Applied Mathematics 3 79-733 http://d.doi.org/.436/am..377 Published Online July (http://www.scirp.org/journal/am) Periodicity and Solution of Rational Recurrence Relation of Order Si Tarek F. Ibrahim Department of Mathematics Faculty of Sciences and Arts (S. A.) King Khalid University Abha KSA Department of Mathematics Faculty of Science Mansoura University Mansoura Egypt Email: tfibrahem@mans.edu.eg Received April 7 ; revised May 7 ; accepted June 3 ABSTRACT Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and applied mathematics. Every dynamical system a n f an determines a difference equation and vise versa. We obtain in this paper the solution and periodicity of the following difference equation. n nnn4 n n3n 5 () n where the initial conditions 5 4 3 and are arbitrary real numbers with and 3 5 not equal to be zero. On the other hand we will study the local stability of the solutions of Equation (). Moreover we give graphically the behavior of some numerical eamples for this difference equation with some initial conditions. Keywords: Difference Equation; Solutions; Periodicity; Local Stability. Introduction Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and applied mathematics. Every dynamical system a n f an determines a difference equation and vise versa. Recently there has been great interest in studying difference equations. One of the reasons for this is a necessity for some techniques whose can be used in investigating equations arising in mathematical models decribing real life situations in population biology economic probability theory genetics psychology...etc. Difference equations usually describe the evolution of certain phenomenta over the course of time. Recently there are a lot of interest in studying the global attractivity boundedness character the periodic nature and giving the solution of nonlinear difference equations. Recently there has been a lot of interest in studying the boundedness character and the periodic nature of nonlinear difference equations. Difference equations have been studied in various branches of mathematics for a long time. First results in qualitative theory of such systems were obtained by Poincaré and Perron in the end of nineteenth and the beginning of twentieth centuries. For some results in this area see for eample [-3]. Although difference equations are sometimes very simple in their forms they are etremely difficult to understand throughly the behavior of their solutions. Many researchers have investigated the behavior of the solution of difference equations for eamples. Cinar [] investigated the solutions of the following difference equations n n n n n an bnn Karatas et al. [4] gave that the solution of the difference equation n n5 n n5 G. Ladas M. Kulenovic et al. [] have studied period two solutions of the difference equation AB C n n n n n Simsek et al. [3] obtained the solution of the difference equation n n3 n Ibrahim [5] studied the third order rational difference Equation n nn n nn In this paper we obtain the solution and study the periodicity of the following difference equation Copyright SciRes.
73 T. F. IBRAHIM n nnn4 n n3n 5 () n where the initial conditions 5 4 3 and are arbitrary real numbers with 3 and 5 not equal to be zero. On the other hand we will study the local stability of the solutions of Equation (). Moreover we give graphically the behavior of some numerical eamples for this difference equation with some initial conditions. Here we recall some notations and results which will be useful in our investigation. Let I be some interval of real numbers and Let k F : I I be a continuously differentiable function. Then for every set of initial conditions k k I the difference equation F n () n n n nk has a unique solution n []. n k Definition (.) A point I is called an equilibrium point of Equation () if F. That is n for n is a solution of Equation () or equivalently is a fied point of F. Definition (.) The difference Equation () is said to be persistence if there eist numbers m and M with < m M < such that for any initial k k there eists a positive integer N which depends on the initial conditions such that m n M for all n N. Definition (.3) (Stability) Let I be some interval of real numbers. ) The equilibrium point of Equation () is locally stable if for every ε > there eists δ > such that for I with k k k k we have for all n k. ) The equilibrium point of Equation () is locally asymptotically stable if is locally stable solution of Equation () and there eists γ > such that for all I with k k k k we have lim n n. 3) The equilibrium point of Equation () is global attractor if for all k k I we have lim n n. 4) The equilibrium point of Equation () is globally asymptotically stable if is locally stable and is also a global attractor of Equation (). 5) The equilibrium point of Equation () is unstable if not locally stable. The linearized equation of Equation () about the equilibrium is the linear difference equation k n i y F y n ni i Theorem (.4) [] Assume that bers) and k Then p q pq (real num- is a sufficient condition for the asymptotic stability of the difference equation n pn qnk n. Remark (.5) Theorem (.4) can be easily etended to a general linear equations of the form nk pnk pkn n where p p pk (real numbers) and k. Then Equation (4) is asymptotically stable provided that k pi. i Definition (.6) (Periodicity) A sequence n is said to be periodic with period p if n p n for all n nk k.. Solution and Periodicity In this section we give a specific form of the solutions of the difference Equation (). Theorem (.) Let be a solution of Equation (). Then n n k Equation () have all solutions and the solutions are 4n5 k 4n4 h 4n3 L 4n 4n 4n 4n h Lk 4n k 4n3 h 4n4 L 4n5 4n6 4n7 4n8 Lk h where 5 k 4 h 3 L. For n = the result holds. Now suppose that n > and that our assumption holds for n. We shall show that the result holds for n. By using our assumption for n we have the following: 4n 9 k 4n 8 h 4n 7 L 4n 6 4n 5 4n 4 4n 3 h Lk 4n k 4n h 4n L 4n9 4n 8 4n7 4n6 Lk h Now it follows from Equation () that 4n5 4n64n 84n 4n74n94n Lk h L l h k Copyright SciRes.
