Global asymptotic stability in a rational dynamic equation on discrete time scales

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1 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] Global asyptotic stability i a atioal dyaic euatio o discete tie scales a( t) b( ( t)) ( ( t)), t T c ( ( ( t))) Sh. R. Elzeiy Depatet of Matheatics, Faculty of Sciece, Al-Baha Uivesity, Kigdo of Saudi Aabia E-ail: shelzeiy@yahoo.co Abstact- I this pape, we study the global stability, peiodicity chaacte ad soe othe popeties of solutios of the atioal dyaic euatio o discete tie scales whee a, b, c ad. a( t) b( ( t)) ( ( t)), t T, c ( ( ( t))) Keywods-Ratioal dyaic euatio, Tie scales, Euilibiu poit, Global attacto, Peiodicity, Boudedess, Ivaiat iteval. I. INTRODUCTION The theoy of tie scales, which has ecetly eceived a lot of attetio, was itoduced by Stefa Hilge i his PhD thesis [5] i ode to uify cotiuous ad discete aalysis. The theoy of dyaic euatios ot oly uifies the theoies of diffeetial euatios ad diffeece euatios, but also it eteds these classical cases to cases i betwee, e.g., to so-called -diffeece euatios. Sice the seveal authos have epouded o vaious aspects of this ew theoy, see the suvey pape by Agawal et al. [] ad the efeeces cited theei. May othe iteestig tie scales eist, ad they give ise to ay applicatios, aog the the study of populatio dyaic odels (see [8]). A book o the subject of tie scales by Bohe ad Peteso [5] suaizes ad ogaizes uch of the tie scales calculus (see also [4]). The study of atioal dyaic euatios o tie scales goes to back to Elzeiy [3]. Fo the otios used below we efe the eade to [5] ad to the followig a shot itoductio to the tie scale calculus. Defiitio.: A tie scale is a abitay oepty closed subset of the eal ubes. Thus,,,, i. e., the eal ubes, the iteges, the atual ubes, ad the oegative iteges ae eaples of tie scales, as ae [,] [,3], [,], ad thecato set, while, \,,(,), i. e., the atioal ubes, the iatioal ubes, the cople ubes, ad the ope iteval betwee ad, ae ot tie scales. Thoughout this pape, a tie scale is deoted by the sybol T ad has the topology that it iheits fo the eal ubes with the stadad topology. To efeece poits i the set T, the fowad ad backwad jup opeatos ae defied. Defiitio.: Fo t T,, the fowad opeato :T T ( t) if{ st : s t}, ad backwad opeato :T T is defied by ( t) sup{ st : s t}. If T has a aiu t ad a iiu t, the is defied by Page

2 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] ( t ) t, ad ( t ) t. Whe ( t) t, the t is called ight scatteed. Whe ( t) t, the t is called left scatteed. Poits t such that ( t) t ( t), ( t) t sup T, o ( t) t if T, ae called isolated poits. If a tie scale cosists of oly isolated poits, the it is a isolated (discete) tie scale. Also, if t sup T ad ( t) t, thet is called ight-dese, ad if t if T ad ( t) t, the t is called left-dese. Poits t that ae eithe left-dese o ight-dese ae called dese. Fially, the gaiiess opeato : T [, ) is defied by ( t) ( t) t ad if : f T is a fuctio, the the fuctio f : T f ( t) f ( ( t)) fo all t T, is defied by i. e., f f is the copositio fuctio of f with. Eaple.: Let us biefly coside the two eaples T ad T. (i) If T, the we have fo ay t, ( t) if{ s : s t} if( t, ) t, ad siilaly ( t) t. Hece evey poit t is left-dese ad ight-dese. The gaiiess opeato fo all t. (ii) If T, the we have fo ay t ( t) if{ s : s t} if{ t, t,...} t, ad siilaly ( t) t. Hece, evey poitt is leftscatteed ad ight-scatteed. The gaiiess opeato fo all t. Eaple.: Coside the tie scale T { : }. The, we have ( ) ( ), ( ) ( ), ad ( ) fo. Thus, ( t) ( t ), ( t) ( t ), ad ( t) t. Eaple.3: Let T / 4 { k / 4: k }. The, we have fo t T, ( t) if{ st : s t} if{ t / 4: } t / 4, ad ( t) t h/ 4. Hece, evey poit t so that i this eaple is costat. T is isolated ad ( t) ( t) t t / 4t / 4 fo t T, t Eaple.4: Coside the tie scale T, we get ( t) t, ( t), ad ( t) t fo t T. Cotiuity o tie scales is defied i the followig ae. Defiitio.3: Assue f : T is a fuctio ad let t T. If t is a isolated poit, the we defie li f ( s) f ( t) st Page

