On the Third Order Rational Difference Equation

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1 Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 27, On the Third Order Rational Difference Equation x n x n 2 x (a+x n x n 2 ) T. F. Irahim Department of Mathematics, Faculty of Science Mansoura University, Mansoura 3556, Egypt tfirahem@mans.edu.eg Astract In this paper we will give the solutions of the rational difference equation n x n 2 x x (a+x n x n 2 ),n 0,,..where initial values,, and are nonnegative real numers with a and 0. Moreover we investigate some properties for this difference equation such as the local staility and the oundedness for the solutions. Mathematics Suject Classification: 39A0 Keywords: staility, oundedness, solvaility, periodic Introduction Recently there has een a lot of interest in studying the oundedness character and the periodic nature of nonlinear difference equations. For some results in this area, see for example -. Difference equations have een studied in various ranches of mathematics for a long time. First results in qualitative theory of such systems were otained y Poincaré and Perron in the end of nineteenth and the eginning of twentieth centuries. The systematic description of the theory of difference equations one can find in ooks,6,0. Cinar 5gave that the solution of the difference equation x +x n x. Karatas et al.8 gave that the solution of the difference equation x n 5 +x n 2 x n 5

2 322 T. F. Irahim Aloqeili 2 has otained the solutions of the difference equation x, a x n x In this paper we will give the solutions of the rational difference equation x n x n 2 x (a + x n x n 2 ), () n 0,,..where initial values,, and are nonnegative real numers with a and 0. Moreover we investigate some properties for this difference equation such as the local staility and the oundedness for the solutions. 2 Solvaility of the difference eq() when a, : The following theorem give the solution of the difference equation x n x n 2 x ( + x n x n 2 ) (*) Theorem Suppose that x n } e a solution of equation (*) where the initial values,, and are nonnegative real numers. Then the solutions of equation (*) have the form x 4n 3 ( ) ( + (4i 5) x0 ) ( + (4i 3) ) (2) x 4n 2 x 4 n n ( + (4i 4) x0 ) ( + (4i 2) ) ( + (4i 3) x0 ) ( + (4i ) ) (3) (4) x 4n n ( + (4i 2) x0 ) ( + (4i) ) (5)

3 Rational difference equation 323 where n, 2,.... Proof. For n the result holds. Now suppose that n>0 and that our assumption holds for n. That is; x 4n 7 ( ) ( + (4i 5) x0 ) ( + (4i 3) ) (6) x 4n 6 ( + (4i 4) x0 ) ( + (4i 2) ) (7) x 4n 5 x 4n 4 ( + (4i 3) x0 ) ( + (4i ) ) ( + (4i 2) x0 ) ( + (4i) ) (8) (9) Now we will try to prove that the eqs(2-5) hold at n. Using Eq(7), Eq(8) and Eq(9) we have x 4n 3 x 4n 4 x 4n 6 x 4n 5 ( + x 4n 4 x 4n 6 ) (+(4i 2)x0 ) (+(4i) ) } (+(4i 3)x0 ) (+(4i ) ) (+(4i 4)x0 } ) (+(4i 2) ) } ( + (+(4i 2)x0 ) (+(4i) ) } (+(4i 4)x0 }) ) (+(4i 2) ) / ( + (4n 4) ) } (+(4i 3)x0 ) (+(4i ) )

4 324 T. F. Irahim ( ) x0 ( + (4n 4) )(+ / ( + (4n 4) )) ( + (4i ) x0 ) ( + (4i 3) ) ( ) x0 +(4n 4) + ( + (4i ) x0 ) ( + (4i 3) ) ( ) x0 +(4n 3) ( + (4i ) x0 ) ( + (4i 3) ) ( ) x0 ( + (4i 5) x0 ) ( + (4i 3) ) Using Eq(2), Eq(8) and Eq(9) we have x 4n 2 x 4n 3 x 4n 5 x 4n 4 ( + x 4n 3 x 4n 5 ) ( ) (+(4i 5)x0 ) (+(4i 3) ) } (+(4i 2)x0 ) (+(4i) ) } (+(4i 3)x0 } ) (+(4i ) ) + ( ) (+(4i 5)x0 ) (+(4i 3) ) } (+(4i 3)x0 } ) (+(4i ) ) (+(4i 2)x0 ) (+(4i) ) x0 (+(4n 5) ) ( )(+(4n 3) ) } + } ( ) (+(4n 5) ) x0 (+(4n 5) ) ( )(+(4n 3) ) } ( ) (+(4n 5) )

