Nonlinear Piecewise-Defined Difference Equations with Reciprocal Quadratic Terms

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1 Joural of Matematcs ad Statstcs Orgal Researc Paper Nolear Pecewse-Defed Dfferece Equatos wt Recprocal Quadratc Terms Ramada Sabra ad Saleem Safq Al-Asab Departmet of Matematcs, Faculty of Scece, Jaza Uversty, Saud Araba Departmet of Matematcs, Faculty of Scece, Al-Albayt Uversty, Jorda Artcle story Receved: -9-5 Revsed: --5 Accepted: --5 Correspodg Autor: Ramada Sabra Departmet of Matematcs, Faculty of Scece, Jaza Uversty, Saud Araba E-mal: Abstract: I ts study we cosder te several types of pece-wse defed sequeces. We prove some results cocerg te beavor of te sequece. I some cases we prove tat te terms repeat temselves certa patters ad we prove oter cases tat te terms grow mootocally after few terms ad approaces fty. Keywords: Dfferece Equatos, Dscrete Dyamcal Systems, Bfurcato, Perodc Sequeces Itroducto I te teory of dfferece equatos te estece of perodc solutos plays a mportat role. Applcatos of tese classcal estece teorems are well ow ow. Recetly, tese teorems ave bee geeralzed. For detal, see te researc papers (Hou et al., ; Papascopoulos et al., 7; Stevc, 7; Zag et al., ). I ts study we preset some deftos from dscrete dyamcal systems ad troduce ew some deftos ad teores. Oe aspect of ts study s to study te estece of fed pots ad pots avg fte perod of certa systems. Furter, we fd solutos for some dffcult equatos. Ha ad Ceg () cosdered te followg olear dfferece equato: = a bf ( ) c () λ were, te fucto f λ s a pece-wse defed fucto as follows: f ( ) = for (, λ) f ( ) = for ( λ, ) Tey studed cases we te sequece s perodc or ubouded. Ts dfferece equato s equvalet to te system: u = y y = au bf ( y ) c () λ Usg te otato: Tey foud: c, b p = q = c a a If <λ<p, te all solutos of te system coverge to te pot (p; p). If λ = p, te all solutos of te system suc tat u, y (p, ) coverge to (p; p) ad for te remag start values coverges to te lmt -cycle (p; q), (q; p) If p<λ<q, te all solutos of te system coverge to te lmt -cycle (p; q), (q; p) If λ = q, te all solutos of te system suc tat u, y Φ a specfc subset of R coverge to (q; q) ad for te remag start values coverges to te lmt -cycle (p; q), (q; p) If λ>q, te all solutos of te system coverge to (q; q) Al-Asab ad Guyer () started researc o a ew type of olear dfferece equatos. Tey cosdered a ad sde of te dfferece equatos, wc cossts of a pecewse defed fucto two ways. Ts fucto s lear te two ways. Ts made tgs easer to treat. Some results were prove, wle oter results are just cojectured. Ts s a ew approac of dealg wt dfferece equatos, wc posses perodc solutos. Qea et al. () too furter steps te drecto of cosderg olear equatos. Tey ave a rgt ad sde of te dfferece equatos, wc cossts of a pece-wse defed fucto two ways. But, ts fucto s olear oe of te two ways. Te teory developed (Qea et al., ) s more 5 Ramada Sabra ad Saleem Safq Al-Asab. Ts ope access artcle s dstrbuted uder a Creatve Commos Attrbuto (CC-BY). lcese.

2 Ramada Sabra ad Saleem Safq Al-Asab / Joural of Matematcs ad Statstcs 5, (): 88.9 DOI:.8/jmss dffcult to uderstad ta (Al-Asab ad Guyer, ) ad uses more complcated matematcal tecques. I (Qea et al., ; Sabra ad Al-Asab, 5a; 5b) ad (Al-Asab ad Sabra, 5) we fd cases, wc ubouded solutos ests. Te researcers dd ot eplore all possble cases we te pece-wse defed fucto cossts of lear braces. Tey left some room for researc ts drecto. Also, te case tat te fucto s olear oe of te two braces we see oly quadratc terms. Of course, quadratc equatos are a atural sese te et step we we wat to ave olear terms ad provde more room for provg teorems rater tat cubc or ger order equatos. Geeral Sequeces I ts study we troduce te sequece defed as follows: τ, v, for =,,... = θ j, > v Te we obta te sequece:,,,..., = Proof: We ote frst tat: Sce: < > () Wc s true accordg to our coce of ad. Sce te fucto: Is decreasg, we get: f ( ) = for > Te umbers:,, j, v, τ, θ m m v < for all m =,..., () Are determed accordg to certa specfcatos. Ts type of sequeces was studed by te autors oter papers (Al-Asab ad Sabra, 5). Sequeces wt τ = - Te umber τ s set to mus oe. We ca see tat some cases te sequeces s ot well-defed. For eample, f we set: Te we obta: =, v =, j =, =, = * =, = udefed I case we te geerated sequece cotues welldefed to fty we ecouter dfferet beavor types. I some cases we obta smple perodc beavor as we sall sow ow: Proposto : et ad. If we coose: ( )( ) j =, < v <, Now, we compute te terms of te sequece based o () ad () as follows: v, v, j = < = > = = ( )( ) ( )( ) v > Smlarly, we ca sow tat: m m m = > v for all m =,..., Tus, we obta from (5): We cosder a complemetary case for v. Proposto : et ad. If we coose: v > ma, Te we obta for all values of j te sequece:,, (5) 89

