ch (for some fixed positive number c) reaching c

Transcription

1 GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 DOI 0.60/s z Nonlinear Piecewise-defined Difference Equations wit Reciprocal and Cubic Terms Ramadan Sabra and Saleem S. Al-asab Received 05 Mar 05 Accepted Apr 05 i. If 0 < λ < p ten all solutions of te system converge to te point (p; p). ii. If λ = p ten all solutions of te system suc tat Abstract In tis paper we consider te several types of piecewise defined sequences. We prove some results concerning te beavior of te sequence. In some cases we prove tat te terms repeat temselves in certain patterns and we prove in oter cases tat te terms grow monotonically after few terms and approaces infinity. Keywords-component; difference equations; discrete dynamical systems; bifurcation; periodic sequences I. u 0 y 0 ( p ) converge to (p; p) and for te remaining start values converges to te limit -cycle (p; q) (q; p). iii. If p < λ < q ten all solutions of te system converge to te limit -cycle (p; q) (q; p). iv. If λ = q ten all solutions of te system suc tat INTRODUCTION In te teory of difference equations te existence of periodic solutions plays an important role. Applications of tese classical existence teorems are well known now. Recently tese teorems ave been generalized. For detail see te researc papers [ 4]. In tis paper we present some definitions from discrete dynamical systems and introduce new some definitions and teories. One aspect of tis study is to study te existence of fixed points and points aving finite period of certain systems. Furter we find solutions for some difficult equations. By using matematical programs we study te beavior of teir solutions in case te solution is unknown. u0 y0 a specific subset of R converge to (q; q) and for te remaining start values converges to te limit -cycle (p; q) (q; p). v. If λ > q ten all solutions of te system converge to (q; q). Al-asab and Guyker (see [5]) started in 0 researc on a new type of nonlinear difference equations. Tey considered a and side of te difference equations wic consists of a piecewise defined function in two ways. Tis function is linear in te two ways. Tis made tings easier to treat. Some results were proven wile oter results are just conjectured. Tis is a new approac of dealing wit difference equations wic posses periodic solutions. Qena Al-Asab and Guyker (see [6]) took furter steps in te direction of considering nonlinear equations. Tey ave a rigt and side of te difference equations wic consists of a piece-wise defined function in two ways. But tis function is nonlinear in one of te two ways. Te teory developed in [6] is more difficult to understand tan in [5] and uses more complicated matematical tecniques. In [6] [] and [8] tere were cases in wic unbounded solutions exists. Te researcers did not explore all possible cases wen te piece-wise defined function consists of linear brances. Tey left some room for researc in tis direction. Also in te case tat te function is nonlinear in one of te two brances we see only quadratic terms. Of course quadratic equations are in a natural sense te next step wen we want to ave nonlinear terms and provide more room for proving teorems rater tat cubic or iger order equations. Hou Han and Ceng (see []) considered te following nonlinear difference equation xn axn bf ( xn ) c were te function follows f is a piece-wise defined function as f ( x ) for x (0 ] f ( x ) 0 for x ( ) Tey studied cases wen te sequence is periodic or unbounded. Tis difference equation is equivalent to te system un y n y n aun bf ( y n ) c Using te notation p tey found: c bc q a a DOI: 0.56/5-88_..56 Te Autor(s) 05. Tis article is publised wit open access by te GSTF

