Existence and Global Attractivity of Positive Periodic Solutions in Shifts δ ± for a Nonlinear Dynamic Equation with Feedback Control on Time Scales

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1 Exience and Global Araciviy of Poiive Periodic Soluion in Shif δ ± for a Nonlinear Dynamic Equaion wih Feedback Conrol on Time Scale MENG HU Anyang Normal Univeriy School of mahemaic and aiic Xian ge Avenue 436, Anyang CHINA humeng200@26.com LILI WANG Anyang Normal Univeriy School of mahemaic and aiic Xian ge Avenue 436, Anyang CHINA ay wanglili@26.com Abrac: Thi paper i concerned wih a nonlinear dynamic equaion wih feedback conrol on ime cale. Baed on he heory of calculu on ime cale, by uing a fixed poin heorem of ric-e-conracion, ome crieria are eablihed for he exience of poiive periodic oluion in hif δ ± of he yem; hen, by uing ome differenial inequaliie and conrucing a uiable Lyapunov funcional, ufficien condiion which guaranee he global araciviy of he yem are obained. Finally, wo numerical example are preened o illurae he feaibiliy and effecivene of he reul. Key Word: poiive periodic oluion; global araciviy; hif operaor; feedback conrol; ime cale. Inroducion In he real world, lo of yem are coninuouly diurbed by unpredicable force which can reul in change in parameer. Of pracical inere i he queion of wheher or no a yem can wihand hoe unpredicable diurbance which peri for a finie period of ime. In he language of conrol variable, we call he diurbance funcion a conrol variable. During he la decade, differen ype of funcional differenial and difference equaion wih feedback conrol have been exenively udied; ee, for example, -5 and he reference herein. However, in applicaion, here are many yem whoe developing procee are boh coninuou and dicree. Hence, uing he only differenial equaion or difference equaion can accuraely decribe he law of heir developmen; ee, for example, 6,7. Therefore, here i a need o eablih correponden dynamic model on new ime cale. The heory of calculu on ime cale (ee 8 and reference cied herein) wa iniiaed by Sefan Hilger 9 in order o unify coninuou and dicree analyi. Therefore, he udy of dynamic equaion on ime cale, which unifie differenial, difference, h- difference, and q-difference equaion and more, ha received much aenion; ee 0-4. The exience problem of periodic oluion i an imporan opic in qualiaive analyi of funcional dynamic equaion. Up o now, here are only a few reul concerning periodic oluion of dynamic e- quaion wih feedback conrol on ime cale; ee, for example, 5,6. In hee paper, auhor conidered he exience of periodic oluion for dynamic equaion on ime cale aifying he condiion here exi a ω > 0 uch ha ± ω T, T. Under hi condiion all periodic ime cale are unbounded above and below. However, here are many ime cale uch a q Z {q n : n Z} {0} and N { n : n N} which do no aify he condiion. Adıvar and