3 Ramada Sabra ad Saleem Safq Al-Asab / Joural of Matematcs ad Statstcs 5, (): 88.9 DOI:.8/jmss Proof: As proof of Proposto we ow tat: Hece: Now, we compute te terms as follows: = < v, = < v, Sequeces wt τ = - (6) I ts secto we set te value of τ as mus two. We troduce te sequece: j ( D j ) j = = D j Sce D < D we obta: > D j = v (9) Equato (9) represets te ducto step. Proposto : et,, τ = - ad θ =. If we coose: f f { } j a a f =, =,,,,..., f =, =,, Proposto : et,, τ - ad θ =. If we coose: =, j (, ), v < f f f Te we obta a perodc sequece. Proof: We start le ts: f f =, =, =,..., =, f f =, f = > v = j = a Te we get a ubouded sequece. Proof: We ote frst tat: f =, = < v, f < v, = = v = < v f = = f f We cotue ts maer tll we reac : (7) But a s a elemet of te set {,,,, }. Hece we obta a perodc sequece. Proposto 5: et,, τ = - ad θ =. If we coose: =, j >, v < f f f Te we get a ubouded sequece. Proof: As before we reac: f f f j = < < < v = > v, = j j > > v (8) f f f f Now we use te otato: We deote by: f = >, D = j > f Ts allows us to wrte te followg epresso: Now we prove by ducto tat: D j > v, =, We tae as bass step: < < < < < v 6 ( ) ( ) ( j) ( j ) f f 5 f = D j = j Accordg to our assumptos: We suppose ow tat for some we ave: <, >, f 6 j j Hece: 9

4 Ramada Sabra ad Saleem Safq Al-Asab / Joural of Matematcs ad Statstcs 5, (): 88.9 DOI:.8/jmss Sce: 5 ( j j) > j j j 6 j j j > j j > > 5.5 j j j j j Sce j > v we coclude tat 6 = 5 - j 5 j > 5. We use te otato D = 6 5 >. We prove by ducto tat: 6 () for = 6,7,. As bass step we tae = 6. Ts yelds 6 Dj. But: = D > j D > v 6 5 We tur atteto ow to ducto step: Suppose tat for some 6 te followg relato olds: 6 Hece, accordg to defto: = j j 6 6 ( D j) j = D j > v From () we deduce tat te lmt of s fte. I some cases tere wll be perodc beavor of te sequece. For eample, te settg: = j = = v Geerates te sequece: 9 [8, 7),, 8,,8 We ca prove te followg. Proposto 6: et ad θ. If we coose: θ v <, j = ±, [, v] Te we get te sequece:. Proof: We ote tat:,,,, θ θ j = ± ± C Sce: We obta: v v v = <, = >, j j j θ θ θ = = ( ) = = ±, Cocluso We developed teory for te ew troduced cocept of olear pecewse-defed sequeces, wc are dfferet ta te refereces. We determed some codtos uder wc te sequeces ave some perodc beavor. But, te sequece becomes sometmes ubouded. Tere are stll ope problems to solve ts drecto. Actually, our observatos usg te computer lead us to te guess tat te bouded sequeces are suc tese sequeces, wc we cosdered ere. I ts paper we pave a way ow to start a bfurcato aalyss wt respect to some parameters le j or v sce te sequeces cages beavor for dfferet settgs. For eample, we we cosder: =, j (, ), v < f f f We obta a ubouded sequece. Wle te coce: f f { } j a a f =, =,,,,..., eads to a perodc sequece. But, we caot tll ow state wat appes for te oter cases of j. Acowledgemet Te autors would le to ta Academc Deasp Jaza Uversty for support. Autor s Cotrbutos All autors equally cotrbuted ts wor Etcs Ts artcle s orgal ad cotas upublsed materal. Te correspodg autor cofrms tat all of te oter autors ave read ad approved te mauscrpt ad o etcal ssues volved. Refereces Al-Asab, S. ad J. Guyer,. Pecewse defed recursve sequeces wt applcato matr teory. J. Mat. Computatoal Sc. 9

5 Ramada Sabra ad Saleem Safq Al-Asab / Joural of Matematcs ad Statstcs 5, (): 88.9 DOI:.8/jmss Al-Asab, S. ad R. Sabra, 5. Nolear pecewsedefed ad cubc dfferece equatos. Proceedgs of te t Aual Iteratoal Coferece o Computatoal Matematcs, Computatoal Geometry ad Statstcs (AIC 5), Sgapore. Hou, C.,. Ha ad S.S. Ceg,. Complete asymptotc ad bfurcato aalyss for a dfferece equato wt pecewse costat cotrol. Adv. Dfferece Equatos. Papascopoulos, G., C.J. Scas ad G. Stefadou, 7. O a -order system of yess-type dfferece equatos. Adv. Dfferece Equatos. DOI:.55/7/7 Qea, M., S. Al-Asab ad J. Guyer,. Nolear pecewse defed dfferece equatos. It. Mat. Forum, 7: Sabra, R. ad S. Al-Asab, 5a. Bfurcatos of quadratc pece-wse defed recursve sequeces. It. Mat. Forum, : 7-7. Sabra, R. ad S. Al-Asab, 5b. Nolear pecewse-defed dfferece equatos wt recprocal ad cubc terms. GSTF J. Mat. Statst. Operatos Res., : -9. Stevc, S., 7. Asymptotcs of some classes of gerorder dfferece equatos. Dscrete Dyamcs Nature Socety. DOI:.55/7/568 Zag,., G. Zag ad H. u,. Perodcty ad attractvty for a ratoal recursve sequece. J. Appled Mat. Computatos, 9: 9-. DOI:.7/BF