2 GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 II. SEQUENCES WITH POSITIVE POWERS ten we obtain te sequence suc tat In tis section we introduce te sequence defined as follows for L k Lk v Lk L k j Lk v Te numbers Lu c Lur c F (i ) G (i ) Lurs c F ( p ) G ( p ) k... Lu r s y c F ( p ) G ( p ). for some positive integers r s and y. Proof: Our assumptions ensure tat te function J is increasing in u and ence L0 j v are determined according to certain specifications. L0 J (t b a) v A. Periodic sequences In [] Hamadne talked about te possibility of starting wit c (for some fixed positive number c) reacing c F (i ) Tus according to definition we obtain L L0 j J (t u b a) G (i ) Due to te properties of J and our coice for a and b we get by making similar calculations G(P) and so fort. We are going and ten getting back to c to generalize tese ideas. We will prove a result wic as anoter starting point but leads to tis periodic sequence after a couple of terms. We introduce te following new functions F ( p) Li J (t u i b a) v for i... u Lu J (t b a) j a c. i F (i ) i G (i ) k We deduce te rest of te result beyond tis point using Hamadne's work (see [] for details) k 0 J (t u b a) We use te notation We note tat (b a) u a t B. Example Let ten a( ) B t F () G() 40 j ( 8 40 ) V 6. 0 J (68 40 ) 8 ( 40 6 (6 (8 40 ))) (b a) u t J (t u b a) B J (t u b a) J (t u b a) J (t b a) B b. We formulate some results in te typical matematical way as propositions but will omit some proofs. We illustrate tem instead of proofs by examples. Proposition Let c and. Assume tat i p t and u are positive integers wit p i. c c F (i) G (i)t c F ( p ) G ( p ) 0. We set V c Assume tat (c F ( i ) G ( i )t c F ( p ) G ( P ) )( ) t F ( i ) G ( i ) t ( 8 40 (6 ( 8 40 ))) v t j. V c F (i ) G (i ) c F ( i ) G ( i ) J (t b a). If we take J (8 40 ) b c a c F ( i ) G ( i )t c F ( p ) G ( p ) j c i 4 p u 6 t. We ave L0 J (t u b a ) We obtain te sequence c F ( i )G ( i ) v min V J (t b a) } Te sponsor is te Deansip of Scientific Researc of Jazan University (Jizan Kingdom of Saudi Arabia) Te Autor(s) 05. Tis article is publised wit open access by te GSTF

3 GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 L0 we ave L L0 j Tis inequality is equivalent to ( ) c c c c c L L L j L4 L j L5 L4 j c c c ( c c ) c But since j c we get j j c c L6 * L5 L * 8 4 L8 * 4 j j According to our assumptions L0 c v c L UNBOUNDED SEQUENCES Te function We consider now te oter beavior of te sequences wic we study. If tey do not become periodic ten tey start growing and tend to infinity. We distinguis two cases regarding te nature of τ. L (c j ). f ( x) (x j ) j is positive outside te following interval j j j j A. Even Powers By even power we mean tat We proved already tat {46...} c Trougout tis section we assume tat we ave an even power. Let be a constant. We use te notation c0 ( ) tus f (c) 0 j i or (c j ) j We distinguis between two cases: and. In te latter case we do not put many restrictions on te value of L0. i.e. L j c v. By definition we obtain Proposition : Let and. Let c be a real number suc tat We prove now by induction tat L L j L j L j j Ln Ln for n 4... c c0. If we coose We ave seen te validity of tis inequality in case n=. Tis is te basis step. Now in te induction step we suppose tat L0 c j c 0 v c j Lk Lk ten te sequence increases and tends to infinity. Proof: We note first tat c0. Hence we get L L0 j c j c c 0 L L j III. In oter words L L0 40 (( ) c ) c c c0 Hence according to definition is te nontrivial solution of te Lk Lk j Lk j Lk j following equation We turn our attention to ( ) x x lim Lk as k Due to te increasing properties of te function ( ) x x Let M be arbitrary positive number wic is witout loss of generality greater tan j. We use te notation Te Autor(s) 05. Tis article is publised wit open access by te GSTF

4 GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 d L j 0 d L j 0 We note tat We note tat L j d L L j L j j d L j d L L j L j j d L4 L j L j j d. L4 L j L j j d. Continuing in tis manner we reac for k = Continuing in tis manner we reac for k = Lk j (k )d Lk j (k )d Hence for all k suc tat Hence for all k suc tat Mj k max{4 } d we obtain k max{4 we obtain Lk M Proposition : Let and. Assume tat te following relation olds Lk M. Remark: If we coose suc tat ( c0. Let c be a real number suc tat ten c max{ c0 }. If we coose ) ). c0 ( L0 c j c 0 v c B. Even Powers By odd power we mean tat ten te sequence increases and tends to infinity. Proof: According to our assumptions Mj } d L0 c v {5...}. Hence we get Trougout tis section we assume tat we ave an odd power. L L 0 j c j c c 0 Furter we obtain since c Proposition 4: Let i and L (c j ) (c j ). If we coose L0 c j c F (i) G (i) Using te ideas of te previous proof we conclude tat ten te sequence decreases after te term Li for all L j c v c F (i ) G (i ) v c F (i) G (i) But we ave according to assumptions L j c c0 and tends to minus infinity. Proof: We know tat by assumption By definition we obtain L0 c v L (c ) c F () G () v L L j L j L j We prove now by induction tat... Li c F ( i ) G ( i ) v j Ln Ln for n Hence according to te coice of j We ave seen te validity of tis inequality in case n=. Tis is te basis step. Now in te induction step we suppose tat L i L i j v j Lk Lk According to definition we obtain Li ( Li ) Li Li v. Hence according to definition due to j We use te notation Lk L k j Lk j Lk j d j c F (i) G (i) We turn our attention to Hence we get lim Lk as k Let M be arbitrary positive number wic is witout loss of generality greater tan j. We use te notation Li (d ) According to definition we obtain Li ( Li ) ( Li ) (d ) v Te Autor(s) 05. Tis article is publised wit open access by te GSTF 4

5 GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 4 A 54 AD D A D T ( A) E. 8 E By induction we prove now Li n ( d ) F ( n ) v Li n Li n n... For example if A 8 D 0 we get te solution 6.5. Te basis step as been proved. Suppose tat A. Sequences wit Li m ( d ) F ( m ) We set D 0 in (). We assign te value of te solution in () to j. We illustrate our idea as follows: If we take Li m Li m for some m Hence we ave Lim (d ) F ( m) v A 8 j 6.5 L0 and according to definition we obtain.4 v 4 Li m ( Li m ) ( Li m )F ( m ) ( Li m ) ten we get te sequence F ( m ) L 4 L 8 j A j.4 v Lim ( Lim ) Lim Lim L ( A j )) f ( j ) j j (d ) F ( n ) as n Since te value of j was te root. We see tat we got a constant sequence i. e. L4 L j j j j... Since we are done. Remark : If we coose Li j i 4 0 j c F (i ) G (i ) c F (i ) G (i ) v c We want to generalize tis result for different starting values. In te rest of tis section we set = and c. We take F ( i ) G ( i ) L0 c. ten we imitate te proof of proposition..4 by Hamadne (see []) and find tat we get a monotone increasing sequence. We set C. Example Let c i. Since A 4c. According to () te root is A T ( A) 6 T ( A) were F () G() 40 F () G() T ( A) we coose j v We obtain te sequence We denote tis root by b(a). Wen we study te function L0 L 6 L 8 L Wen we study te function A R We seek a solution for te equation g ( x) f ( x) D D R 6 Using Matematica we determine te following real solution of te equation A T ( A) T ( A) x x x x x f ( x) 0 for x We consider te cubic function we find tat SEQUENCES WITH CUBIC TERMS f ( x ) ( A x ) x f ( x) L4 46 L IV. A A x x x x x we find tat g ( x) x for x 0. () Hence we ave te following inequality were Te Autor(s) 05. Tis article is publised wit open access by te GSTF 5

