Solution and Behavior of a Rational Recursive Sequence of order Four

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1 Australia Joural of Basic ad Applied Scieces, 7(4): , 2013 ISSN Solutio ad Behavior of a Ratioal Recursive Sequece of order Four Tarek F. Ibrahim Departmet of Mathematics, Faculty of Sciece, Masoura Uiversity, Masoura 35516, Egypt. Abstract: We obtai i this paper the solutios of the followig recursive sequeces +1= -2-3 / (±1± -2-3), =0,1,... where the iitial coditios are arbitrary real umbers. Also, we study the behavior of the solutios. Key words: differece equatios, recursive sequeces, stability, periodic solutio. INTRODUCTION Differece equatios or discrete dyamical systems is diverse field which impact almost every brach of pure ad applied mathematics. Every dyamical system a = + 1 f ( a) determies a differece equatio ad vise versa. Recetly,there has bee great iterest i studyig differece equatios systems. Oe of the reasos for this is a ecessity for some techiques whose ca be used i ivestigatig equatios arisig i mathematical models describig real life situatios i populatio biology, ecoomic, probability theory, geetics, psychology,...etc. The theory of differece equatios occupies a cetral positio i applicable aalysis. There is o doubt that the theory of differece equatios will cotiue to play a importat role i mathematics as a whole. Noliear differece equatios of order greater tha oe are of paramout importace i applicatios. Such equatios also appear aturally as discrete aalogues ad as umerical solutios of differetial ad delay differetial equatios which model various diverse pheomea i biology, ecology, physiology, physics, egieerig ad ecoomics. It is very iterestig to ivestigate the behavior of solutios of a system of oliear differece equatios ad to discuss the local asymptotic stability of their equilibrium poits. I this paper we obtai the solutios of the followig recursive sequeces +1= -2-3 / (±1± -2-3), =0,1,... (1) where the iitial coditios are arbitrary real umbers. Also, we study the behavior of the solutios. Here, we recall some otatios ad results which will be useful i our ivestigatio. Let I be some iterval of real umbers ad let k+1 f:i I, be a cotiuously differetiable fuctio. The for every set of iitial coditios -k,-k+1,...,0 I, the differece equatio =f(,,..., ), =0,1,..., (2) k has a uique solutio { } (Kocic, V.L. ad G. Ladas, 1993). =-k Defiitio 1. (Equilibrium Poit) A poit Iis called a equilibrium poit of Eq.(2) if =f(,,...,). That is, = for 0,is a solutio of Eq.(2), or equivaletly, is a fied poit of f. Correspodig Author: Tarek F. Ibrahim, Departmet of Mathematics, Faculty of Sciece, Masoura Uiversity, Masoura 35516, Egypt tfibrahem@mas.edu.eg 816

2 Defiitio 2. (Stability) (i) The equilibrium poit δ > 0 such that for all k, k+ 1,..., 0 I with -k - + -k < δ, we have of Eq.(2) is locally stable if for everyε > 0, there eists < ε for all k. (ii) The equilibrium poit solutio of Eq.(2) ad there eists 0 with -k - + -k < γ, we have lim =. of Eq.(2) is locally asymptotically stable if is locally stable γ >, such that for all k, k+ 1,..., ₁, 0 I,,...,, I (iii) The equilibrium poit of Eq.(2) is global attractor if for all k k 1 ₁ 0 we have lim =. (iv) The equilibrium poit also a global attractor of Eq.(2). (v) + of Eq.(2) is globally asymptotically stable if is locally stable, ad is The equilibrium poit of Eq.(2) is ustable if is ot locally stable. The liearized equatio of Eq.(2) about the equilibrium is the liear differece equatio y = +1 k (,,..., ) f i = 0 i y i Theorem A [33]: Assume that p R, i 1,2,...,k i = ad k { 0,1, 2,... }. The k i = 1 p i < 1 is a sufficiet coditio for the asymptotic stability of the differece equatio +k +p1 +k pk =0, =0,1,... Defiitio 3. (Periodicity) A sequece { } is said to be periodic with period p if =-k p + = for all k. The ature of may biological systems aturally leads to their study by meas of a discrete variable. 817