Raffoul inroduced a new periodiciy concep on ime cale which doe no oblige he ime cale o be cloed under he operaion ± ω for a fixed ω > 0. They defined a new periodiciy concep wih he aid of hif operaor δ ± which are fir defined in 7 and hen generalized in 8. Recenly, by uing he cone heory echnique, many reearcher udied he exience and mulipliciy of poiive periodic oluion in hif δ ± for ome nonlinear fir-order funcional dynamic equaion on ime cale; ee However, o he be of our knowledge, here are few paper publihed on he exience and global araciviy of poiive periodic oluion in hif δ ± for nonlinear dynamic yem wih feedback conrol on ime cale. Moivaed by he above aemen, in he preen paper, baed on he heory of calculu on ime cale, we hall udy a nonlinear fir-order dynamic equa- E-ISSN: Volume 0, 205

2 ion wih feedback conrol on ime cale a follow: x () x()r() a()x() b()x(σ()) c()y(), () y () η()y() + g()x(), where T, T R be a periodic ime cale in hif δ ± wih period P, ) T and T i nonnegaive and fixed. r, a, b, c, η, g C(T, (0, )) are -periodic funcion in hif δ ± wih period ω and η R +. The iniial condiion of yem () in he form x( ) x 0, y( ) y 0, T, x 0 > 0, y 0 > 0. (2) The aim of hi paper i, by uing a fixed poin heorem of ric-e-conracion, o obain ufficien condiion for he exience of poiive periodic oluion in hif δ ± of yem (); and by uing ome differenial inequaliie and conrucing a uiable Lyapunov funcional, o obain ufficien condiion for he global araciviy of yem (). For convenience, we inroduce he noaion f u up f(), f l inf f(),,δ+ ω () T,δ+ ω () T where f i a poiive and bounded funcion. Throughou hi paper, we aume ha (H ) Θ : e r (, δ ω +( )) < ; (H 2 ) min{r l, a l, b l, c l, η l, g l } > 0, max{r u, a u, b u, c u, η u, g u } < ; (H 3 ) r l > a u M + c u M 2, where M ru b l, M 2 gu M η l ; (H 4 ) a l g u > 0; (H 5 ) η l c u > 0. 2 Preliminarie Le T be a nonempy cloed ube (ime cale) of R. The forward and backward jump operaor σ, ρ : T T and he grainine µ : T R + are defined, repecively, by σ() inf{ T : > }, ρ() up{ T : < } µ() σ(). A poin T i called lef-dene if > inf T and ρ(), lef-caered if ρ() <, righ-dene if < up T and σ(), and righ-caered if σ() >. If T ha a lef-caered maximum m, hen T k T\{m}; oherwie T k T. If T ha a righ-caered minimum m, hen T k T\{m}; oherwie T k T. For he baic heorie of calculu on ime cale, ee 8. A funcion p : T R i called regreive provided + µ()p() 0 for all T k. The e of all regreive and rd-coninuou funcion p : T R will be denoed by R R(T, R). Define he e R + R + (T, R) {p R : + µ()p() > 0, T}. If r i a regreive funcion, hen he generalized exponenial funcion e r i defined by { } e r (, ) exp ξ µ(τ) (r(τ)) τ for all, T, wih he cylinder ranformaion ξ h (z) { Log(+hz) h, if h 0, z, if h 0. Le p, q : T R be wo regreive funcion, define p q p+q+µpq, p p, p q p ( q). + µp Lemma. (ee 8) If p, q : T R be wo