3 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] ad we say f is cotiuous at t. Whe t is ot isolated poit, the whe we wite li f ( s) L, st it is udestood that s appoaches t i the tie scale ( st, s t). We say f is cotiuous o T. povided li f ( s) f ( t) fo all t T. st I paticula, we have that ay fuctio defied o a isolated tie scale (sice all of its poits ae isolated poits) is cotiuous. We say f :[ a, b] T is cotiuous povided f is cotiuous at each poit i ( ab, ) T cotiuous at a, ad f is ight cotiuous at b. I the tie scale calculus, the fuctios ae ight-dese cotiuous which we ow defie., f is left Defiitio.4: A fuctio f : T is called ight-dese cotiuous o biefly d-cotiuous povided it is cotiuous at ight-dese poits i T. ad its left-sided liits eist (fiite) at left-dese poits i T.. The set of d-cotiuous fuctios f : T will be deoted i this pape by C C ( T) C ( T, ) d d d I the et theoe we see that the jup opeato is d-cotiuous. Theoe. (Bohe et al. [5]): The fowad opeato :T T is iceasig, d-cotiuous, ad ( t) t fo all t T, ad the jup opeato is discotiuous at poits which ae left-dese ad ight-scatteed. Reak.: The gaiiess fuctio : T [, ) is d-cotiuous ad is discotiuous at poits i T that ae both left-dese ad ight-scatteed. Whe T, the atioal dyaic euatio o discete tie scales a( t) b( ( t)) ( ( t)), t T, c ( ( ( t))) whee a, b, c ad. becoes the ecusive seuece whee a, b, c ad.. a b c,,,..., (.) Now, the diffeece euatios (as well as diffeetial euatios ad delay diffeetial euatios) odel vaious divese pheoea i biology, ecology, physiology, physics, egieeig ad ecooics, etc.[]. The study of oliea diffeece euatios is of paaout ipotace ot oly i thei ow ight but i udestadig the behavio of thei diffeetial coutepats. Thee is a class of oliea diffeece euatios, kow as the atioal diffeece euatios, each of which cosists of the atio of two polyoials i the seuece tes. Thee has bee a lot of wok coceig the global asyptotic behavio of solutios of atioal diffeece euatios [, 3, 6, 7, 9-, 4, 6, 8, 9, -6]. This pape addesses, the global stability, peiodicity chaacte ad boudedess of the solutios of the atioal dyaic euatio o discete tie scales a( t) b( ( t)) ( ( t)), t T, (.) c ( ( ( t))) Page 3

4 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] whee a, b, c ad. Whe T ad, ou euatio educes to euatio which eaied by Yag et al. [4]. Hee, we ecall soe otatios ad esults which will be useful i ou ivestigatio. Let I be soe iteval of eal ubes ad let f be a cotiuous fuctio defied o I I ( ( t )), ( t ) I coditios, it is easy to see that the dyaic euatio o discete tie scales. The, fo iitial ( ( t)) f ( ( t), ( ( t))), t T, (.3) has a uiue solutio { ( t) : t T}, which is called a ecusive seuece o tie scales. Defiitio.5: A poit is called a euilibiu poit of euatio (.3) if f (, ). That is, () t fo t T, is a solutio of euatio (.3), o euivaletly, is a fied poit of f. Assue is a euilibiu poit of euatio (.3) ad u f( t) (, ), ad v f( ( t)) (, ). The the lieaized euatio associated with euatio (.3) about the euilibiu poit is The chaacteistic euatio associated with euatio (.4) is Theoe. (Lieaized stability theoe [7]): () If u v ad v, the is locally asyptotically stable. () If u v ad v, the is a epelle. (3) If u v ad u 4 v, the is a saddle poit. (4) If u v, the is a o hypebolic poit. Defiitio.6: We say that a solutio { ( t) : t T} of euatio (.3) is bouded if ( t) A fo all t T. z( t) uzt vz( t). (.4) u v. (.5) Defiitio.7: (a) A solutio { ( t) : t T} of euatio (.) is said to be peiodic with peiod if ( t ) ( t) fo all t T. (.6) (b) A solutio { ( t) : t T} of euatio (.3) is said to be peiodic with pie peiod, o -cycle if it is peiodic with peiod ad is the least positive itege fo which (.6) holds. Defiitio.8: A iteval J with iitial coditios I is called ivaiat fo euatio (.3) if evey solutio { ( t) : t T} ( ( ( t )), ( t )) J J satisfies ( t) J fo all t T. Fo a eal ube t ( ) ad a positive ube R, let O( ( t), R) { ( t) : ( t) ( t) R}. Fo othe basic teiologies ad esults of diffeece euatios the eade is efeed to [7]. of euatio (.3) Page 4

