha for ϕ h R 1,wehave max [,ω] f (,ϕ) ϕ h <α 1 + ε = 1 Bω, i.e., when ϕ h R 1 and [,ω],wehave f (,ϕ)< ϕ h /(Bω) R 1 /(Bω). Hence (H 4 ) is saisfied. For ε = β 1 1/(Aδω) >, here exiss a sufficienly large R 2 > such ha, when ϕ h δr 2,wehave min [,ω] f (,ϕ) ϕ h >β 1 ε = 1 Aδω. Therefore, when ϕ h [δr 2, R 2 ] and [,ω], wehave f (,ϕ) > R 2 δ/(aδω) = R 2 /(Aω), i.e., (H 3 ) is saisfied. Lemma 5.1 ells us ha he claim (i) is rue. Now we show (ii). Assume ha f = α 2 (1/(Aδω), ) and f = β 2 [, 1/Bω). For any ε = α 2 1/(Aδω) >, here exiss a sufficienly small R 3 <δr 2 such ha, when < ϕ h R 3,wehave min [,ω] f (,ϕ) ϕ h >α 2 ε = 1 Aδω. This implies ha, when ϕ h [δr 3, R 3 ] and [,ω], wehave f (,ϕ) > δr 3 /(Aδω) = R 3 /(Aω), which shows ha (H 3 ) holds. For ε = 1/(Bω) β 2 >, f = β 2 implies ha here exiss a sufficienly large R 4 > R 1 such ha, when ϕ h > R 4,wehave max [,ω] f (,ϕ) ϕ h <β 2 + ε = 1 Bω. (5.4) In order o prove ha (H 4 ) is saisfied, we consider wo cases. Case 1. If max [,ω] f (,ϕ)is unbounded, hen here exis ϕ D δ wih ϕ h = R 5 > R 4 and [,ω] such ha f (,ϕ) f (,ϕ ) for < ϕ h ϕ h = R 5. (5.5)

15 124 L. Bi e al. / Nonlinear Analysis 68 (28) Since ϕ h = R 5 > R 4,by(5.4) and (5.5), we obain f (,ϕ) f (,ϕ )< ϕ h Bω = R 5 Bω for < ϕ h R 5 and [,ω], i.e., (H 4 ) holds. Case 2. If max [,ω] f (,ϕ)is bounded, hen here exiss M 5 > such ha f (,ϕ) M 5 for (,ϕ) [,ω] D δ. (5.6) Here we can choose R 5 such ha R 5 > M 5 Bω. When < ϕ h R 5, [,ω], by(5.6), one has f (,ϕ) M 5 < R 5 /(Bω), i.e., (H 4 ) holds. Lemma 5.1 implies he conclusion (ii). Theorem 5.3. (i) If (H 4 ) is saisfied and f, f (1/(Aδω), ), hen (4.1) has a leas wo posiive ω-periodic soluions u 1 and u 2 wih < u 1 < K 2 < u 2, where K 2 is defined in (H 4 ). (ii) If (H 3 ) is saisfied and f, f [, 1/(Bω)), hen (4.1) has a leas wo posiive ω-periodic soluions u 1 and u 2 wih < u 1 < K 1 < u 2, where K 1 is defined in (H 3 ). Proof. We only show (i) since he proof of (ii) is very similar. From f (1/(Aδω), ) and he proof of Theorem 5.2(i), we know ha here exiss a sufficienly large R 2 > K 2 such ha f (,ϕ)> R 2 Aω for ϕ D δ wih ϕ h [δr 2, R 2 ]. From f (1/(Aδω), ) and he proof of Theorem 5.2(ii), we know ha here exiss a sufficienly small R 2 (, K 2) such ha f (,ϕ)> R 2 Aω for ϕ D δ wih ϕ h [δr2, R 2 ]. By Lemma 5.1, (4.1) has a leas wo posiive ω-periodic soluions u 1 and u 2, which saisfy R2 < u 1 < K 2 < u 2 < R 2. The proof is complee. Theorem 5.4. If one of he condiions (i) f = and f [, 1/(Bω)), (ii) f = and f [, 1/(Bω)), (iii) f = and f (1/(Aδω), ), (iv) f = and f (1/(Aδω), ), is saisfied, hen (4.1) has a leas one posiive ω-periodic soluion. Proof. Assume ha he condiion in (i) is saisfied. We can choose M 6 so ha M 6 > 1/(Aδω). Since f =, here exiss a consan r 4 such ha f (,ϕ) M 6 ϕ h for ϕ D δ wih < ϕ h r 4. (5.7) Le Ω r4 ={u P ω : u < r 4 }. Then, for any u K δ Ω r4,wehaveδ u h δ u u (θ). I follows ha u D δ and δr 4 u h r 4.By(4.5) and (5.7), we obain +ω Fu A f (s, u s ) s AM 6 u s h s AM 6 δr 4 ω r 4 = u, i.e., for u K δ Ω r4,wehave Fu u. On he oher hand, he fac ha f [, 1/(Bω)) and Theorem 5.2(ii) imply ha here exiss R 5 > r 4 > such ha f (,ϕ)< R 5 /(Bω) for ϕ D δ wih ϕ h R 5. Le Ω R5 ={u P ω : u < R 5 }. Then, for any u K δ Ω R5, we have +ω Fu B f (s, u s ) s < BωR 5 Bω = R 5 = u, i.e., for u K δ Ω R5,wehave Fu u. Conclusion (i) is valid by Lemma 4.1. By argumens similar o hose above, we can prove he claim for he cases (ii) (iv). The deails are omied here.