6 GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 A T ( A) 6 T ( A) Li G ( i ) c F ( i ) for all A. Tis insures tat Li B j B v olds for all A. since te following inequality T ( A) A 6 T ( A) We formulate a general result as follows: Lemma 5: Let ten. If we coose L0 c j b(4c ) olds for all A. Hence we obtain Li (B b (B)) b (B) j B v 4c v c In general we obtain Li j for i Lk j j j for k i Proof: We know tat by assumption In oter cases we can get a real periodic beavior. We formulate tis in te following proposition: L c v Proposition : Let and L 4c j T (4c ) L0 c 4c 4 c v 6 T ( 4c ) ten since te following inequality j c c c v c Li Li Hence we obtain L c j c v. L c v L ( 4c b ( 4c )) b ( 4c ) j 4c c v B. Sequences wit X F (i ) Y G(i ) Li j j j for i Specifically wen we consider te equation We formulate next te more general result. Te previous lemma is te result wen we set i=. Proposition 6: Let. Let i be an integer wit i >. We set ten (c X Y j ) j c F ( P ) G ( P ) According to () one of te solutions is B G (i) c F (i ) L0 c j b( B) c X Y R were B v B G ( i ) since i >. We obtain by assumptions L c B G ( 0) G ( i ) c F () R 4 c X Y 54c X F ( P ) Y G ( p ) c F ( p ) G ( p ) ) R (c X Y c F ( P ) G ( p ) E ). 8 If c and satisfy Proof: We note first tat c B E Lk j for all k i. We use te notation In general we obtain If we coose for i... L c v A.4. Hence we obtain c. If we coose Proof : We know tat by assumption T ( A) A 6 T ( A) olds for all c F ( i ) v Li B v A 0 j b ( A) B G (i ) c F (i ) c ten we obtain E 4c X Y c X Y 54 c X Y c F ( i ) v... c X Y c X Y We conclude terefore Te Autor(s) 05. Tis article is publised wit open access by te GSTF 6

7 GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 E 6 c X Y c X Y R (c X Y c X Y c X Y ) 8 X Y X Y X Y c 4c c 8 X If (c X Y c F ( p ) G ( p ) E ) 8 c Y [c X Y c F ( P ) G ( P ) c X Y ] 8 c X Y c F ( P ) G ( P ) E 8 (c X Y 5c X Y ) 8 c X Y c F ( p ) G ( p ) c X Y 8 c F ( i ) G ( i ). is a positive integer c X Y R c X Y c F (i ) G ( i ) 0. R We use te notation V c F ( i ) G ( i ) G ( i ). We note now tat for positive integers : V c F ( i ) G ( i ) G ( i ) c F ( i ) G ( i ) c F ( i ) G ( i ).8c F ( i ) G ( i ) c F ( i ) G ( i ). Proposition 8: Let c. Let i p be nonnegative integers suc tat p i. Let be a positive integer. If we coose L0 c j c X Y R c F ( i ) G ( i ) v V R ten te sequence is periodic and as te form L0 L... L p c F ( p) G( p) Let c p i and. We obtain We ave now E (4 Y 54 Y 5 ) R (Y E ) j Y R R and V Li c F (i ) G (i ) v Li c F ( i ) G ( i ) F ( i ) G ( i ) G ( i ) c V v. Furter according to our assumptions we get We obtain for any v between 4 and 5 te following sequence L0 L 4 L 8 L 6 Li c X Y j. c X Y c F ( P ) G ( P ) E 8 X Y F ( P ) G ( P ) c c E 8 X Y c 8 X Y c X F (i ) F () Y G().... C. Example L0 c v L c F () G () v L c F ( ) G ( ) v c X Y j Li (c X Y j ) j c F ( p ) G ( p ) Proof: We note first tat On te oter and X Y c G ( i ) V v Furter according to te definition of j we get Li c F ( i i ) G ( i ) Li Li c F ( p ) G ( p )... L L4 6. Te Autor(s) 05. Tis article is publised wit open access by te GSTF

8 GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 V. SEQUENCES WITH RECIPROCAL TERMS ( n )( ) j n In tis section we consider te following piece-wise defined sequence Qn v Qn Qn for k... Q j Q v n n v n n n In tis section we take te initial value always as ten we obtain te sequence Te numbers j and v are real numbers determined according to certain specifications. We can see tat in some cases te sequences is not well-defined. For example if we set Proof: We note first tat Q... Qn... Q0. v j 4 n n n n since ten we obtain n n n n Q Q * 4 0 Q undefined n n In some cases we obtain simple periodic beavior. For example if we set wic is true according to our coice of and n. Since te function ( ) v j R ten we obtain f ( x ) x Q0 Q Q... x for x is decreasing we get n n v n m m n for all m... n We want now to produce more complicated periodic beavior: we require v. Hence te first tree terms will be Q0 Q v Q j Now we compute te trems as follows If we set Q 0 v Q v j Q j ten we obtain te periodic sequence... ( n )( ) n n ( n )( ) n n v n If we set j ten we obtain v j v Similarly we can sow tat n m m v n for all m... n Qm In tis case te sequence will be Q0 Q v Q j Tus we obtain Q ( j ) j ( ) j n n n Qn n We generalize tis idea in te following result. Proposition : Let and n. If we coose Te Autor(s) 05. Tis article is publised wit open access by te GSTF 8