regreive funcion, hen (i) e 0 (, ) and e p (, ) ; (ii) e p (σ(), ) ( + µ()p())e p (, ); (iii) e p (, ) e e p(,) p(, ); (iv) e p (, )e p (, r) e p (, r); (v) e p(,) e e q(,) p q(, ); (vi) (e p (, )) p()e p (, ). The following definiion, lemma abou he hif operaor and he new periodiciy concep for ime - cale which can be found in 20,23. Le T be a non-empy ube of he ime cale T and T be a fixed number, define operaor δ ± :, ) T T. The operaor δ + and δ aociaed wih T (called he iniial poin) are aid o be forward and backward hif operaor on he e T, repecively. The variable, ) T in δ ± (, ) i called he hif ize. The value δ + (, ) and δ (, ) in T indicae uni ranlaion of he erm T o he righ and lef, repecively. The e D ± : {(, ), ) T T : δ (, ) T } are he domain of he hif operaor δ ±, repecively. Hereafer, T i he large ube of he ime cale T uch ha he hif operaor δ ± :, ) T T exi. E-ISSN: Volume 0, 205

3 Definiion (Periodiciy in hif δ ± ) Le T be a ime cale wih he hif operaor δ ± aociaed wih he iniial poin T. The ime cale T i aid o be periodic in hif δ ± if here exi p (, ) T uch ha (p, ) D ± for all T. Furhermore, if P : inf{p (, ) T : (p, ) δ ±, T }, hen P i called he period of he ime cale T. Definiion (Periodic funcion in hif δ ± ) Le T be a ime cale ha i periodic in hif δ ± wih he period P. We ay ha a real-valued funcion f defined on T i periodic in hif δ ± if here exi ω P, ) T uch ha (ω, ) D ± and f(δ ω ±()) f() for all T, where δ ω ± : δ ± (ω, ). The malle number ω P, ) T i called he period of f. Definiion ( -periodic funcion in hif δ ± ) Le T be a ime cale ha i periodic in hif δ ± wih he period P. We ay ha a real-valued funcion f defined on T i -periodic in hif δ ± if here exi ω P, ) T uch ha (ω, ) D ± for all T, he hif δ± ω are -differeniable wih rd-coninuou derivaive and f(δ±())δ ω ± ω () f() for all T, where δ± ω : δ ± (ω, ). The malle number ω P, ) T i called he period of f. Lemma δ ω +(σ()) σ(δ ω +()) and δ ω (σ()) σ(δ ω ()) for all T. Lemma Le T be a ime cale ha i periodic in hif δ ± wih he period P. Suppoe ha he hif δ± ω are -differeniable on T where ω P, ) T and p R i -periodic in hif δ ± wih he period ω. Then (i) e p (δ ω ±(), δ ω ±( )) e p (, ) for, T ; (ii) e p (δ±(), ω σ(δ±())) ω e p (, σ()) for, T. ep(,) +µ()p() Lemma Le T be a ime cale ha i periodic in hif δ ± wih he period P, and le f be a -periodic funcion in hif δ ± wih he period ω P, ) T. Suppoe ha f C rd (T), hen f() ± () δ ω ± () f(). Lemma 8. 