16 L. Bi e al. / Nonlinear Analysis 68 (28) Remark 5.1. From Theorem 5.4, one knows ha if f =, f = or f =, f =, hen (4.1) has a leas one posiive ω-periodic soluion. When T = R, his conclusion reduces o he main resul of [19]. Theorem 5.5. Assume (H 4 ).If f =,f (1/(Aδω), ) or f =,f (1/(Aδω), ) holds, hen (4.1) has a leas wo posiive ω-periodic soluions u 1 and u 2 wih < u 1 < K 2 < u 2, where K 2 is defined in (H 4 ). Proof. We only prove he case for f =, f (1/(Aδω), ), since he oher case is similar. Le Ω r4 ={u P ω : u < r 4 } and f = α 3, where r 4 < r 2. I follows from f = and he proof of Theorem 5.4(i) ha Fu u for u K δ Ω r4. Le Ω R2 ={u P ω : u < R 2 } wih R 2 > K 2. I follows from f = α 3 (1/Aδω, ) and Theorem 5.2(i) ha f (,ϕ)> R 2 Aω for ϕ D δ wih ϕ h [δr 2, R 2 ]. By (H 4 ) and he proof of Lemma 5.1, (4.1) has a leas wo posiive ω-periodic soluions u 1 and u 2 wih < u 1 < K 2 < u 2. Theorem 5.6. Assume ha (H 3 ) holds. If f =, f [, 1/(Bω)) or f =, f [, 1/(Bω)), hen (4.1) has a leas wo posiive ω-periodic soluions u 1 and u 2 wih < u 1 < K 1 < u 2, where K 1 is defined in (H 3 ). So far, we have deliberaely explored he exisence and he mulipliciy of posiive periodic soluions. To conclude, we have proved he following heorem. Theorem 5.7. If f (1/(Aδω), ], f [, 1/(Bω)) or f [, 1/(Bω)), f (1/(Aδω), ] or (H 3 ), (H 4 ) holds, hen (4.1) has a leas one posiive ω-periodic soluion. If (H 3 ), f, f [, 1/(Bω)) or (H 4 ), f, f (1/(Aδω), ] is saisfied, hen (4.1) has a leas wo posiive ω-periodic soluions. Now we coninue wih he nonexisence of periodic soluions of (4.1). Theorem 5.8. Le R i,i {1, 2, 3, 4}, be as in he proof of Theorem 5.2. If ( ) f (,ϕ) 1 f, f, min R 3 ϕ h δr 2 ϕ h Aδω, or f, f, max R 1 ϕ h R 4 f (,ϕ) ϕ h [ ) 1, Bω is saisfied, hen (4.1) has no posiive ω-periodic soluion. Proof. We only prove he firs claim. From he fac ha f, f (1/(Aδω), ) and he proof of Theorem 5.2, i follows ha here exis R 3, R 2 > such ha f (,ϕ)> 1 Aδω ϕ h for < ϕ h R 3 and f (,ϕ)> 1 Aδω ϕ h for ϕ h δr 2. f (,ϕ) Then by min R3 ϕ h δr 2 ϕ h > Aδω 1,wehave f (,ϕ)> 1 Aδω ϕ h for any ϕ h (, ). If (4.1) has a posiive ω-periodic soluion, say v, hen Fv = v. Hence +ω +ω v = Fv = G(, s) f (s,v s ) s > A 1 Aδω v s h s +ω 1 +ω ωδ v 1 +ω s() s = ωδ v(s) s 1 δ v s = v, ωδ which is a conradicion. The claim (i) is valid. The proof of (ii) will be omied since i is similar o ha of (i).