5 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] II. MAIN RESULTS. Local asyptotic stability of the euilibiu poits Coside the atioal dyaic euatio o discete tie scales a( t) b( ( t)) ( ( t)), t T, (.) c ( ( ( t))) whee a, b, c ad.. Let a c a, c, ad d. The euatio (.) ca be ewitte as b b b a( t) ( ( t)) ( ( t)), t T. (.) c d( ( ( t))) The chage of vaiables / y( t) ( d) ( t), followed by the chage ( t) y( t) educes the above euatio to p( t) ( ( t)) ( ( t)), t T, (.3) ( ( ( t))) whee a c p a, ad c. Heeafte, we focus ou attetio o euatio (.3) istead of euatio (.). b b Now, whe is the atio of odd positive iteges, euatio (.3) has oly two euilibiu poits, ad p, ad whe is positive atioal ube ad ueato's eve positive iteges, euatio (.3) has uiue euilibiu poit ad thee euilibiu poits: if p,, p, ad p if p. Futheoe, suppose that, The, i this case, we coside ( t) fo all t T. k, k,. Whe p, euatio (.3) has uiue euilibiu poit. Whe p, howeve, euatio (.3) has the followig two euilibiu poits:, ad p. The local asyptotic behavio of is chaacteized by the followig esult. THEOREM.: () If a{, p }, the is locally asyptotically stable. () If p, the is a epelle. (3) If p, the is a saddle poit. Page 5

6 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] Poof: The Lieaized euatio associated with euatio (.3) about the euilibiu poit is p z( ( t)) z( t) z( ( t)), t T. (.4) Now, whe T, the euatio (.4) becoes p z z z,. (.5) The chaacteistic euatio associated with euatio (.5) is p t, whee z( t), (.6) whe T,, the euatio (.4) becoes p z z z,. (.7) The chaacteistic euatio associated with euatio (.7) is p log ( t), whee z( t), (.8) whe T h, the euatio (.4) becoes p zh( ) zh zh( ),. (.9) The chaacteistic euatio associated with euatio (.9) is p t/ h, whee z( t), (.) ad whe T the euatio (.4) becoes, p z z z, ( ) ( ). (.) The chaacteistic euatio associated with euatio (.9) is p Hece, the chaacteistic euatio associated with euatio (.4) is p Let u, ad v. t, whee z( t). (.) p () The esult follows fo Theoe. () ad the followig elatios, fo all t T. (.3) Page 6

7 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] p ( p ) u ( v) ( ), ad v. () The esult follows fo Theoe.() ad the followig elatios (3) Note that p p ( p ) u v, ad v. p p / 4. The esult follows fo Theoe. (3) ad the followig elatios p p ( p ) p 4 p 4 u v, ad u 4 v ( ). Now, the local asyptotic behavio of ad ae chaacteized by the followig esult. Theoe. : Assue that p. (i) If eithe ( p ) p o 3p, whee, the both ad ae locally asyptotically stable. (ii) If eithe ( p ) p o a{, 3p }, whee, the both ad ae epelle. (iii) If eithe p 3p, whee, the is a saddle poit. Poof : The Lieaized euatio associated with euatio (.3) about the euilibiu is Hece, the chaacteistic euatio associated with euatio (.4) is p ( p ) z( ( t)) z( t) z( ( t)). (.4) p p p ( p ), tt. p p Let u p ( ) ad v p. p p (i) Assue ( p ) p, the, p. The, fo Theoe. () ad the followig elatios p ( p ) p ( p ) p ( p ) u ( v) ( ) ( ), p p p p p ad ( p ) v, p we coclude that is locally asyptotically stable. Assue 3p, whee, the, p. The, fo Theoe. () ad the followig elatios Page 7

8 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] p ( p ) p ( p ) p 3p u ( v) ( ) ( ), p p p p p ad v p p p p, we coclude that is locally asyptotically stable. Siilaly, we ca pove that is locally asyptotically stable. (ii) Assue ( p ) p, the, p. The, fo Theoe. () ad the followig elatios p ( p ) p ( p ) p ( p ) u ( v) ( ) ( ), p p p p p ad ( p ) v, p we coclude that is a epelle. Assue that a{,3 p }, whee. The, fo Theoe. () ad the followig elatios p ( p ) p p (3p ) u ( v), p p p p p ad whe p, we have p p, so whe p, wehave p, so p p v, p p p p v, p p we coclude that is a epelle. Siilaly, we ca pove that is a epelle. (iii) If. p Note that the give coditio is euivalet to p p. Besides, we have p(3p 4) p. 4( p ) The, fo Theoe. (3) ad the followig elatios p p p p u v, ad p p p p we coclude that is a saddle poit. p 4( p ) 4 ( ) [4( ) (3 4)], p p ( p ) u v p p p Page 8