8 Suppoe ha r i regreive and f : T R i rd-coninuou. Le T, y 0 R, hen he unique oluion of he iniial value problem i given by y r()y + f(), y( ) y 0 y() e r (, )y 0 + e r (, σ(τ))f(τ) τ. Lemma 9. (ee ) Aume ha a > 0, b > 0 and a R +. Then y () ()b ay(), y() > 0,, ) T implie y() () b a +(ay() )e b ( a) (, ),, ) T. Lemma 0. (ee ) Aume ha a > 0, b > 0. Then y () ( )y()(b ay(σ())), y() > 0,, ) T implie y() ( ) b a +( b ay( ) )e b(, ),, ) T. 3 Exience reul In hi ecion, we hall udy he exience of a lea one poiive periodic oluion in hif δ ± of yem () via a fixed poin heorem of ric-e-conracion. Se X {x : x C(T, R), x(δ ω +()) x()} wih he norm defined by x 0 up x(),,δ+ ω () T hen X i a Banach pace. By uing Lemma, 5 and 8, we can obain he following lemma. Lemma. x() X i an ω-periodic oluion in hif δ ± of yem () if and only if x() i an ω- periodic oluion in hif δ ± of x() where and +c() G(, )x() a()x() + b()x(σ()) + () G(, ) H(, ) H(, θ)g(θ)x(θ) θ, (3) e r (, σ()) e r (, δ ω + ()), e η (, σ()) e η (, δ ω + ()). E-ISSN: Volume 0, 205

4 Proof. If y() i an ω-periodic oluion in hif δ ± of he econd equaion of yem (). By uing Lemma 8, for, δ+() ω T, we have y() e η (, )y() + e η (, σ(θ))g(θ)x(θ) θ. Le δ ω +() in he above equaliy, we have y(δ ω +()) e η (δ ω +(), )y() + e η (δ ω +(), σ(θ))g(θ)x(θ) θ. Noicing ha y(δ ω +()) y() and e η (, δ ω +()) e η (, δ ω +( )), hen y() where H(, ) H(, )g()x() : (Ψx)(), (4) e η (, σ()) e η (, δ ω + ()). Le y() be an ω-periodic oluion in hif δ ± of (4). By Lemma and 5, we have y () η()y() +H(σ(), δ ω +())g(δ ω +())δ ω x(δ ω +()) H(σ(), )g()x() η()y() + g()x(). Therefore, he exience problem of ω-periodic oluion in hif δ ± of yem () i equivalen o ha of he following equaion x () x()r() a()x() b()x(σ()) c() H(, )g()x(). (5) Repeaing he above proce, i follow from (5) ha x() X i an ω-periodic oluion in hif δ ± of yem () if and only if x() i an ω-periodic oluion in hif δ ± of x() G(, )x() a()x() + b()x(σ()) where +c() + () G(, ) Thi proof i complee. H(, θ)g(θ)x(θ) θ, e r (, σ()) e r (, δ ω + ()). I i eay o verify ha he Green funcion G(, ) and H(, ) aify he propery 0 < Θ G(, ) Θ Θ,, δω +() T, (6) and 0 < H(, ) Γ Γ Γ. where Θ : e r (, δ ω +( )), Γ : e η (, δ ω +( )). By Lemma 6, we have G(δ ω +(), δ ω +()) G(, ), H(δ ω +(), δ ω +()) H(, ), T,, δ ω +() T. (7) In order o obain he exience of periodic oluion in hif δ ± of yem (), we fir make he following preparaion: Le E be a Banach pace and K be a cone in E. The emi-order induced by he cone K i denoed by, ha i, x y if and only if y x K. In addiion, for a bounded ube A E, le α E (A) denoe he (Kuraowki) meaure of non-compacne defined by α E (A) inf { d > 0 : here i a finie number of ube A i A uch ha A i and diam(a i ) d }, where diam(a i ) denoe he diameer of he e A i. Le E, F be wo Banach pace and D E, a coninuou and bounded map Φ : Ω F i called k-e conracive if for any bounded e S D we have α F (Φ(S)) kα E (S). Φ i called ric-e-conracive if i i k-econracive for ome 0 k <. Lemma 2. 24, 25 Le K be a cone of he real Banach pace X and K r,r {x K r x R} wih R > r > 0. Suppoe ha Φ : K r,r K i ric-e-conracive uch ha one of he following - wo condiion i aified: (i) Φx x, x K, x r and Φx x, x K, x R. (ii) Φx x, x K, x r and Φx x, x K, x R. Then Φ ha a lea one fixed poin in K r,r. A i E-ISSN: Volume 0, 205