3 Particular eamples iclude populatio dyamics ad geetics. Some elemetary models of biological pheomea, icludig a sigle species populatio model, harvestig of fish, the productio of red blood cells, vetilatio volume ad blood CO₂ levels, a simple epidemics model ad a model of waves of disease that ca be aalyzed by differece equatios are show i (Mickes, R.E., 1987). Recetly, there has bee iterest i so-called dyamical diseases, which correspod to physiological disorders for which a geerally stable cotrol system becomes ustable. Oe of the first papers o this subject was that of Mackey ad Glass (1977). I that paper they ivestigated a simple first order differece-delay equatio that models the cocetratio of blood-level CO₂. They also discussed models of a secod class of diseases associated with the productio of red cells, white cells, ad platelets i the boe marrow. The dyamical characteristics of populatio system have bee modelled, amog others by differetial equatios i the case of species with overlappig geeratios ad by differece equatios i the case of species with o-overlappig geeratios. I practice, oe ca formulate a discrete model directly from eperimets ad observatios. Sometimes, for umerical purposes oe wats to propose a fiite-differece scheme to umerically solved a give differetial equatio model, especially whe the differetial equatio caot be solved eplicitly. For a give differetial equatio, a differece equatio approimatio would be most acceptable if the solutio of the differece equatio is the same as the differetial equatio at the discrete poits (Kuleovic, M.R.S. ad G. Ladas, 2001). But uless we ca eplicitly solve both equatios, it is impossible to satisfy this requiremets. Most of the time, it is desirable that a differetial equatio, whe derived from a differece equatio, preserves the dyamical features of the correspodig cotiuous-time model such as equilibria, their local ad global stability characteristics ad bifurcatio behaviors. If such discrete models ca be derived from cotiuous-time models ad it will preserve the cosidered realities; such discrete-time models ca be called `dyamically cosistet' with the cotiuous-time models. The study of asymptotic stability ad oscillatory properties of solutios of differece equatios is etremely useful i the behavior of mathematical models of various biological systems ad other applicatios. This is due to the fact that differece equatios are appropriate models for describig situatios where the variable is assumed to take oly a discrete set of values ad they arise frequetly i the study of biological models, i the formulatio ad aalysis of discrete time systems, the umerical itegratio of differetial equatios by fiite-differece schemes, the study of determiistic chaos, etc. For eample, (Ladas, G., 1989) the study of oscillatio of positive solutios about the positive steady state N i the delay logistic differece equatio m N =N ep[r(1- p N ], +1 j -j j = 0 where r, p m (0, ), p 0, p ₁,..., p m 1 [0, ) ad m + r 1, which describes situatios where populatio growth is ot cotiuous but seasoal with o-overlappig geeratios, leads to the study of oscillatios about zero of a liear differece equatio of the form m - + p =0, =0,1, i -ki i=0 Also, differece equatios are appropriate models for describig situatios where populatio growth is ot cotiuous but seasoal with overlappig geeratios. For eample, the differece equatio, y +1=yep[r(1-y /k)], has bee used to model various aimal populatios. This equatio is cosidered by some to be the discrete aalogue of the logistic differetial equatio 818