17 1242 L. Bi e al. / Nonlinear Analysis 68 (28) Alhough we have esablished sufficien crieria for he nonexisence of a periodic soluion of (4.1), he crieria depend on he parameers in he proof of Theorem 5.2. Asin[22], consider he sysem x () = a(, x())x(σ ()) + λf (, x ), T, (5.8) where a parameer λ is isolaed from he funcional f. By argumens similar o hose above, one can easily reach he following claim. Theorem 5.9. If f >, f > or f, f <, hen (5.8) has no posiive ω-periodic soluion for sufficienly large or small λ>, respecively. Remark 5.2. By argumens similar o hose above, one can also derive sufficien crieria for he exisence of a leas one or wo periodic soluions for he funcional dynamic equaions wih infinie delay on ime scales of he form (5.8). Usually, sufficien crieria for he exisence or nonexisence of periodic soluions of he equaions of form (5.8) involve he saemen... for sufficienly large or small λ>... (one can see his from Theorem 5.9 and [16,22]), which are a lile bi less concree. So we prefer o sudy he funcional dynamic equaions of form (4.1) insead of form (5.8) in his paper. One can also observe ha he equaions of form (5.8) can faciliae he sudy of he nonexisence of periodic soluions as we have seen in Theorems 5.8 and 5.9. Remark 5.3. In his subsecion, we have sysemaically sudied he exisence and he nonexisence of posiive periodic soluions of (4.1). In fac, by exacly he same argumens, one can easily esablish he corresponding (same) crieria for he exisence and nonexisence of posiive periodic soluions of x () = a(, x())x(σ ()) f (, x ). (5.9) The only difference is ha G(, s) should be replaced by G(, s) = e a( + ω,s). k a Remark 5.4. As menioned in he inroducion, one of our principal aims is o unify he exisence of periodic soluions of some differenial equaions and heir corresponding discree analogues. If T = R and T = Z, hen (4.1) reduces o x () = a(, x())x() + f (, x ), R (5.1) and x( + 1) x() = a(, x())x( + 1) + f (, x ), Z, respecively. Anoher discree analogue of (5.1) reads x( + 1) x() = a(, x())x() + f (, x ), Z. In order o include his equaion in our sudy, i suffices o explore he exisence and nonexisence of periodic soluions of he sysem x () = a(, x())x() f (, x ), T. (5.11) By exacly he same argumens as were used for (4.1), i is no difficul o see ha Theorems 5.7 and 5.8 are valid for (5.11). The only difference is ha G,γ 1,γ 2, A, B here should be replaced by G(, s) = e a( + ω,s), γ 1 = inf k e α( + ω,s), γ 2 = sup e β ( + ω,s), A = γ 1, B = γ 2. a s ω s ω k β k α In addiion, on he basis of he exac same argumens, one can also derive he same crieria for he exisence and nonexisence of periodic soluions of x () = a(, x())x() + f (, x ) (5.12) provided ha 1/(1 µ(s)a(s)) is posiive and bounded. For breviy, deails are omied here.

18 5.2. Periodic soluions of higher dimensional dynamic sysems L. Bi e al. / Nonlinear Analysis 68 (28) In he previous subsecion, we focused on scalar dynamic equaions wih infinie delay on ime scales. In his subsecion, we urn o invesigaing he n-dimensional sysem X () = A()X (σ ()) + G(, X ), (5.13) where A() = diag[a 1 (), a 2 (),...,a n ()] wih a i C rd, a i ( + ω) = a i (), and k ai > for all i {1,...,n}. G = (g 1, g 2,...,g n ) T is defined on R C h, and G(,φ) is rd-coninuous in and is coninuous in φ wih G( + ω,φ) = G(,φ). In addiion, g i (,ϕ) maps bounded ses ino bounded ses and g i (,φ) for φ C h wih φ i (θ) and θ R. For convenience, we inroduce he following noaion: G i (, s) = e a i (s, ) 1, δ i = k ai e ai (ω, ), δ = min {δ i}, A i = δ i, B i = 1, 1 i n 1 δ i 1 δ i A = A i, B = B i, P ={u C rd (T, R n ) : u( + ω) = u(), R, u i C rd (T, R)}, i=1 u = max [,ω] i=1 i=1 u i () for u P, E δ ={φ C h : φ i (θ) δ φ i h,θ R }, K ={x P : x i (), x i () δ x i }, gi = lim max g i (,ϕ), g i ϕ h [,ω] ϕ = lim min g i (,ϕ), h ϕ h [,ω] ϕ h gi = lim max g i (,ϕ), g i ϕ h [,ω] ϕ = lim min g i (,ϕ), h ϕ h [,ω] ϕ h G = max 1 i n g i, G = max 1 i n g i, G = min 1 i n gi, G = min 1 i n gi. We firs lis below some conclusions wihou proof since he proofs are very similar o hose in he above secion. Lemma 5.2. Le A, Q P, A() = diag[a 1 (), a 2 (),...,a n ()], Q() = (q 1 (), q 2 (),...,q n ()) T. Then X () = A()X (σ ()) + Q() has a unique ω-periodic soluion given by X () = (x 1 (), x 2 (),...,x n ()), where x i () = For any u P, consider he equaion +ω G i (, s)q i (s) s. X () = A()X (σ ()) + G(, u ). (5.14) Lemma 5.2 ells us ha he unique ω-periodic soluion of (5.14) is given by X u () = (x 1 (), x 2 (),...,x n ()), where x i () = +ω Define he operaor F : K P wih componens (F 1, F 2,...,F n ) by (F i u)() = +ω G i (, s)g i (s, u s ) s for u P ω and T. G i (, s)g i (s, u s ) s. One can easily show ha X is an ω-periodic soluion of (5.13) if and only if X is a fixed poin of F in K.Iisno difficul o show ha F(K ) K. Le η be a posiive consan and Ω ={x P : x η}. Then F : K Ω K is compleely coninuous. On he basis of G, G, G, G, one can esablish exacly he same sufficien crieria (Theorems 5.7 and 5.8) for he exisence and nonexisence of ω-periodic soluions of (5.13). For simpliciy, as an example, we only presen he deails of he proof for he case where G =, G =.