5 Define K, a cone in X, by K {x X : x() Θ x 0,, δ ω +( ) T }, (8) and an operaor Φ : K X by (Φx)() G(, )x() a()x() + b()x(σ()) +c() + () H(, θ)g(θ)x(θ) θ. (9) In he following, we hall give ome lemma concerning K and Φ defined by (8) and (9), repecively. Lemma 3. Aume ha (H ) hold, hen Φ : K K i well defined. Proof. For any x K. In view of (4), for T, we obain hen (Ψx)(δ ω +()) + (δ ω ) δ+ ω () δ ω x(δ ω +(u)) u (Ψx)(), δ ω H(δ ω +(), )g()x() H(δ ω +(), δ ω +(u))g(δ ω +(u))δ ω + (u) H(, u)g(u)x(u) u (Φx)(δ+()) ω + (δ+ ω ()) G(δ+(), ω )x() a()x() +b()x(σ()) + c()(ψx)() G(δ+(), ω δ+(u))x(δ ω +(u)) ω a(δ+(u))δ ω + ω (u)x(δ+(u)) ω +b(δ+(u))δ ω + ω (u)x(σ(δ+(u))) ω +c(δ+(u))δ ω + ω (u)(ψx)(δ+(u)) ω u G(, u)x(u) a(u)x(u) +b(u)x(σ(u)) + c(u)(ψx)(u) u (Φx)(), ha i, (Φx)(δ ω +()) (Φx)(), T. So Φx X. Furhermore, for x K,, δ ω +( ) T, we have and Φx 0 Θ +c() (Φx)() Θ Θ +c() Θ Θ +c() + ( ) + () + ( ) + () + ( ) + () x() a()x() + b()x(σ()) H(, θ)g(θ)x(θ) θ x() a()x() + b()x(σ()) H(, θ)g(θ)x(θ) θ x() a()x() + b()x(σ()) H(, θ)g(θ)x(θ) θ Θ Φx 0, (0) ha i, Φx K. The proof i complee. Lemma 4. Aume ha (H ) (H 2 ) hold, hen Φ : K ΩR K i ric-e-conracive, where Ω R {x X : x 0 < R}. Proof. I i eay o ee ha Φ i coninuou and bounded. Now we prove ha α X (Φ(S)) kα X (S) for any bounded e S Ω R and 0 < k <. Le η α X (S). Then, for any poiive number ε < η, here i a finie family of ube {S i } aifying S i S i wih diam(s i ) η + ε. Therefore x y 0 η + ε for any x, y S i. () A S and S i are precompac in X, i follow ha here i a finie family of ube {S ij } of S i uch ha S i j S ij and x y 0 ε for any x, y S ij. (2) In addiion, for any x S and, δ ω +( ) T, we have (Φx)() G(, )x() a()x() + b()x(σ()) +c() + () R 2 Π : A, Θ H(, θ)g(θ)x(θ) θ E-ISSN: Volume 0, 205

6 where Π : δ+ ω () a()+b()+c() δ+ ω () H(, θ) g(θ) θ. And (Φx) () a()x() r()(φx)() x() + b()x(σ()) +c() H(, θ)g(θ)x(θ) θ r u A ( +R 2 a u + b u + c u Γ Γ + ( ) ) g(θ) θ. Applying he Arzela-Acoli Theorem, we know ha Φ(S) i precompac in X. Then, here i a finie family of ube {S ijk } of S ij uch ha S ij k S ijk and Φx Φy 0 ε for any x, y S ijk. (3) A ε i arbirary mall, i follow ha α X (Φ(S)) kα X (S). Therefore, Φ i ric-e-conracive. complee. The proof i Theorem 5. Aume ha (H ) (H 2 ) hold, hen yem () ha a lea one poiive ω-periodic oluion in hif δ ±. Proof. Le R Θ Θ( Θ) and 0 < r < Θ 3 Π Π, hen we have 0 < r < R. From Lemma 3 and 4, we know ha Φ i ric-e-conracive on K r,r. In view of (9), we ee ha if here exi x K uch ha Φx x, hen x i one poiive ω-periodic oluion in hif δ ± of yem (). Now, we hall prove ha condiion (ii) of Lemma 2 hold. Fir, we prove ha Φx x, x K, x 0 r. Oherwie, here exi x K, x 0 r uch ha Φx x. So x > 0 and Φx x K, which implie ha (Φx)() x() Θ Φx x 0 0,, δ ω +( ) T. (4) Moreover, for, δ ω +( ) T, we have (Φx)() G(, )x() a()x() + b()x(σ()) +c() + () Θ r x 0 H(, θ)g(θ)x(θ) θ + ( ) a() + b() +c() + () H(, θ)g(θ) θ r Θ Π x 0 < Θ x 0. (5) In view of (4) and (5), we have x 0 Φx < Θ x 0 < x 0, which i a conradicion. Finally, we prove ha Φx x, x K, x 0 R alo hold. For hi cae, we only need o prove ha Φx x x K, x 0 R. Suppoe, for he ake of conradicion, ha here exi x K and x 0 R uch ha Φx < x. Thu x Φx K \ {0}. Furhermore, for any, δ+( ω 0 ) T, we have x() (Φx)() Θ x Φx 0 > 0. (6) In addiion, for any, δ ω +( ) T, we find (Φx)() G(, )x() a()x() + b()x(σ()) +c() + () Θ Θ Θ2 x 2 0 +c() + () H(, θ)g(θ)x(θ) θ + ( ) a() + b() H(, θ)g(θ) θ Θ 3 Θ ΠR2 R. (7) From (6) and (7), we obain x > Φx 0 R, which i a conradicion. Therefore, condiion (i) and (ii) hold. By Lemma 2, we ee ha Φ ha a lea one nonzero fixed poin in K. Therefore, yem () ha a lea one poiive ω-periodic oluion in hif δ ±. The proof i complee. 4 Global araciviy In hi ecion, we hall udy he global araciviy of poiive periodic oluion in hif δ ± of yem () wih iniial condiion (2). Applying Lemma 9 and 0, we can obain he following lemma. E-ISSN: Volume 0, 205

7 Lemma 6. Le (x(), y()) be any poiive periodic oluion in hif δ ± of yem () wih iniial condiion (2). If (H 3 ) hold, hen yem () i permanen, ha i, any poiive periodic oluion in hif δ ± (x(), y()) of yem () aifie m lim inf m 2 lim inf x() lim up x() M, (8) y() lim up y() M 2, (9) epecially if m x 0 M, m 2 y 0 M 2, hen m x() M, m 2 y() M 2,, ) T, where M ru b l, M 2 gu M η l, m rl a u M c u M 2 b u, m 2 gl m η u. Lemma 7. Aume ha (H 3 ) (H 5 ) hold, hen yem () i globally aracive. Proof. Le z () (x (), y ()) and z 2 () (x 2 (), y 2 ()) be any wo poiive periodic oluion in hif δ ± of yem (). I follow from (8)-(9) ha for ufficien mall poiive conan ε 0 (0 < ε 0 < min{m, m 2 }), here exi a T > 0 uch ha m ε 0 < x i () < M + ε 0, m 2 ε 0 < y i () < M 2 + ε 0, (20) where T, + ) T, i, 2. Since x i (), i, 2 are poiive, bounded and differeniable funcion on T, hen here exi a poiive, bounded and differeniable funcion m(), T, uch ha x i ()(+m()), i, 2 are ricly increaing on T. Then L T(x i () + m()) x i () + m() + x i(σ())m () x i () + m() x i () x i () + x i(σ())m (), i, 2. x i () + m() Here, we can chooe a funcion m() uch ha m () +m() i bounded on T, ha i, here exi wo poiive conan ζ > 0 and ξ > 0 uch ha < ζ < m () +m() < ξ, T. Se V () e δ (, T ) ( L T (x ()( + m())) L T (x 2 ()( + m())) + y () y 2 () ), where δ 0 i a conan (if µ() 0, hen δ 0; if µ() > 0, hen δ > 0). I follow from he mean value heorem of differenial calculu on ime cale for T, + ) T, x () x 2 () M + ε 0 L T (x ()( + m())) L T (x 2 ()( + m())) x () x 2 (). (2) m ε 0 Le γ min{(m ε 0 )(a l g u ), η l c u }. We divide he proof ino wo cae. Cae I. If µ() > 0, e δ > max{(b u + ξ m )M, γ} and µ()δ < 0. Calculaing he upper righ derivaive of V () along he oluion of yem (), i follow from (20), (2), (H 4 ) and (H 5 ) ha for T, + ) T, D + V () x e δ (, T ) gn(x () x 