4 y (t)=ry(t)(1-y(t)/k), where r ad k are the growth rate ad the carryig capacity of populatio, respectively. Grove, et al. (2000) studied the stability ad the semicycles of solutios of the biological model: =a +b e El-Metwally et al. (2003) ivestigated the asymptotic behavior of the populatio model: = α+ β e whereα is the immigratio rate ad β is the populatio growth rate. Dig et al. (2008) studied the followig discrete delay Mosquito populatio equatio =( + )e +1 α β -1 - The geeralized Beverto-Holt stock recruitmet model has ivestigated i (DeVault, R., 1998; Beverto, R.J. ad S.J. Holt, 1957): +1=a +(b -1)/(1+c -1+d ) See also (Che, J. ad D. Blackmore, 2002; Elabbasy, E.M., 2007; Elettreby, M.F. ad H. El-Metwally, 2007; El-Metwally, H., 2007; El-Metwally, H. ad M.M. El-Afifi, 2008; El-Metwally, H., 2001; Grove, E.A., 2000; Huo, H.F. ad W.T. Li, 2005; Pielou, E.C., 1974; Kuag, Y. ad J.M. Cushig, 1996; Rafiq, A., 2006; Saleh, M. ad S. Abu-Baha, 2006; Sedaghat, H., 2003; Summers, D., 2000). The log term behavior of the solutios of oliear differece equatios of order greater tha oe has bee etesively studied durig the last decade. For eample, various results about boudedess, stability ad periodic character of the solutios of the secod-order oliear differece equatio see (Abu-Saris, R., 2008; Agarwal, R.P., 2000; Agarwal, R.P. ad E.M. Elsayed, 2008; Aloqeili, M., 2006; Aloqeili, M., 2006; Atalay, M., 2005; Beverto, R.J. ad S.J. Holt, 1957; Che, J. ad D. Blackmore, 2002; Ciar, C., 2004; DeVault, R., 1998; Dig, X. ad R. Zhag, 2008; Elabbasy, E.M., 2007; Elabbasy, E.M. ad E.M. Elsayed, 2008; Elabbasy, E.M., 2007; Elabbasy, E.M., 2006; Elabbasy, E.M., 2007; Elabbasy, E.M., 2007). May researchers have ivestigated the behavior of the solutio of differece equatios for eample: Agarwal et al. (Agarwal, R.P. ad E.M. Elsayed, 2008) ivestigated the global stability, periodicity character ad gave the solutio of some special cases of the differece equatio +1=a+(d -l -k )/(b-c -s ) Aloqeili (2006) has obtaied the solutios of the differece equatio +1= -1/(a- -1) Ciar (2004) ivestigated the solutios of the followig differece equatio +1=a -1/(1+b -1) Elabbasy et al. (Elabbasy, E.M., 2006; Elabbasy, E.M., 2007) ivestigated the global stability, boudedess, periodicity character ad gave the solutio of some special cases of the differece equatios =a -b /(c -d ), +1-1 k = α /( β+ γ ) +1 -k -i i = 1 819

5 Ibrahim (Ibrahim, T.F., 2009) get the solutios of the ratioal differece equatio +1= -2 / -1(a+b -2) Karatas et al. (2006) get the form of the solutio of the differece equatio _{+1}=((_{-5})/(1+_{-2}_{-5})) Other related results o ratioal differece equatios ca be foud i refs. (Elsayed, E.M., 2009; Grove, E.A., 2000; Grove, E.A. ad G. Ladas, 2005; Grove, E.A., 2000; Hamza, A.E., S.G. Barbary, 2008; Huo, H.F. ad W.T. Li, 2005; Ibrahim, T.F., 2009; Karatas, R., 2006; Kocic, V.L. ad G. Ladas, 1993; Yalçıkaya, I. ad C. Ciar, 2009; Zayed, E.M.E. ad M.A. El-Moeam, 2005). O the Recursive Sequece +1= -2-3 / ( ) I this sectio we give a specific form of the solutio of the equatio i the form +1= -2-3 / ( ), =0,1,... (3) where the iitial values are arbitrary positive real umbers. Theorem Let { } = d = c = = a be a solutio of Eq.(3). The for =0,1,... =-3 1 ( 1+ 2iab )( 1+ ( 2i + 1) bc )( 1+ 2icd ) i = 0 ( 1+ ( 2i + 1) ab )( 1+ ( 2i ) bc )( 1+ ( 2i + 1) 1 ( 1+ ( 2i + 1) ab )( 1+ ( 2i ) bc )( 1+ ( 2i + 1) i = 0 ( 1+ 2iab )( 1+ ( 2i + 1) bc )( 1+ ( 2i + 2) 1 ( 1+ ( 2i ) ab )( 1+ ( 2i + 1) bc )( 1+ ( 2i + 2) b i = 0 ( 1+ ( 2i + 1) ab )( 1+ ( 2i + 2) bc )( 1+ ( 2i + 1) 1 ( 1+ ( 2i + 1) ab )( 1+ ( 2i + 2) bc )( 1+ ( 2i + 1) i = 0 ( 1+ ( 2i + 2) ab )( 1+ ( 2i + 3) bc )( 1+ ( 2i + 2) 1 ( 1+ ( 2i + 2) ab )( 1+ ( 2i + 1) bc )( 1+ ( 2i + 2) i = 0 ( + ( i + ) ab )( + ( i + ) bc )( + ( i + ) 1 ( 1+ ( 2i + 1) ab )( 1+ ( 2i + 2) bc )( 1+ ( 2i + 3) i = 0 ( + ( i + ) ab )( + ( i + ) bc )( + ( i + ) cd = a(1+cd) ab(1+cd) = d(1+bc) = d, = c, = b, = a. where Proof: For = 0the result holds. Now suppose that > 0ad that our assumptio holds for 1. That is; = d = c 2 i = 0 2 i = 0 ( 1+ 2iab )( 1+ ( 2i + 1) bc )( 1+ 2icd ) ( 1+ ( 2 + 1) )( 1+ ( 2 ) ) 1+ ( 2 + 1) ( 1+ ( 2 + 1) )( 1+ ( 2 ) ) 1+ ( 2 + 1) ( 1+ 2 ) 1+ ( 2 + 1) ( ) ( ) ( )( ( ) ) i ab i bc i cd i ab i bc i cd iab i bc i cd 820

