A System of Difference Equations with Solutions Associated to Fibonacci Numbers

Transcription

1 International Journal of Difference Equations ISSN Volume Number pp ) A System of Difference Equations with Solutions Associated to Fibonacci Numbers Yacine Halim University Center of Mila Department of Mathematics and Computer Sciences BP 6 Mila Algeria and LMAM Laboratory Jijel University Algeria halyacine@yahoofr Abstract This paper deals with form the periodicity and the stability of the solutions of the system of difference equations x n+ + y n y n+ + x n n N 0 where N 0 N {0} and the initial conditions x x x 0 y y and y 0 are real numbers AMS Subject Classifications: 39A0 40A0 Keywords: System of difference equations general solution stability Fibonacci numbers Introduction The theory of difference equations developed greatly during the last twenty-five years of the twentieth century The applications of the theory of difference equations is rapidly increasing to various fields such as numerical analysis biology economics control theory finite mathematics and computer science Thus there is every reason for studying the theory of difference equations as a well deserved discipline Recently there have been a growing interest in the study of solving systems of difference equations see for example [ ] and the references cited therein Received June 8 0; Accepted March 7 06 Communicated by Ewa Schmeidel

2 66 Y Halim In these researches authors are interested in investigating the form periodicity boundedness local and global) stability and asymptotic behavior of solutions of various systems of difference equations of nonlinear types In [4] Tollu et al investigated the dynamics of the solutions of the two difference equations x n+ n N 0 ± + x n which were then extended by Halim to systems of two nonlinear difference equations x n+ ± y n y n+ ± x n n N 0 where N 0 N {0} and the initial conditions are real numbers in [7] In paper [] Stević show that the following systems of difference equations x n+ u n + v n y n+ w n + s n n N 0 where u n v n w n s n are some of the sequences x n or y n with real initial values x 0 and y 0 are solvable in fourteen out of sixteen possible cases Motivated by these fascinating results we shall determine the form and investigate the periodicity and global character of the solutions of the systems x n+ + y n y n+ where the initial conditions are real numbers Preliminaries + x n n N 0 ) A well-known recurrence sequence of order two is the widely studied Fibonacci sequence {F n } n recursively defined by the recurrence relation F F F n+ F n + F n n ) ) As a result of the definition ) it is conventional to define F 0 0 Various problems involving Fibonacci numbers have been formulated and extensively studied by many authors Different generalizations and extensions of Fibonacci sequence have also been introduced and thoroughly investigated see for example [8]) An interesting property of this integer sequence is that the ratio of its successive terms converges to the wellknown golden mean or the golden ratio) φ For more fascinating properties of Fibonacci numbers we refer the readers to [] Now in the rest of this section we shall present some basic notations and results on the study of nonlinear difference equation which will be useful in our investigation for more details see for example []

3 System of Difference Equations 67 Let f and g be two continuously differentiable functions: f : I 3 J 3 I g : I 3 J 3 J I J R and for n k N 0 consider the system of difference equations { xn+ f x n x n x n y n y n y n ) y n+ g x n x n x n y n y n y n ) ) where x x x 0 ) I 3 and y y y 0 ) J 3 Define the map H : I 3 J 3 I 3 J 3 by where Let HW ) f 0 W ) f W ) f W ) g 0 W ) g W ) g W )) W u 0 u u v 0 v v ) T f 0 W ) fw ) f W ) u 0 f W ) u g 0 W ) gw ) g W ) v 0 g W ) v W n x n x n x n y n y n y n ) T Then we can easily see that system ) is equivalent to the following system written in vector form W n+ HW n ) n 0 3) that is x n+ f x n x n x n y n y n y n ) x n x n x n x n y n+ g x n x n x n y n y n y n ) y n y n y n y n Definition Equilibrium point) An equilibrium point x y) I J of system ) is a solution of the system { x f x x x y y y) y g x x x y y y) Furthermore an equilibrium point W I 3 J 3 of system 3) is a solution of the system W HW )