2 ()) () x () x 2 () x 2 () ( + m () x (σ()) x ) 2(σ()) + m() x () x 2 () δ e δ (, T ) L T (x (σ())( + m(σ()))) L T (x 2 (σ())( + m(σ()))) + e δ (, T ) gn(y () y 2 ())(y () y2 ()) δ e δ (, T ) y (σ()) y 2 (σ()) e δ (, T ) gn(x () x 2 ()) a()(x () x 2 ()) b()(x (σ()) x 2 (σ())) c()(y () y 2 ()) + m () x (σ())x 2 () x ()x 2 (σ()) + m() x ()x 2 () δ e δ (, T ) L T (x (σ())( + m(σ()))) L T (x 2 (σ())( + m(σ()))) + e δ (, T ) gn(y () y 2 ()) η()(y () y 2 ()) + (x () x 2 ()) δ e δ (, T ) y (σ()) y 2 (σ()) e δ (, T ) gn(x () x 2 ()) a()(x () x 2 ()) b()(x (σ()) x 2 (σ())) c()(y () y 2 ()) ( + m () x (σ())(x 2 () x ()) + m() x ()x 2 () + x ) ()(x (σ()) x 2 (σ())) x ()x 2 () E-ISSN: Volume 0, 205

8 δ e δ (, T ) L T (x (σ())( + m(σ()))) L T (x 2 (σ())( + m(σ()))) + e δ (, T ) gn(y () y 2 ()) η()(y () y 2 ()) + (x () x 2 ()) δ e δ (, T ) y (σ()) y 2 (σ()) e δ (, T ) a() g() + m () x (σ()) x () x 2 () + m() x ()x 2 () δ e δ (, T ) b() M + ε 0 m () x (σ()) x 2 (σ()) + m() x 2 () e δ (, T ) (η() c()) y () y 2 () δ e δ (, T ) y (σ()) y 2 (σ()) e δ (, T ) (a l g u ) x () x 2 () e δ (, T ) (η l c u ) y () y 2 () e δ (, T ) ((m ε 0 )(a l g u ) L T (x ()( + m())) L T (x 2 ()( + m())) +(η l c u ) y () y 2 () ) γ e δ (, T ) ( L T (x ()( + m())) L T (x 2 ()( + m())) + y () y 2 () ) γv (). (22) By he comparion heorem and (22), we have ha i, hen V () e γ (, T ) V (T ) ( ) M + ε 0 < 2 + M 2 + ε 0 e γ (, T ), m ε 0 e δ (, T ) ( L T (x ()( + m())) L T (x 2 ()( + m())) + y () y 2 () ) ( ) M + ε 0 < 2 + M 2 + ε 0 e γ (, T ), m ε 0 x () x 2 () + y () y 2 () M + ε ( 0 ) M + ε 0 < 2 + M 2 + ε 0 m ε 0 e ( γ) ( δ) (, T ). (23) Since µ()δ < 0 and 0 < γ < δ, hen ( γ) ( δ) < 0. I follow from (23) ha lim x () x 2 () 0, + lim y () y 2 () 0. + Cae II. If µ() 0, e δ 0, hen σ() and e δ (, T ). Calculaing he upper righ derivaive of V () along he oluion of yem (), i follow from (20), (2), (H 4 ) and (H 5 ) ha for T, + ) T, D + V () ( x gn(x () x 2 ()) () x () x 2 () ) x 2 () +gn(y () y 2 ())(y () y2 ()) gn(x () x 2 ()) (a() + b()) (x () x 2 ()) c()(y () y 2 ()) +gn(y () y 2 ()) η()(y () y 2 ()) +g()(x () x 2 ()) (a() + b() g()) x () x 2 () (η() c()) y () y 2 () ((m ε 0 )(a l + b l g u ) L T (x ()( + m())) L T (x 2 ()( + m())) +(η l c u ) y () y 2 () ) γ( L T (x ()( + m())) L T (x 2 ()( + m())) + y () y 2 () ) γv (), (24) where γ min{(m ε 0 )(a l + b l g u ), η l c u }. By he comparion heorem and (24), we have V () e γ (, T ) V (T ) ( ) M + ε 0 < 2 + M 2 + ε 0 e γ (, T ), m ε 0 ha i, hen L T (x ()( + m())) L T (x 2 ()( + m())) + y () y 2 () ( ) M + ε 0 < 2 + M 2 + ε 0 e γ (, T ), m ε 0 x () x 2 () + y () y 2 () M + ε ( 0 ) M + ε 0 < 2 + M 2 + ε 0 e γ (, T ). (25) m ε 0 I follow from (25) ha lim x () x 2 () 0, + lim y () y 2 () 0. + From he above dicuion, we can ee ha yem () i globally aracive. Thi complee he proof. E-ISSN: Volume 0, 205