T. F. IBRAHIM 73 4n4 4n54n74n9 4n64n 84n h f u u u3 u4 u5 u6 uu 3u5 uu4u6 4n3 4n44n64n 8 4n54n74n9 L Therefore it follows that 4n 4n34n54n7 4n44n64n 8 f u u3u5 uu4u6 similarly we can derive f u uuu 3 5uu 4 6 uuu 4 6 4n 4n uuu 3 5 uuu 4 6. 4n h Lk 4n k f u3 uu 5 uu4u6 4n3 h 4n4 L 4 n5 4n6 4n7 4n8 Lk h. f u4 uu 3u5 uu4u6 Thus the proof is completed. f u5 uu 3 uu4u6 Theorem (.) Suppose that n f u6 uu 3u5 be a solution of Equation (). uu4u6 nk Then all solutions of Equation () are periodic with period At the equilibrium point we have fourteen. f u p f u p From Equation () we see that f u3 p3 f u4 p4 n nnn4 nn3n5 f u5 p5 f u6 p6 n n nn3 nnn4 n5 Then the linearized equation of Equation () about n3 nnn nnn3 is n4 yn pyn pyn p3yn p4yn3 n4 n3n n nnn n3 py 5 n4 py 6 n5 n5 n & n6 n & n7 n i.e. n8 n & n9 n n5 & n n4 yn yn yn yn yn3 yn4 yn5 n n3& n n & n 3 n & n 4 n Whose characteristic equation is which completes the proof. 6 5 4 3 3. Stability of Solutions In this section we study the local stability of the solutions By the generalization of theorem (.4) we have of Equation (). which is impossible. This means that the equilibrium Lemma (3.) point is unstable. Similarly we can see that the Equation () have two equilibrium points which are equilibrium point is unstable. and. For the equilibrium points of Equation () we can write 4. Numerical Eamples For confirming the results of this section we consider numerical eamples which represent different types of solutions to Equation (). 4 3 4 3 Then i.e. Thus the equilibrium points of Equation () is are and. Theorem (3.) The equilibrium points and are unstable. We will prove the theorem at the equilibrium point and the proof at the equilibrium point by the Eample 4. Consider 5 = 4 = 3 = 4 = = and = 7. See Figure. Eample 4. Consider 5 = 4 = 3 = = 6 = 5 and = 4. See Figure. Eample 4.3 Consider 5 = 3 4 = 5 3 = 7 = 3 = same way. and =. See Figure 3. 6 Let f : be a continuous function defined Eample 4.4 by Consider 5 = 4 4 = 3 3 = = 9 = 7 Copyright SciRes.
73 T. F. IBRAHIM Figure. The periodicity of solutions with period 4 with unstable equilibrium points = and =. Figure 4. The periodicity of solutions with period 4 with unstable equilibrium points = =. and = 6. See Figure 4. 5. Acknowledgements We want to thank the referee for his useful suggestions. Figure. Periodicity of solutions with period 4 with unstable equilibrium points = =. Figure 3. Periodicity of solutions with period 4 with unstable equilibrium points = and =. REFERENCES [] C. Cinar On the Positive Solutions of the Difference Equation n n ann Applied Mathematics and Computation Vol. 58 No. 3 4 pp. 793-797. doi:.6/j.amc.3.8.39 [] C. Cinar On the Positive Solutions of the Difference Equation n an bnn Applied Mathematics and Computation Vol. 56 No. 4 pp. 587-59. doi:.6/j.amc.3.8. [3] S. N. Elaydi An Introduction to Difference Equations Springer-Verlag Inc. New York 996. [4] R. Karatas C. Cinar and D. Simsek On Positive Solutions of the Difference Equation n n5 n n5 International Journal of Contemporary Mathematical Sciences Vol. No. 6 pp. 495-5. [5] T. F. Ibrahim On the Third Order Rational Difference Equation n nn n nn International Journal of Contemporary Mathematical Sciences Vol. 4 No. 7 9 pp. 3-334. [6] T. F. Ibrahim Global Asymptotic Stability of a Nonlinear Difference Equation with Constant Coefficients Mathematical Modelling and Applied Computing Vol. No. 9. [7] T. F. Ibrahim Dynamics of a Rational Recursive Sequence of Order Two International Journal of Mathematics and Computation Vol. 5 No. D9 9 pp. 98-5. [8] T. F. Ibrahim Solvability and Attractivity of the Solu- Copyright SciRes.
T. F. IBRAHIM 733 tions of a Rational Difference Equation Journal of Pure and Applied Mathematics: Advances and Applications Vol. No. 9 pp. 7-37. [9] T. F. Ibrahim Periodicity and Analytic Solution of a Recursive Sequence with Numerical Eamples Journal of Interdisciplinary Mathematics Vol. No. 5 9 pp. 7-78. [] V. L. Kocic and G. Ladas Global Behavior of Nonlinear Difference Equations of Higher Order with Applications Kluwer Academic Publishers Dordrecht 993. [] M. R. S. Kulenovic and G. Ladas Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures Chapman & Hall/CRC Press Boca Raton. doi:./978435384 [] G. Ladas and M. Kulenovic On Period Two Solutions of n n n ABn Cn Journal of Difference Equations and Applications Vol. 6 pp. 64-646. [3] D. Simsek C. Cinar and I. Yalcinkaya On the Recursive Sequence n n 3 n International Journal of Contemporary Mathematical Sciences Vol. No. 6 pp. 475-48. Copyright SciRes.