4 68 Y Halim Definition Stability) Let W be an equilibrium point of system 3) and be any norm eg the Euclidean norm) The equilibrium point W is called stable or locally stable) if for every ɛ > 0 there exists δ > 0 such that W 0 W < δ implies W n W < ɛ for n 0 The equilibrium point W is called asymptotically stable or locally asymptotically stable) if it is stable and there exists γ > 0 such that W 0 W < γ implies W n W 0 n + 3 The equilibrium point W is said to be global attractor respectively global attractor with basin of attraction a set G I 3 J 3 ) if for every W 0 respectively for every W 0 G) W n W 0 n + 4 The equilibrium point W is called globally asymptotically stable respectively globally asymptotically stable relative to G) if it is asymptotically stable and if for every W 0 respectively for every W 0 G) W n W 0 n + The equilibrium point W is called unstable if it is not stable Remark 3 Clearly x y) I J is an equilibrium point for system ) if and only if W x x x y y y) I 3 J 3 is an equilibrium point of system 3) From here on by the stability of the equilibrium points of system ) we mean the stability of the corresponding equilibrium points of the equivalent system 3) The linearized system associated to system 3) about the equilibrium point is given by W x x x y y y) W n+ AW n n 0 where A is the Jacobian matrix of the map H at the equilibrium point W given by f 0 f 0 f 0 f 0 f 0 f 0 W ) W ) W ) W ) W ) W ) u 0 u u v 0 v v f f f f f f W ) W ) W ) W ) W ) W ) u 0 u u v 0 v v f f f f f f W ) W ) W ) W ) W ) W ) A u 0 u u v 0 v v g 0 g 0 g 0 g 0 g 0 g 0 W ) W ) W ) W ) W ) W ) u 0 u u v 0 v v g g g g g g W ) W ) W ) W ) W ) W ) u 0 u u v 0 v v g g g g g g W ) W ) W ) W ) W ) W ) u 0 u u v 0 v v

5 System of Difference Equations 69 Theorem 4 See [0]) If all the eigenvalues of the Jacobian matrix A lie in the open unit disk λ < then the equilibrium point W of system 3) is asymptotically stable On the other hand if at least one eigenvalue of the Jacobian matrix A have absolute value greater than one then the equilibrium point W of system 3) is unstable Now we are in the position to investigate the form and behavior of solutions of the system ) and this is the content of the next section 3 Main Result 3 Form of the Solutions In this section we give the explicit form of solutions of the system of difference equations x n+ y n+ 3) + y n + x n where the initial values are arbitrary real numbers with the restriction that { x y x y x 0 y 0 / F } n+ ; n F n The following theorem describes the form of the solutions of system 3) Theorem 3 Let {x n y n } n k be a solution of 3) Then for n 0 x 6n+i F n+ + F n y i 3 F n+ + F n+ y i 3 i 3 y 6n+i F n+ + F n x i 3 F n+ + F n+ x i 3 i 3 x 6n+i F n+ + F n+ x i 6 F n+3 + F n+ x i 6 i 4 6 y 6n+i F n+ + F n+ y i 6 F n+3 + F n+ y i 6 i 4 6 Proof From 3) we have and x y x + y y + x x 3 + y + y 0 y 3 + x + x 0 x 4 + x + x x + x + x x 6 + x 0 + x 0 y 4 + y + y y + y + y y 6 + y 0 + y 0