9 Togeher wih Theorem 5 and Lemma 7, we can obain he following heorem. Theorem 8. Aume ha he condiion (H ) (H 5 ) hold, hen yem () wih iniial condiion (2) ha a unique globally aracive poiive periodic oluion in hif δ ±. 5 Numerical example Example. Conider he following yem on ime cale T R: x () x() in π 2 ( in π 2 )x() x(σ()) 0.2y(), (26) y () ( co π 2 )y() +( in π 2 )x(). Le 0, hen δ ω +() +2. By a direc calculaion, we can ge r u, r l 0.6, a u 0.05, a l 0.04, b u b l, c u c l 0.2, η u 0.5, η l 0.3, g u 0.02, g l 0.0, e r (0, 2) 0.209, M, M , a u M + c u M Obviouly, e r (, δ+( ω 0 )) <, r l > a u M + c u M 2, a l g u > 0, η l c u > 0, ha i, he condiion (H ) (H 5 ) hold. According o Theorem 8, yem (26) ha a unique globally aracive poiive periodic oluion in hif δ ±. Example 2. Conider he following yem on ime cale T 2 N 0 : x () x() x() x(σ()) 20 y(), y () 3 y() + 5 x(). (27) Le, hen δ ω +() 4. By a direc calculaion, we can ge r u 0.2, r l 0.05, a u 2, a l 0.5, b u 40, b l 0, c u , c l 0.025, η u 0.025, η l , g u 0.2, g l 0.05, e r (, 4) 0.688, M 0.02, M , a u M + c u M Obviouly, e r (, δ ω +( )) <, r l > a u M + c u M 2, a l g u > 0, η l c u > 0, ha i, he condiion (H ) (H 5 ) hold. According o Theorem 8, yem (27) ha a unique globally aracive poiive periodic oluion in hif δ ±. 6 Concluion Thi paper udied a nonlinear dynamic equaion wih feedback conrol on ime cale. Baed on he heory of calculu on ime cale, by uing a fixed poin heorem of ric-e-conracion, ome crieria are eablihed for he exience of poiive periodic oluion in hif δ ± of he yem; hen, by uing ome differenial inequaliie and conrucing a uiable Lyapunov funcional, ufficien condiion which guaranee he global araciviy of he yem are obained. The reul obained in hi paper can be applied o yem on more general ime cale, no only ime cale are unbounded above and below. So he field of applicaion of one dynamic yem i widened. Beide, he mehod ued in hi paper can be applied o udy many oher dynamic yem. Acknowledgemen: Thi work i uppored by he Key Projec of Scienific Reearch in College and U- niveriie in Henan Province (Gran No.5A0004). Reference: F. Chen, Poiive periodic oluion of neural Loka-Volerra yem wih feedback conrol, Appl. Mah. Compu., 62(3), 2005, pp Y. Xia, J. Cao, H. Zhang, F. Chen, Almo periodic oluion in n-pecie compeiive yem wih feedback conrol, J. Mah. Anal. Appl., 294, 2004, pp Q. Liu, R. Xu, Perience and global abiliy for a delayed nonauonomou ingle-pecie model wih diperal and feedback conrol, Diff. Equ. Dyn. Sy., (3-4), 2003, pp F. Chen, Permanence of a ingle pecie dicree model wih feedback conrol and delay, Appl. Mah. Le., 20(7), 2007, pp Y. Fan, L. Wang, Permanence for a dicree model wih feedback conrol and delay, Dicree Dyn. Na. Soc., Vol. 2008, Aricle ID V. Spedding, Taming naure number, New Scieni, 2404, 2003, pp E-ISSN: Volume 0, 205

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