9 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] Assue that. p Note that the give coditio is euivalet to p p. Besides, we have p(4 3 p) 3p. 4( p) The, fo Theoe. (3) ad the followig elatios p p p p u v, ad p p p p we coclude that is a saddle poit. p 4( p ) 4 ( ) [ (4 3 ) 4( ) ], p p ( p) u v p p p. Boudedess of solutios of euatio (.3) I this sectio, we study the boudedess of solutio of euatio (.3). Theoe.3: Suppose that p ad is positive atioal ube ad ueato's eve positive iteges, i,. The the solutio of euatio (.3) is bouded fo all t T. Poof: We ague that ( t) A fo all t T by iductio o t. Case (): If T. Give ay iitial coditios A ad A, we ague that A fo all by iductio o. It follows fo the give iitial coditios that this assetio is tue fo,. Suppose the assetio is tue fo ad ( ). That is, A ad A. Now, we coside, whee we put istead of ( ) i euatio (.3), p p ( p) A (.5) Sice,,. (.6) Fo (.5) ad (.6), we obtai, ( p) A A fo all. Case(): If T h { hk : h ad k }. Give ay iitial coditios h A ad A, we ague that h A fo all by iductio o. It follows fo the give iitial coditios that this assetio is tue fo,. Suppose the assetio is tue fo ad ( ). That is, Now, we coside A ad A. h( ) h( ) h, whee we put h istead of h ( ) i euatio (.3), Page 9

10 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] p p ( p) A h( ) h( ) h( ) h( ) h. (.7) h( ) h( ) h( ),. (.8) Sice, h( ) h( ) h ( ) Fo (.7) ad (.8), we obtai, Case (3): If k, ( p) A h A fo all. T { : }. Give ay iitial coditios A ad A, we ague that A fo all k by iductio o k. It follows fo the give iitial coditios that this assetio is tue fo k,. Suppose the assetio is tue fo ( ) ad ( ). That is, A ad A. ( ) ( ) Now, we coside, whee we put istead of ( ) i euatio (.3), p p ( p) A. (.9) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Sice,,. (.) ( ) ( ) Fo (.9) ad (.), we obtai, Case(4): If T, ( p) A k A fo all k. ( ). Give ay iitial coditios / A ad A, we ague that A fo all k, by iductio o k. It follows fo the give iitial coditios that this assetio is tue k fo k /,. Suppose the assetio is tue fo k ad k, whee. That is, A ad A. Now, we coside, whee we put istead of ( ) i euatio (.3), p p A. (.) ( ) p Sice,,. (.) Fo (.) ad (.), we obtai, ( p) A k A fo all k. This copletes the poof. Page

11 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6].3 Peiodic solutios of euatio I this sectio, we study the eistece of peiodic solutios of euatio (.3). The followig theoe states the ecessay ad sufficiet coditios that this euatio has a peiodic solutios. Theoe.4: Euatio (.3) has pie peiod two solutios if ad oly if p whee p p ad 3,. Poof: Fist, suppose that, thee eists a pie peiod two solutio...,,,,,..., of euatio (.3). We will pove that 3p. We see fo euatio (.3) that p p ad. The, p, ad p, hece, ( ) ( )( ) p( ) ( ). Sice,, ( ) p. Theefoe, ( p ), (.3) ad so, p p ( )( ) ( ) ( ) ( )( )., [ Sice ( ) ] p p ( ). Hece, ( p ) ( p )( ). (.4) Fo (.3) ad (.4), we obtai ( p ) ( p )( p ). The, p( p ). (.5) It is clea ow, fo euatio (.3) ad (.5) that So, ( p )( 3 p). The, 3p. ad ae the two distict oots of the uadatic euatio t p t p p ( ) ( ), Secod, suppose that 3p. We will show that euatio (.3) has a pie peiod two solutios. Assue that Theefoe ( p ) ( p )( 3 p) ( p ) ( p )( 3 p), ad. ad ae distict eal ubes. Set, We wish to show that ( ( t )) ad ( t ), t T. ( ( t )) ( ( t )). Page

12 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] It follows fo euatio (.3) that ( ( t )) p( t) ( ( t)) ( ( t )) p[ ( p ) ( p )( 3 p) [( p ) ( p )( 3 p)] ( p ) ( p )( 3 p) ( p ) ( p )( 3 p) ( p )( p) ( p ) ( p )( 3 p) ( p ) ( p )( 3 p) ( p)( p) ( p ) ( p )( 3 p) [( p ) ( p )( 3 p)] ( p ) ( p )( 3 p) ( p p )[( p ) ( p )( 3 p)] 4( p p ) ( p ) ( p )( 3 p) t p p Siilaly as befoe oe ca easily show that Thus euatio (.3) has the pie peiod two solutio whee ad ( ( )),. ( ( t)), whee ( ( t)) ad ( t)....,,,,,..., ae the distict oots of the uadatic euatio (.3). The poof is copleted. To eaie the global attactivity of the euilibiu poits of euatio (.3), we fist eed to deteie the ivaiat itevals fo euatio (.3)..4 Ivaiat itevals ad global attactivity of the zeo euilibia I this subsectio, we deteie the faily of ivaiat itevals ceteed at. Theoe.5: Assue p. The fo ay positive eal ube A p The iteval O(, A) ( A, A) is ivaiat fo e. (.3). / ( ( )), Poof: We coside T, h { hk : h ad k }, { : }, ad,. Case (): If T. Give ay iitial coditios A ad A,we ague that A fo all by iductio o. It follows fo the give iitial coditios that this assetio is tue fo,. Suppose the assetio is tue fo ad ( ). That is, A ( ( p )), ad A ( ( p )). / / Now, we coside, whee we put istead of ( ) i euatio (.3). Sice, A ( ( p )) p, Page

