Aled Mathematcs -8 htt://d.do.org/.6/am..9a Publshed Ole Setember (htt://www.scr.org/joural/am) O the Behavor of Postve Solutos of a Dfferece Equatos Sstem * Decu Zhag Weqag J # Lg Wag Xaobao L Isttute of Sstems Scece ad Mathematcs Naval Aeroautcal ad Astroautcal Uverst Yata Yata Cha Emal: # jwqkeshu@6.com Receved March 5 ; revsed Arl 5 ; acceted Arl Corght Decu Zhag et al. Ths s a oe access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese whch ermts urestrcted use dstrbuto ad reroducto a medum rovded the orgal work s roerl cted. ABSTRACT Motvated b a oe roblem the lterature Damcs of Secod Order Ratoal Dfferece Equatos wth Oe Problems ad Cojecture we troduce a dfferece equato sstem: * where. We tr to fd out some codtos such that the soluto of sstem coverges to erodc soluto. Ths model ca be aled to the two seces cometto ad oulato bolog. Kewords: Dfferece Equato; Sstem; Mootoct; Perodct; Oscllator. Itroducto I the moograh of Damcs of Secod Order Dfferece Equato [] M. R. S. Kuleovć ad G. Ladas gave a oe roblem (see []. 99) as followg: Oe roblem..8: Determe whether ever ostve solutos of the followg equato coverges to a erodc soluto of the corresodg equato:. () Motvated b the Oe Problem we troduce the dfferece equato sstem:. () where the tal ots Recetl there has bee great terest studg dfferece equato sstems. Oe of the reasos for ths s the ecesst for some techques that ca be used - * Research suorted b Dstgushed Eert Foudato ad Youth Scece Foudato of Naval Aeroautcal ad Astroautcal Uverst. # Corresodg author. vestgatg equatos arsg mathematcal models descrbg real lfe stuatos oulato bolog ecoomcs robablt theor etc. There are ma aers related to the dfferece equatos sstem such as [-9]. I [] Car studed the solutos of the sstem of dfferece equatos: () I [] E. Camouzs ad Paaschooulos studed the global asmtotc behavor of ostve soluto of the sstem of ratoal dfferece equatos: () m m I [] Ahmet Yasar Ozba studed the sstem of ratoal dfferece equatos: a b (5) q q I [5] Abdullah Selcuk Kurbal et al. studed the behavor of ostve solutos of the sstem of ratoal dfferece equatos: Corght ScRes.
D. C. ZHANG ET AL.. (6) I ths aer we tr to fd out some codtos such that the soluto of sstem () coverges to erodc soluto. At the same tme we ca get the oscllator of sstem (). Before gvg some results of the sstem () we eed some deftos as follows [6]: Defto. A ar of sequeces of ostve real umbers that satsfes sstem () s a os tve soluto of sstem (). If a ostve soluto of sstem () s a ar of ostve costats that soluto s the equlbrum soluto. Defto. A strg of cosecutve terms s m (res. s m) ( s m ) s sad to be a ostve semccle f (res. ) s m s (res. s ) ad m (res. m ). Otherwse that s sad to be a egatve semccle. A strg of cosecutve terms s s m m s sad to be a ostve(res.egatve) semccle f are ostve (res.egatve) s m s m semccle. A soluto (res. ) oscllates about (res. ) f for ever N there est s m N s m such that s m (res. s cm c ). We sa that a soluto of sstem oscllates about f oscllates about or oscllates about.. Some Lemmas Lemma. The sstem () has a uque ostve equlbrum. The roof of lemma. s ver eas so we omt t. Lemma. If q q q The ever ostve soluto of sstem () wth rme erod two takes the forms q q q q q q q or q q q q q q q s a erod-two soluto of sstem (). Proof: Let m qr m qr be a erod-two soluto of sstem (). The b sstem () we get or m r q q m r We ca see that (7) ca be chaged to (7) q m r mr q (8) Form (8) we ca obta m r m r ad q q q q q q q q q q q q q q Therefore we comlete the roof. Lemma. Assume that the tal ots ad s a ostve soluto of sstem (). The the followg cases are true: (a) If ; the ad are both creasg. (b) If ; the ad are both decreasg. Proof: (a) B sstem () we ca get Corght ScRes.
D. C. ZHANG ET AL. 5.e. where. B codto ad (9) we get: 5 6 8 9 (9) () B codto ad (9) we get: () B codto 6 7 9 Equall we ca get: ad (9) we get: () 5 7 8 5 6 8 9 6 7 9 () () (5) 5 7 8 Hece b ducto ad ()-(5) we roof that ad are both creasg. Usg the same method we ca rove that case (b) holds. Therefore we com lete the roof. Lemma. Assume that. The there does ot est a ostve soluto of sstem () such that ad are both - creasg or both decreasg. Proof: B Equato ( 9) we ca get ad have the same mootoous. Frstl we roof that there does ot est ostve soluto such that ad are both creasg. Assume for the sake of cotradcto that we have the followg results: () () s creasg; B sstem () we obta s also creasg. ma (6) ma Equatos (6) ad (7) t mles that:.e ma ma 5 6 7 Because of 7 5 8 6 we ca get ma ma 5 6 7 5 6 ma ma 6 7 8 5 6 ma 5 ma 5 6 Also we ca get ma 5 ma 5 6 6 6 (7) (8) (9) Because of the assumtos () ad () t s eas to see th at (8) ad (9) do ot hold. Ths s a cotradcto ad we roof the case of creasg does ot hold. Net we roof there does ot est ostve soluto of sstem () such that ad are both decreasg. Assume for the sake of cotradcto that we have the followg results: () s decreasg; () s also decreasg. B the Lmtg Theorem we kow that ad are both decreasg to a ar of costats. We set lm q lm m lm r lm ad q m r. B sstem () we kow that these costats satsf the sstem ().e. mr q q m r However f q m r Equato () do ot holds whch s cotradcto. Hece we comlete the roof of lemma. Lemma.5 Assume that. The there does ot est a ostve soluto of sstem () such () Corght ScRes.