6 70 Y Halim So the result hold for n 0 Suppose now that n and that our assumption holds for n That is x 6n )+i F n + F n y i 3 F n + F n y i 3 i 3 3) y 6n )+i F n + F n x i 3 F n + F n x i 3 i 3 33) x 6n )+i F n + F n x i 6 F n+ + F n x i 6 i ) y 6n )+i F n + F n y i 6 F n+ + F n y i 6 i 4 6 3) For i 3 it follows from 3) 3) and 33) that and x 6n+i + y 6n 3+i + +x 6n )+i F n +F n )+F n +F n )y i 3 F n +F n y i 3 F n +F n +F n y i 3 +F n y i 3 F n +F n y i 3 F n+ + F n y i 3 F n+ + F n + F n + F n )y i 3 F n+ + F n y i 3 F n+ + F n+ y i 3 y 6n+i + x 6n 3+i + +y 6n )+i F n +F n )+F n +F n )x i 3 F n +F n x i 3 F n +F n +F n x i 3 +F n x i 3 F n +F n x i 3 F n+ + F n x i 3 F n+ + F n + F n + F n )x i 3 F n+ + F n x i 3 F n+ + F n+ x i 3

7 System of Difference Equations 7 Similarly for i 4 6 from 3) 34) and 3) we get and This completes the proof x 6n+i + y 6n +k)+i + +x 6n )+i F n+ +F n +F n x i 6 +F n x i 6 F n+ +F n x i 6 F n+ +F n +F n x i 6 +F n x i 6 F n+ +F n x i 6 F n+ + F n+ x i 6 F n+ + F n+ + F n x i 6 + F n+ x i 6 F n+ + F n+ x i 6 F n+3 + F n+ x i 6 y 6n+i + x 6n 3+i + +y 6n )+i F n+ +F n +F n y i 6 +F n y i 6 F n+ +F n y i 6 F n+ +F n +F n y i 6 +F n y i 6 F n+ +F n y i 6 F n+ + F n+ y i 6 F n+ + F n+ + F n y i 6 + F n+ y i 6 F n+ + F n+ y i 6 F n+3 + F n+ y i 6 3 Global Stability of Positive Solutions In this section we study the asymptotic behavior of positive solutions of the system 3) Let I J 0 + ) and consider the functions f : I 3 J 3 I g : I 3 J 3 J defined by fu 0 u u v 0 v v ) + v gu 0 u u v 0 v v ) + u

8 7 Y Halim Lemma 3 System 3) has a unique positive equilibrium point in I J namely + E : + ) Proof Clearly the system x + y y + x has a unique solution in I J which is + x y) + ) The proof is complete Theorem 33 The equilibrium point E is locally asymptotically stable Proof The linearized system about the equilibrium point + E ) is given by I 3 J 3 X n+ AX n X n x n x n x n y n y n y n ) T 36) where A is 6 6 matrix and given by A We can find the eigenvalues of the matrix A from the characteristic polynomial P λ) deta λi 6 ) λ ) 0 Now consider the two functions defined by aλ) λ ) bλ) <

9 System of Difference Equations 73 We have bλ) < aλ) λ : λ Thus by Rouche s Theorem all zeros of P λ) aλ) bλ) 0 lie in λ < So by Theorem 4) we get that E is locally asymptotically stable Theorem 34 The equilibrium point E is globally asymptotically stable Proof Let {x n y n } n k be a solution of 3) By Theorem 33) we need only to prove that E is global attractor that is lim x n x n x n y n y n y n ) E n or equivalently lim x n y n ) E n To do this we prove that for i 6 we have lim x 6n+i lim y 6n+i + For i 3 it follows from Theorem 3) that and Using Binet s formula where α + lim x 6n+i lim y 6n+i lim lim lim lim β we get F n+ + F n y i 3 F n+ + F n+ y i 3 + F n F n+ y i 3 F n+ F n+ + y i 3 37) F n+ + F n x i 3 F n+ + F n+ x i 3 + F n F n+ x i 3 F n+ F n+ + x i 3 38) F n αn β n α β n N 0 39) lim F n F n+ lim α n β α) n α β α n+ β α) n+ α β α 30)