13 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] Thus, p. (.6) p p p ( ) a{, } a{, } A fo all. Case (): If T h { hk : h ad k }. Give ay iitial coditios h A ad A, we ague that h A fo all by iductio o. It follows fo the give iitial coditios that this assetio is tue fo h,. Suppose the assetio is tue fo ad ( ). That is, Now, we coside Thus, A ( ( p )), ad A ( ( p )). / / h( ) h( ) h, whee we put h istead of h ( ) i euatio (.3). Sice, h ( ) A ( ( p )) p, p p p p. (.7) h ( ) h( ) h( ) h( ) h( ) h ( ) a{ ( ), ( ) } h h h( ) h( ) h( ) a{, } A fo all. h( ) h( ) Case (3): If T { : }. Give ay iitial coditios k, A ad A, we ague that A fo all k by iductio o k. It follows fo the give iitial coditios that this assetio is tue fo k,. Suppose the assetio is tue fo ( ) ad ( ). That is, A ( ( p )), ad A ( ( p )). / / ( ) ( ) Now, we coside, whee we put istead of ( ) i euatio (.3). Sice, Thus, A p p ( ( )), ( ) p. (.8) ( ) p p p ( ) a{, } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a{, } A fo all k. ( ) ( ) Page 3

14 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] Case (4): If T, A ad A, we ague that. Give ay iitial coditios / A fo all k, by iductio o k. It follows fo the give iitial coditios that this assetio is tue k fo k /,. Suppose the assetio is tue fo k ad k, whee. That is, A ( ( p )), ad A ( ( p )). / / Now, we coside, whee we put istead of ( ) i euatio (.3). Sice, Thus, A ( ( p )) p, p. (.9) p p p ( ) a{, } a{, } A fo all. This copletes the poof. Now, we ivestigate the global attactivity of the euilibiu poit. Lea.: Assue that T, p, ad is positive atioal ube ad ueato's eve positive iteges. Futheoe, suppose that ad coside euatio (.3) with the estictio that Let { } be a solutio of this euatio ad The, R (,), ad R p l ( ( )), (.3) f : O(, R) O(, R) O(, R). p R. (.3) (a{, }) R a{, },,,.... (.3) / Poof: Fo Theoe.5, we have A R,,,,,..., whee we put istead of ( ) i euatio (.3). The, (a{, }) R. Thus, Hece, (a{, }) R p. Page 4

15 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] p. This easthat R (,). (a{, }) Now, we pove that (.3), by iductio o. Fo euatio (.3), we have p ( p ) ( ) a{, } (.33) The, fo (.33), we obtai a{, }, fo (.6). (.34) p ( ) a{, }. (.35) But (a{, }) R p. The, (a{, }). Fo (.35), we have p p ( ) a{, } a{, } (a{, }) R a{, } R a{, }, / ad fo (.33) ad (.34), we obtai p p ( ) a{, } a{, } (a{, }) p (a{a{, }, }) p (a{, }) a{a{, }, } a{, } R a{, }. Thus, the ieuality (.3) holds fo,. Suppose that the ieuality (.3) holds fo ad ( 3), espectively. By (.34), we have So, The, fo (.33), we have a{, } fo all. p p p R. (a{, }) p ( ) a{, } R a{, } R a{ R, R } a{, } R a{, }. ( )/ ( )/ / Page 5

16 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] This copletes the iductive poof of (.3). Lea.: Assue that T h { hk : h(,) ad k }, p, ad is positive atioal ube ad ueato's eve positive iteges. Futheoe, suppose that (.3) holds ad coside euatio (.3) with the estictio that f : O(, R) O(, R) O(, R). Let { : h(,) ad } be a solutio of this euatio ad k p R. (.36) (a{, }) h The, R (,), ad R a{, },,,.... (.37) h h/ h Poof: Fo Theoe.5, we have Whee we put h istead of ( ) A R,,,,..., h h i euatio (.3). The, (a{ h, }) R. Thus, (a{, }) R p. Hece, h p. This eas that R (,). (a{, }) Now, we pove that (.37), by iductio o. Fo euatio (.3), we have The, fo (.38), we obtai h p p h( ) h( ) h ( ) a{ ( ), ( ) } (.38) h h h( ) h( ) a{, }, fo (.7). (.39) h( ) h( ) p h ( ) a{, h }. (.4) h But (a{, }) R p. The, (a{, }). h h h h Fo (.4), we have p p ( ) a{, } a{, } h h h h (a{, h }) R a{, } R a{, }, h/ h ad fo (.38) ad (.39), we obtai h Page 6