6 = b = a 2 ( 1+ ( 2i ) ab )( 1+ ( 2i + 1) bc )( 1+ ( 2i + 2) i = 0 ( 1+ ( 2i + 1) ab )( 1+ ( 2i + 2) bc )( 1+ ( 2i + 1) 2 ( 1+ ( 2i + 1) ab )( 1+ ( 2i + 2) bc )( 1+ ( 2i + 1) i = 0 ( 1+ ( 2i + 2) ab )( 1+ ( 2i + 3) bc )( 1+ ( 2i + 2) 2 ( 1+ ( 2i + 2) ab )( 1+ ( 2i + 1) bc )( 1+ ( 2i + 2) i = 0 ( + ( i + ) ab )( + ( i + ) bc )( + ( i + ) 2 ( 1+ ( 2i + 1) ab )( 1+ ( 2i + 2) bc )( 1+ ( 2i + 3) i = 0 ( + ( i + ) ab )( + ( i + ) bc )( + ( i + ) cd = a(1+cd) ab(1+cd) = d(1+bc) Now, it follows from Eq.(3) ad the above assumptios that = / (1+ ) = d ( 1+ 2iab )( 1+ ( 2i + 1) bc )( 1+ 2icd ) ( ) i = 0 ( 1+ 2i + 1 ab )( 1+ ( 2i ) bc )( 1+ ( 2i + 1) Similarly, we ca easily obtai the other relatios. Thus, the proof is completed. Theorem Eq.(3) has a uique equilibrium poit which is the umber zero ad this equilibrium poit is ot locally asymptotically stable. Proof: For the equilibrium poits of Eq.(3), we ca write =²/(1+²) The we have ²(1+²)=² ²(1+²-1)=0, or, ⁴ = 0. Thus the equilibrium poit of Eq.(3) is 0 =. Let f:(0, )³ (0, ) be a fuctio defied by f(u,v,w)=((vw)/(u(1+vw))) Therefore it follows that f_{u}(u,v,w)=-((vw)/(u²(1+vw))), f_{v}(u,v,w)=(w/(u(1+vw)²)), f_{w}(u,v,w)=(v/(u(1+vw)²)), we see that f_{u}(,,)=-1, f_{v}(,,)=1, f_{w}(,,)=1. The proof follows by usig Theorem A. Numerical Eamples: For cofirmig the results of this sectio, we cosider umerical eamples which represet differet types of solutios to Eq. (3). 821

7 Eample 1. We assume ₃=6, ₂=2, ₁=10, ₀=7 See Fig. 1. Fig. 1: Eample 2. See Fig. 2, sice ₃=0.8, ₂=0.6, ₁=0.5, ₀=1.3. Fig. 2: O the Recursive Sequece _{+1}=((_{-2}_{-3})/(_{}(-1+_{-2}_{-3}))) I this sectio we obtai the solutio of the differece equatio i the form _{+1}=((_{-2}_{-3})/(_{}(-1+_{-2}_{-3}))),=0,1,..., (4) where the iitial values are arbitrary o zero real umbers with ₃ ₂ 1, ₂ ₁ 1, ₁₀

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10 Fig. 3: Eample 4. See Fig. 4, sice we suppose that ₃=11, ₂=3, ₁=2/3, ₀=3. Fig. 4: The followig cases ca be proved similarly. O the Recursive Sequece _{+1}=((_{-2}_{-3})/(_{}(1-_{-2}_{-3}))) I this sectio we get the solutio of the third followig equatio _{+1}=((_{-2}_{-3})/(_{}(1-_{-2}_{-3}))),=0,1,..., (5) where the iitial values are arbitrary positive real umbers. 825