9 GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 [] VI. Conclusion We developed teory for te new introduced concept of nonlinear piecewise -defined sequences. We determined some conditions under wic te sequences ave some periodic beavior. But te sequence becomes sometimes unbounded. Tere are still open problems to solve in tis direction. Actually our observations using te computer lead us to te guess tat te bounded sequences are suc tese sequences wic we considered ere. In tis paper we pave a way ow to start a bifurcation analysis wit respect to some parameters like j or v since te sequences canges beavior for different settings. For example wen we consider L0 and [8] A. Hamadne On a study of some difference equations Ms. C. tesis AL al-bayt University 0 Te tesis appearedas "A Study of some Difference Equations including Bifurcation Analysis"in te book ISBN by lap-publising.com. M. Qena A study on te fixed and periodic points of certain discrete dynamical systems equations Ms. C. tesis AL al-bayt University 0 Te tesis appeared as "New concepts in sequences" wit te project number 500 and ISBN by lappublising.com. AUTHOR S PROFILE take a positive integer m. We assume furter tat G ( m ) v G ( m) By setting c= = and i=m in proposition 4 we conclude tat te sequence is unbounded for Dr. Ramadan Sabra obtained te P. D. in Matematics (functional analysis) from University of Minsk / Belarussia in. j G ( m). By setting c== and i=m+ in proposition 6 we te sequence is bounded (becomes constant after some term) for j b( G ( m) ). Moreover if we coose 0 j c F (i ) G (i ) ten we imitate te proof of proposition..4 by Hamadne (see []) and find tat we obtain a monotone increasing sequence wit infinite limit. Dr. Saleem Al-asab studied Diplom Matematics (86-) and finised is P. D. in Matematics at te Humboldt University in Berlin (-). Te specialization is matematical analysis. Since 4 e was working as lecturer at many jordanian universities. Since 00 e is working al Al-albayt university. In 0 e beacme associate professor. He publised several works about te topic of magic squares. He supervised many Ms. C. students. ACKNOWLEDGMENT Saleem al-asab tanks te Al-albayt University for support since tis paper was prepared during te year of sabbatical leave offered by te university. Tis article is distributed under te terms of te Creative Commons Attribution License wic permits any use distribution and reproduction in any medium provided te original autor(s) and te source are credited. REFERENCES [] [] [] [4] [5] [6] C. Hou L. Han and S. S. Ceng Complete asymptotic and bifurcation analysis for a difference equation wit piecewise constant control Hindawi Publising Corporation Advances In Difference Equations Volume 0. G. Papascinopoulos C. J. Scinas and G. Stefanidou On a k-order system of Lyness-type difference equations Hindawi Publising Corporation Advances in Difference Equations Volume 00 Article ID. S. Stevic Asymptotics of some classes of iger-order difference equations Hindawi Publising Corporation Discrete Dynamics in Nature and Society Vo lume 00 Article ID 568. L. Zang G. Zang and H. Liu Periodicity and attractivity for a rational recursive sequence J. Appl. Mat. Comput. pp S. Al-asab J. Guyker Piecewise defined recursive sequences wit application in matrix teory Journal of Matematical and Computational Science Vol. No M. Qena S. Al-Asab and J. Guyker Nonlinear piecewise defined difference equations International Matematical Forum Vol. No. 0 pp Te Autor(s) 05. Tis article is publised wit open access by te GSTF