17 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] p p ( ) a{, } a{, } h h h (a{ h, }) p (a{a{, }, }) p (a{, }) R h h/ a{ h h, }. a{a{, }, } h a{, } R a{, } h h Thus, the ieuality (.37) holds fo,. Suppose that the ieuality (.37) holds fo h ( ) ad h( )( 3), espectively. By (.39), we have So, The, fo (.38), we have a{, } fo all. p p p R. (a{, }) h h( ) h( ) h p ( ) a{, } R a{, } h h( ) h( ) h( ) h( ) h ( ) R a{, } R a{ R, R } a{, } h h h( )/ h( )/ h( ) h( ) h R a{, }. h / h This copletes the iductive poof of (.37). Lea.3: Assue that T { : }, p, ad is positive atioal ube ad ueato's eve positive iteges. Futheoe, suppose that (.3) holds ad coside euatio (.3) with the estictio that f : O(, R) O(, R) O(, R). Let { : k ad } be a solutio of this euatio ad The, k Poof: Fo Theoe.5, we have p R. (.4) (a{, }) R a{, }, k,4,.... (.4) k k / A R, k,,..., whee we put istead of ( ) i euatio (.3). The, (a{, }) R. Thus, Hece, k (a{, }) R p. Page 7

18 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] p. This eas that R (,). (a{, }) Now, we pove that (.4), by iductio o k. Fo euatio (.3), we have The, fo (.43), we obtai p ( ) ( ) ( ) p ( ) a{, } (.43) ( ) ( ) ( ) a{, }, fo (.8). (.44) ( ) ( ) p ( ) a{, }. (.45) But (a{, }) R p. The, (a{, }). Fo (.45), we have p p ( ) a{, } a{, } (a{, }) R a{, } R a{, }, / ad fo (.43) ad (.44), we obtai p p ( ) a{, } a{, } 4 (a{, }) p (a{a{, }, }) p (a{, }) R 4/ a{, }. a{ R a{, }, } R a{, } R a{, } Thus, the ieuality (.4) holds fo,. Suppose that the ieuality (.4) holds fo espectively. By (.44), we have a{, } fo all. ( ) ( ) ( ) ad ( ), Page 8

19 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] p p p So, R. (a{, }) The, fo (.43), we have ( ) ( ) p ( ) a{, } R a{, } ( ) ( ) ( ) ( ) ( ) ( ) / ( ) / ( ) / a{ R, R } a{, }} R a{, }, 3,4,... / R a{, },,,.... This copletes the iductive poof of (.4). Lea.4: Assue that T,, p, ad is positive atioal ube ad ueato's eve positive iteges. Futheoe, suppose that (.3) holds ad coside euatio (.3) with the estictio that f : O(, R) O(, R) O(, R). Let { : k, ad } be a solutio of this euatio ad k p R. (.46) (a{, }) / The, R (,), ad R a{, }. (.47) k k / / Poof: Fo Theoe.5, we have A R, k,, ad, k whee we put istead of ( ) i euatio (.3). The, (a{ /, }) R. Thus, p (a{ /, }) R p. Hece,. This eas that (a{, }) R (,). Now, we pove that (.47), by iductio o. Fo euatio (.3), we have The, fo (.48), we obtai ( p ) / p ( ) a{, } (.48) a{, }, fo (.9). (.49) p ( ) a{ /, }. (.5) / But Page 9

20 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] (a{, }) R p. The, / / / (a{, }). / / / Fo (.5), we have p p ( ) a{, } a{, } / / / (a{ /, }) R a{, } R a{, }, / / / ad fo (.48) ad (.49), we obtai p p ( ) a{, } a{, } (a{, }) p (a{a{, }, }) p (a{, }) / / a{ R a{, }, } / / R a{, } R R a{ / / / / / / / R a{, } R a{, }. Thus, the ieuality (.47) holds fo k,. Suppose that the ieuality (.47) holds fo k ad k,, espectively. By (.49), we have So, The, fo (.48), we have a{, } fo all. /, } / p p p R. (a{, }) p ( ) a{, } R a{, } / / / / a{ R a{, }, R a{, }} / / R a{, }}, 3,4,... / / R a{, },,,3,.... This copletes the iductive poof of (.47). Theoe.6: Assue that p, ad is positive atioal ube ad ueato's eve positive iteges. Futheoe, suppose that (.3) holds, ad coside euatio (.3) with the estictio that f : O(, R) O(, R) O(, R). The the euilibiu poit of the euatio (.3) is a global attacto. Page