19 1244 L. Bi e al. / Nonlinear Analysis 68 (28) Theorem 5.1. If G = and G =, hen (5.13) has a leas one posiive ω-periodic soluion. Proof. Since G =, we have gi =. Choose ε>such ha εb ω<1. Then here exiss a consan s 1 such ha g i (,φ) ε φ h,< φ h s 1, φ E δ. Le Ω 1 ={x P : x < s 1 }. Then for any u K Ω 1,wehave u i (θ) δ ui h and δ u u h u,sou E δ. Therefore Fu = i=1 +ω G i (, s)g i (s, u s ) s B i ε u s h s ε i=1 B i g i (s, u s ) s i=1 B i u s < u. i=1 On he oher hand, suppose G =. We may choose M such ha MδωA > 1 and we can observe ha here exiss a consan s wih s > s 1 such ha g i (,ϕ)> M ϕ h, ϕ h s, ϕ E δ. Le s 2 = s /δ and Ω 2 ={x P : x < s 2 }. Then, for any u K Ω 2,wehaveu E δ and u h = h(s) sup u i (θ) [s,] s h(s) sup δ u i s = δ u =δs 2 = s. Hence Fu = i=1 +ω s θ i=1 G i (, s)g i (s, u s ) s s θ i=1 A i g i (s, u s ) s i=1 A i M u s h s Mδ u ωa u. i=1 I follows from (4.1), (5.5) and (5.6) ha F has a fixed poin u 1 K (Ω 2 \ Ω 1 ) such ha (Fu 1 )() = u 1 () and u i 1 () δ u i 1 δs1 >. Hence u 1 is an ω-periodic soluion of (5.13). Remark 5.5. In his paper, we have sysemaically explored he exisence of periodic soluions of some dynamic equaions wih infinie delay on ime scales, which incorporae as special cases he differenial and he difference equaions invesigaed in [6,1,14 19,22,24,25], and hence he dynamic equaions invesigaed here also incorporae as special cases many well-known models in populaion dynamics, hemaopoiesis, ec. and provide sufficien crieria for he exisence of posiive periodic soluions of hose models. Remark 5.6. The exploraions in his paper reveal ha, when one deals wih he exisence of posiive periodic soluions of differenial equaions and difference equaions, especially by using he mehod of Krasnosel skiĭ s fixed poin heorem, i is unnecessary and useless o prove resuls for differenial equaions and separaely again for heir discree analogues (difference equaions). One can unify such problems in he framework of dynamic equaions on ime scales. Acknowledgemens This work was suppored by NSFC (No ), he Key Projec of he Minisry of Educaion (No. 1662), he projec sponsored by SRF for ROCS, SEM, and he Universiy of Missouri Research Board. References [1] D.R. Anderson, Muliple periodic soluions for a second-order problem on periodic ime scales, Nonlinear Anal. 6 (1) (25) [2] M. Bohner, M. Fan, J. Zhang, Exisence of periodic soluions in predaor prey and compeiion dynamic sysems, Nonlinear Anal. Real World Appl. 7 (5) (26) [3] M. Bohner, M. Fan, J. Zhang, Periodiciy of scalar dynamic equaions on ime scales, J. Mah. Anal. Appl. 26, in press, doi:1.116/j.jmaa [4] M. Bohner, A. Peerson, Dynamic Equaions on Time Scales: An Inroducion wih Applicaions, Birkhäuser, Boson, 21. [5] M. Bohner, A. Peerson, Advances in Dynamic Equaions on Time Scales, Birkhäuser, Boson, 23.

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