11 Fig. 5: Eample 6. Assume that ₃=0.7, ₂=1.2, ₁=0.4, ₀=0.5 see Fig. 6 Fig. 6: 826

12 O the Recursive Sequece _{+1}=((_{-2}_{-3})/(_{}(-1-_{-2}_{-3}))) Here we obtai a form of the solutios of the equatio _{+1}=((_{-2}_{-3})/(_{}(-1-_{-2}_{-3}))),=0,1,..., (6) where the iitial values are arbitrary o zero real umbers with ₃ ₂ -1, ₂ ₁ -1, ₁₀ -1. Fig. 7: Eample 8. Fig. 8. shows the solutios whe ₃=5, ₂=-2/5, ₁=5, ₀=-2/5. 827

13 Fig. 8: ACKNOWLEDGMENTS The first author was fiaced by the Deaship of Scietific Research (DSR), the Kig khalid Uiversity, Abha (uder Project No. KKU_S180_33). The author, therefore, ackowledge with thaks DSR techical ad fiacial support. REFERENCES Abu-Saris, R., C. Ciar ad I. Yalçıkaya, O the asymptotic stability of _{+1}=((a+_{}_{-k})/(_{}+_{-k})), Computers & Mathematics with Applicatios, 56: Agarwal, R.P. ad E.M. Elsayed, Periodicity ad stability of solutios of higher order ratioal differece equatio, Advaced Studies i Cotemporary Mathematics, 17(2): Agarwal, R.P., Differece Equatios ad Iequalities, 1^{st} editio, Marcel Dekker, New York, 2d editio. Aloqeili, M., Dyamics of a kth order ratioal differece equatio, Appl. Math. Comp., 181: Aloqeili, M., Dyamics of a ratioal differece equatio, Appl. Math. Comp., 176(2): Atalay, M., C. Ciar ad I. Yalcikaya, O the positive solutios of systems of differece equatios, Iteratioal Joural of Pure ad Applied Mathematics, 24(4): Beverto, R.J. ad S.J. Holt, O the Dyamics of Eploited Fish Populatios, vol.19, Fish Ivest., Lodo. Che, J. ad D. Blackmore, O the epoetially self-regulatig populatio model, Chaos, Solitos & Fractals, 14(9): Ciar, C., O the positive solutios of the differece equatio _{+1}=((a_{-1})/(1+b_{}_{-1})), Appl. Math. Comp., 156: DeVault, R., G. Dial, V.L. Kocic ad G. Ladas, Global behavior of solutios of _{+1}=a_{}+f(_{},_{-1}), J. Differ. Equatios Appl., 3(3-4): Dig, X. ad R. Zhag, O the differece equatio _{+1}=(α_{}+β_{-1})e^{-_{}}, Advaces i Differece Equatios, Volume, Article ID , 7 pages. Elabbasy, E.M. ad E.M. Elsayed, O the global attractivity of differece equatio of higher order, Carpathia Joural of Mathematics, 24(2):

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15 Ladas, G., Recet developmets i the oscillatio of delay differece equatios, It cof o differetial equatios: Theory ad applicatios i stability ad cotrol, Mackey, M.C. ad L. Glass, Oscillatio ad chaos i physiological cotrol system, Sciece, 197: Mickes, R.E., Differece equatios, New York, Va Nostrad Reihold Comp. Pielou, E.C., Populatio ad Commuity Ecology, Gordo ad Breach, New York. Rafiq, A., Covergece of a iterative scheme due to Agarwal et al., Rostock. Math. Kolloq., 61: Saleh, M. ad S. Abu-Baha, Dyamics of a higher order ratioal differece equatio, Appl. Math. Comp., 181: Sedaghat, H., Noliear Differece Equatios, Theory with Applicatios to Social Sciece Models, Kluwer Academic Publishers, Dordrect. Summers, D., J. Craford ad B. Healey, Chaos i periodically forced discrete-time ecosystem models. Chaos, Solitos & Fractals, 11(14): 23: Yalçıkaya, I. ad C. Ciar, O the dyamics of the differece equatio _{+1}=((a_{-k})/(b+c_{}^{p})), Fasciculi Mathematici, 42. Zayed, E.M.E. ad M.A. El-Moeam, O the ratioal recursive sequece _{+1}=((α+β_{}+γ_{-1})/(A+B_{}+C_{-1})), Commuicatios o Applied Noliear Aalysis, 12(4): Zayed, E.M.E. ad M.A. El-Moeam, O the ratioal recursive sequece _{+1}=A_{}+B_{-k}+((β_{}+γ_{-k})/(C_{}+D_{-k})), Acta Applicadae Mathematicae, I Press. 830