21 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] Poof: Fo Leas. i, i,,3, 4, ( t) whe t, the the euilibiu poit of the euatio (.3) is a global attacto. Net, we deteie the faily of ivaiat itevals ceteed at p, whee p..5 Ivaiat itevals ad global attactivity of the ozeo euilibia I this subsectio, we coside the discete tie scales T, h { hk : h ad k }, { : }, ad,,. This eas that, we deteie the faily of ivaiat itevals ceteed at p. To this ed, we establish the followig elatio. p( t) ( ( t)) p p( t) p ( ( t)) ( ) ( ) ( ( t)) ( ( t)) p( t) p p( t) p p p ( ) ( ) ( ( t)) ( ( t)) ( ( t)) ( ( t)) p p ( ( t) ) ( ( ( t)) ). ( ( t)) ( ( ( t)))( ) I view of p ( )( ) ( p )( ), the above euality is educed to p p ( ( t)) ( ( t) ) ( ( ( t)) ). Thus, ( ( t)) ( ( t)) p p ( ( t)) ( t) ) ( ( t)) ( ( t)) ( ( t)) p p a{ ( ( t )), ( t ) }. (.5) ( ( t)) Now, we ae eady to descibe the faily of ested ivaiat itevals ceteed at. Theoe.7: Assue that p /, ad 3p p. The fo evey positive ube A i{ (3p ),( p), p}, (.5) the iteval O(, A) ( A, A) is ivaiat fo euatio (.3). Poof: Case (): If T we ague that, give ay iitial coditios A ad A, A fo all byiductioo. The poof is siila to the poof of Theoe 4. i [4 ] ad will be oitted. Case : If T h, give ay iitial coditios A ad A, h Page

22 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] we ague that A fo all byiductioo. It follows fo the give iitial assetio is tue fo,. h( ) ad h( ). That is, Now, we coside h ( ) h h( ) h( ) Suppose the asseatio is tue fo A ad A. h whee we put h istead of h ( ) i euatio (.5). Sice, ( A) ( p ) A A ( p) A ( p), we have ( ). Now, the coditio A ( p) is euivalet to h( ) h( ) ( A) p ( p) A, the coditio A (3 p ) is euivalet to p ( ( A) p), ad the coditio A ( p) is euivalet to p ( ( A) p). So, p ( ( A) p). Thus, ( p p ) ( ) p p h( ) h( ) ( A) p p ( A) p ( A) p. Fo this we deduce that p p h ( ). By (.5), we have p p a{, } h h( ) h( ) h ( ) a{, } A. h( ) h( ) This copletes the iductive poof. Whe T ad the poof will be oitted.,, Now, we ivestigate the global attactivity of the euilibiu poit. Lea.8: Assue that T, p /, ad 3p p. Coside euatio (.3) with the estictio that Let { } be a solutio of this euatio ad f : O(, R) O(, R) O(, R), whee R i{ (3p ),( p), p}. (.53) p p M. (.54) p a{, } Page

23 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] The, M (,), ad / M a{, },,,... (.55) The poof is siila to the poof of Lea 6. i [4 ] ad will be oitted. Lea.9: Assue that T h { hk : h(,) ad k }, p /, ad 3p p, ad (.53) holds. Coside euatio (.3) with the estictio that f : O(, R) O(, R) O(, R). Let { : h(,) ad } be a solutio of this euatio ad h The, M (,), ad M p p. (.56) p a{, } h h/ h M a{ h, },,,...(.57) Poof: Note that p ( p p ) p R ( p p ) p p R. h Whe, p, p ( p p ) ( (3p )) R. h Whe, p, p ( p p ) (( p) ) R. h p p So,. p h p p Siilaly,. Hece, M (,). p h Net, we pove euatio (.57) by iductio o. By (.5), whee we put h istead of h ( ), we p p have h a{, h }. Sice, h ( _ ( ) p ( ) p a{, } h h h h Siilaly, we have p R ( p) R, we deive p p p p M p a{, } p p h. The, M a{, } M h M h h / h. a{, }. h Ad we have, Page 3

24 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] p p a{, } h M a{, M a{, }} M a{, } h a{, }. h / M h h h So euatio (.57) holds fo,. Suppose that euatio (.57) holds fo h( ) ad h( -), espectively. By(.5) ad the iductive hypothesis, we have p p h a{ h( ), h( ) } h ( ) p p a{ h( )/, h( )/ M M } a{, h }}. h ( ) Now, fotheoe(.7), we have By iductio o, we ca pove a{, }. h h( ) h( ) a{, }, hece, h ( ) ( ) p ( ) h h( ) h( ) h( ) we coclude p a{, } p R ( p) R, p p M, the h ( ) h( )/ h( )/ h M a{ M, M } a{, h }} h a{, } a{, }} h/ h( )/ h( )/ M M M h a{,} a{, }} h / h/ M M h a{, }}. h / M h This copletes the iductive poof of euatio (.57). Lea.: Assue that euatio (.3) with the estictio that Let { k : k ad } T { : }, p /, ad 3p p, ad (.53) holds. Coside f : O(, R) O(, R) O(, R). be a solutio of this euatio ad M p p. (.58) p a{, } The, M (,) ad / k k M a{, }, k,4,.... (.59) Page 4