6 D. C. ZHANG ET AL. that a d are both de creasg or both creasg. Proof: Frst we roof there does ot est ostve soluto of sstem () such that ad are both decreasg the roof of creasg s smlar so we omt t. Assume for the sake of cotradcto that we have the followg results: () s decreasg; () We set lm lm lm r. s also decreasg. q lm m B emwe kow that ad are both decreasg to a ar of costats. Obvousl the lmts of w Lmt Theor ca ot decrease to zero. B sstem () we ca get m r q () q m r here qmr whch ca be chaged to However f q mr Equato () ca ot hold. Ths s a cotradcto ad we comlete the roof. The the roof of the case of creasg s smlar wth the roof of the the case of decreasg so we omt t. I addto to the method above we ca roof the Lemma.5 b the method of Lemma.. Here we omt t.. Ma Results q m r () mr q Theorem. Assume that ; ad are both decreasg; ad coverges to a erod-two soluto as followg s a ostve soluto of sstem (). The a. d where satsf. Proof: B lemma.(a) we ca obta that ad are both decreasg. The b the Lmt Theorem we ca get lm lm lm tve. We ca set ad lm all est a d are os- lm lm lm lm B lemmas. a d.5 we kow that there does ot est a ostve soluto or such that ad are both decreasg. Hece there s at least oe of satsf ad at least oe of satsf B sstem () we get It s to see that s a erod-two soluto of sstem () ad satsf. We comlete the roof. Corollar. Suose that s a ostve soluto of sstem (). The the followg statemet s true: If ; the soluto of sstem (). evetuall oscllates about equlbrum () Theorem. Assume that ; a d s a ostve soluto of sstem (). The ad are both creasg; ad coverges to a erod-two soluto as followg. Corght ScRes.
D. C. ZHANG ET AL. 7 w here satsf B ducto we ca get. Proof: B lemma.(a) we obta that 6 9 (7) ad are both creasg. 7 We set lm lm lm 7 (8) lm B Equato (9) we ca get whch ca be chaged to: () 5 (5) 7 (6) 5 7 B the we ca get 6 9 5 8 5 8 7 7 6 9 6 9 5 5 8 7 7 6 9 8 5 8 7 7 6 9 6 9 5 8 6 9 5 8 5 8 7 7 6 9 7 6 9 6 9 5 8 5 8 7 5 8 7 7 6 9 6 9 5 8. 5 8 From Lemma we kow that there at least oe. The b Lmtg Theorem we ca get at least oe of the lmtg of must est. Wth o loss geeralt we set the l mt of estwe ca kow. B lmtg Equato (7) we ca get lm lm lm lm lm 7 6 9 Hece we ca get lm.e. Net we tr to roof ad B sstem () we get B () ad () we ca get whch ca be chaged to 5 5. (9) () () () () () Corght ScRes.
8 D. C. ZHANG ET AL. B the both sde of Equato (5) we ca get Assume that lm lm lm (5) (6) b Stolz Theorem we obta lm lm (7) Because the we ca get the lmt of lm However there est such that whch s coducto. Hece the assume does ot hold. We obta lm. Use the same method we ca also get lm. B sstem () we get (8) It s to see that s a erod-two soluto of sstem () ad satsf. Therefore we comlete the roof. Corollar. Suose that s a ostve soluto of sstem (). Th e the followg statemet s true: If ; the the soluto of sstem () oscllates about equlbrum ( ). Theorem. Assume that ; ad s a ostve solu- to of sstem (). The the sstem () has rme erod two solutos ad for. Proof: B the lemma 8 we ca comlete the roof. Here we omt t. REFERENCES [] M. R. S. Kuleovć ad G. Ladas Damcs of Secod Order Ratoal Dfferece Equato wth Oe Problems ad Cojectures CRC Press Chama Hall. [] C. Car O the Postve Solutos of the Dfferece Equato Sst em Aled Mathematcs ad Comutato Vol. 58. - 5. [] E. Camouzs ad G. Paaschooulos Global Asmtotc Behavor of Postve Solutos o the Sstem of Ratoal Dfferece Equatos Aled Mathematcs ad m m Letters Vol. 7. 7-77. [] A. Y. Ozba O the Sstem of Ratoal Dfferece Equa b atos Aled Mathematcs qq ad Comutato Vol. 88 7. 8-87. [5] A. S. Kurbal C. Car ad I. Yalckaa O the Behavor of Postve Solutos of the Sstem of Ratoal Df ferece Equatos Ma thematcal ad Comuter Modellg Vol. 5. 6-67. [6] G. Paaschooulos O a Sstem of Two Nolear Dfferece Equatos Joural of Mathematcsal Aalss ad Alcatos Vol. 9 998. 5-6. [7] G. Paaschooulos M. Rad ad C. J. Schas Stud of the Asmtotc Behavor of the Solutos of Three Sstems of Dfferece Equatos of Eoetal Form Aled Mathematcs ad Comutato Vol. 8. 5-58. [8] E. Camouzs C. M. Ket G. Ladas ad C. D. Ld O the Glob al Character of Solutos of the Sstem: Joural of Df A B C ferece Equatos ad Alcatos Vol. 8. 5-5. [9] E. Camouzs E. Drmos G. Ladas ad W. Tkjha Patters of Boudedess of the Ratoal Sstem A B C Jour- A B C al of Dfferece Equatos ad Alcatos Vol. 8. 89-. Corght ScRes.