10 74 Y Halim similarly we get Thus from 37) 3) we get F n+ lim α 3) F n+ lim x 6n+i + y α i 3 α + y i 3 α + lim y 6n+i + x α i 3 α + x i 3 α + By the same arguments we get for i 4 6: This completes the proof lim x 6n+i Remark 3 If x i0 3 x + some i 0 3 then for n 0 y 3n+i0 + lim y 6n+i + respectively y i0 3 y + ) for respectively x 3n+i0 + ) Using the fact that and Theorem 3) we get x y + x + y y 3n+i0 F n+ + F n x F n+ + F n+ x F n+ +xf n+ +x F n+ + F n+ x + x x F n+ + Fn Fn++Fn+xFn+ +x F n+ + F n+ x +x F n+ + F n+ x and x 3n+i0 F n+ + F n y F n+ + F n+ y F n+ + Fn +y F n+ + F n+ y F n+ +yf n+ +y F n+ + F n+ y + y y F n+ +F n +yf n+ +y F n+ + F n+ y

11 System of Difference Equations 7 Similarly if x i0 6 x + 4 i 0 6 then for n 0 y 3n+i0 + respectively y i0 6 y + ) for some respectively x 3n+i0 + Example 36 We illustrate results of this section we consider the following numerical example Assume ) x x x 0 00 y y 0 and y 0 3 see Fig 3) In figure 3) we see that the sequences xn ) n and x n ) n of solution of equations 3) with given initial conditions converges to the equilibrium + point + ) ) xn) n Figure 3: The sequences x n ) n blue) and y n ) n green) of solution of the equation 3) with initial conditions x x x 0 00 y y 0 and y Conclusion In this study we mainly obtained the relationship between the solutions of system of difference equations 3) and Fibonacci numbers We also presented that the solutions of system in 3) actually converge to equilibrium point The results in this paper can be extended to the following system of difference equations x n+ ± ± y n k y n+ ± ± x n k n k N 0

12 76 Y Halim where N 0 N {0} and the initial conditions x k x k+ x 0 y k y k+ y 0 are real numbers References [] Q Din M Qureshi and A Q Khan Dynamics of a fourth-order system of rational difference equations Adv Difference Equ 0) Article ID [] S Elaydi An introduction to difference equations Undergraduate Texts in Mathematics Springer New York NY USA 996 [3] E M Elsayed On the solutions of higher order rational system of recursive sequences Math Balk )008) [4] E M Elsayed M Mansour and M M El-Dessoky Solutions of fractional systems of difference equations Ars Comb 0)03) [] E M Elsayed Solution for systems of difference equations of rational form of order two J Comput Appl Math 33)04) 7 76 [6] E M Elsayed Solutions of some rational systems of difference equations Util Math 6)03) [7] Y Halim Global character of systems of rational difference equations Electron J Math Analysis Appl 3))0) 04 4 [8] A F Horadam Basic properties of certain generalized sequence of numbers Fibonacci Q 3)96) 6 76 [9] T F Ibrahim Closed form solution of a symmetric competitive system of rational difference equations Stud Math Sci ) 0) 49 7 [0] V L Kocic and G Ladas Global behavior of nonlinear difference equations of higher order with applications Kluwer Academic Publishers Dordrecht 993 [] T Koshy Fibonacci and Lucas numbers with applications John Wiley New York 00 [] S Stević On some solvable systems of difference equations Appl Math Comput 8)9)0) [3] S Stević Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences Electron J Qual Theory Differ Equ 04) No 67 -

13 System of Difference Equations 77 [4] D T Tollu Y Yazlik and N Taskara On the solutions of two special types of Riccati difference equation via Fibonacci numbers Adv Difference Equ 03) Article ID 74 [] N Touafek and E M Elsayed On the periodicity of some systems of nonlinear difference equations Bull Math Soc Sci Math Roum Nouv Sr )03)0) 7 4 [6] N Touafek and E M Elsayed On the solutions of systems of rational difference equations Math Comput Modelling )0) [7] Y Yazlik D T Tollu and N Taskara On the solutions of difference equation systems with Padovan numbers Appl Math 4)A) 03) 0