25 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] Lea.: Assue that T,, p /, ad 3p p, ad (.53) holds. Coside euatio (.3) with the estictio that f : O(, R) O(, R) O(, R). Let { : k,, ad } be a solutio of this euatio ad k The, M (,) ad M p p. (.6) p a{, } / k / k M a{ /, }. (.6) Theoe.9: Assue that p /, ad 3p p, ad (.53) holds. Coside euatio (.3) with the estictio that f : O(, R) O(, R) O(, R). The the euilibiu poit of the euatio (.3) is a global attacto. Poof: Fo Leas. i, i 8,9,,, we obtai ( t) whe t, the the euilibiu poit β of the euatio (.3) is a global attacto. I thak y teache Pof. E. M. Elabbasy. ACKNOWLEDGEMENTS REFERENCES [] M. T. Aboutaleb, M. A. El-Sayed ad A. H. Haza, Stability of the ecusive seuece +, Joual of Matheatical Aalysis ad Applicatios, 6(), [] R. P. Agawal, M. Bohe, D. O'Rega ad A. Peteso, Dyaic euatios o tie scales: A suvey, J. Cop. Appl. Math., Special Issue o (Dyaic Euatios o Tie Scales), edited by R. P. Agawal, M. Bohe ad D. O'Rega, (Pepit i Ule Seiae 5)4(-) (), -6. [3] A. M. Aleh, E. A. Gove ad G. Ladas, O the ecusive seuece, Joual of Matheatical Aalysis ad Applicatios, 33(999), [4] M. Bohe ad A. Peteso, Dyaic Euatios o Tie Scale, A Itoductio with Applicatios, Bikhause, Bosto,. [5] M. Bohe ad A. Peteso, Fist couse o dyaic euatios o tie scales, August 7. [6] C. Cia, O the positive solutios of the diffeece euatio, Applied Matheatics ad Coputatio, + 5(4), -4. [7] K. Cuigha, M. R. S. Kuleovic, G. Ladas ad S. V. Valiceti, O the ecusive seuece, B C Noliea Aalysis, 47(), [8] O. Dosly ad S. Hilge, A ecessay ad sufficiet coditio fo oscillatio of the Stu-Liouville dyaic euatio o tie scales, Special Issue o (Dyaic Euatios o Tie Scales), edited by R. P. Agawal, M. Bohe ad D. O'Rega, J. Cop. Appl. Math. 4(-) (), Page 5

26 Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [ ] [Vol-, Issue-, Decebe- 6] [9] M. M. El- Afifi, O the ecusive seuece 47(4), , Applied Matheatics ad Coputatio, B C [] E. M. Elabbasy, H. El-Metwally ad E. M. Elsayed, O the diffeece euatio a Diffe. Eu., Volue 6, Aticle ID 8579,--. [] H. M. El-Owaidy ad M. M. El- Afifi, A ote o the peiodic cycle of Coputatio, 9(), [] H. M. El-Owaidy, A. M. Ahed ad M. S. Mousa, O the ecusive seueces b c d, Adv., Applied Matheatics ad, Applied Matheatics ad Coputatio, 45(3), [3] Sh. R. Elzeiy, O the Ratioal Dyaic Euatio o Discete Tie at b t Scales ( ( t)), t T, IJOER, Vol., Issue-6, Sept. -5. c d( ( t)) ( ( ( t))) [4] C. Gibbos, M. R. S. Kuleovic ad G. Ladas, O the ecusive seuece, Matheatical Scieces Reseach Hot-Lie, 4 ()(), -. [5] S. Hilge, Aalysis o easue chais- a uified appoach to cotiuous ad discete calculus, Results Math. 8 (99), [6] V. L. Kovic, G. Ladas ad I. W. Rodigues, O atioal ecusive seueces, Joual of Matheatical Aalysis ad Applicatios, 73(993), [7] V. L. Kovic ad G. Ladas, Global Behavio of Noliea Diffeece Euatios of Highe Ode with Applicatios, Kluwe-Acadeic Publishes, Dodecht, 993. p y [8] W. A. Kosala, M. R. S. Kuleovic, G. Ladas ad C. T. Teieia, O the ecusive seuece y, y y Joual of Matheatical Aalysis ad Applicatios, 5(), [9] M. R. S. Kuleovic, G. Ladas ad N. R. Pokup, A atioal diffeece euatio, Coputes ad Matheatics with Applicatios, 4(), [] W. Li ad H. Su, Global attactivity i a atioal ecusive seuece, Dyaic systes ad applicatios (), [] X. Ya ad W. Li, Global attactivity i the ecusive seuece, Applied Matheatics ad Coputatio, 38(3), [] X. Ya ad W. Li, Global attactivity i a atioal ecusive seuece, Applied Matheatics ad Coputatio, 45(3), -. [3] X. Yag, H. Lai, David J. Evas ad G. M. Megso, Global asyptotic stability i a atioal ecusive seuece, Applied Matheatics ad Coputatio,.58(4), [4] X. Yag, B. Che, G. M. Megso ad D. J. Evas, Global attactivity i a ecusive seuece, Applied Matheatics ad Coputatio, 58(4), a b [5] X. Yag, W. Su, B. Che, G. M. Megso ad D. J. Evas, O the ecusive seuece, Applied c d Matheatics ad Coputatio, 6(5), [6] D. Zhag, B. Shi ad M. Cai, A atioal ecusive seuece, Coputes ad Matheatics with Applicatios 4 (), Page 6