5 Rational difference equation (+(4n 3) ) } (+(4n 3) ) } ( + (4i) x0 ) ( + (4i 2) ) +(4n 3) + n Using Eq(2), Eq(3) and Eq(9) we have ( + (4i 4) x0 ) ( + (4i 2) ) ( + (4i) x0 ) ( + (4i 2) ) x 4 x 4n 2 x 4n 4 x 4n 3 ( + x 4n 2 x 4n 4 ) (+(4i 4)x0 ) (+(4i 2) ) ( ) } (+(4i 5)x0 ) (+(4i 3) ) } (+(4i 2)x0 ) (+(4i) ) } + (+(4i 4)x0 ) (+(4i 2) ) } } (+(4i 2)x0 ) (+(4i) ) n ( ) (+(4i 5)x0 ) (+(4i 3) ) (+(4n 4)x0 ) (+(4n 2) ) } + (+(4n 4)x0 ) (+(4n 2) ) } (+(4i 4)x0 ) (+(4i) ) (+(4i 4)x0 ) (+(4i) ) } n ( ) } (+(4i 5)x0 ) (+(4i 3) ) ( ) + (+(4n 4)x0 ) (+(4n 2) ) + (+(4n 4)x0 ) (+(4n 2) ) (+(4n 2) ) (+(4n 2) ) } +(4n 4) } +(4n 4) ( + (4i 3) x0 ) ( + (4i 5) )

6 326 T. F. Irahim ( ) ( + (4n ) ) ( ) ( ) Using Eq(2), Eq(3) and Eq(4) we have ( + (4i 3) x0 ) ( + (4i 5) ) ( + (4i 3) x0 ) ( + (4i ) ) ( + (4i 3) x0 ) ( + (4i ) ) x 4n x 4 x 4n 3 x 4n 2 ( + x 4 x 4n 3 ) ( ) (+(4i 4)x0 ) (+(4i 2) ) n ( + ( ) (+(4i 5)x0 } ) (+(4i ) ) n (+(4i 5)x0 ) (+(4i ) ) } ) (+(4i 4)x0 ) (+(4i 2) ) (+(4) ) ( + (+(4) ) ) (( + (4n ) )+ ) ( + (4i 2) x0 ) ( + (4i 4) ) ( + (4n) ) ( + (4i 2) x0 ) ( + (4i 4) ) n Hence the proof is complete. ( + (4i 2) x0 ) ( + (4i) )

7 Rational difference equation 327 Lemma 2 Let ρ. We have the following relations etween the solutions in eqs(2-5) ρ i) x 4n 3 x 4 + (4) ρ ρ ii) x 4n x 4n 2 +4n ρ iii) x 4n x 4n 2 x 4n 3 x 4 Proof. i) Form Eq(2) and Eq(4) we have x 4n 3 x 4 ( ) ( + (4i 5) x0 } ) ( + (4i 3) ) ( + (4i 3) x0 } ) ( + (4i ) ) ( ) ( + (4i 5) x0 } ) ( + (4i ) ) ρ ( ρ )(+3ρ)... ( + (4n 5) ρ ) ( ρ ) ( + 3ρ )(+7ρ)... ( + (4n ) ρ ) ρ ( ρ ) ( ρ ) ( + (4n ) ρ ) ρ ( + (4n ) ρ ) ii) From Eq(3) and Eq(5) we have x 4n x 4n 2 n } ( + (4i 2) x0 ) ( + (4i) ) ( + (4i 4) x0 } ) ( + (4i 2) ) n } ( + (4i 4) x0 ) ( + (4i) )

8 328 T. F. Irahim ρ () ( + 4ρ)... ( + (4n 4) ρ) ( + 4ρ)(+8ρ)... ( + (4n) ρ) ρ +(4n) ρ iii) By easy calculations from i) and ii). Remark We note that x 4n x 4n 2 x 4n 3 x 4 + (4) ρ +4n ρ. Remark 2 We note that x 4n 3 x 4 0 as n. Remark 3 We note that x 4n x 4n 2 0 as n. 3 Solvaility of the difference eq() when a, : The following theorem give the solution of the difference equation x n x n 2 x ( x n x n 2 ), Theorem 3 Suppose that x n } e a solution of Equation () and h, k and r.let ρ. Then the solutions of equation () have the form x 4n 3 hr k ( + ρ ) ( (4i 5) ρ ) ( (4i 3) ρ) x 4n 2 r x 4 k ( (4i 4) ρ ) ( (4i 2) ρ) ( (4i 3) ρ ) ( (4i ) ρ ) x 4n h ( (4i 2) ρ ) ( (4i) ρ) where n, 2,.... Proof. By using the mathematical induction as in theorem.

9 Rational difference equation Solvaility of the difference eq() when a : The following theorem give the solution of the difference equation() where a and for any value of. Theorem 4 Suppose that x n } e a solution of Equation () and h, k, r and ρ. Then the solutions of equation () have the form x ρ k (a + ρ), x 2 r a 2 +2aρ x 3 k (a + ρ) a 3 +(2a 2 +)ρ x 4 h (a 2 +2aρ) a 4 +(2a 3 + a +)ρ x 4n 3 ρ k (a + ρ) x 4n 2 r a4i 5 + ρ i2 2a 4i 6 + a4i 7 a a 4i 3 + ρ 2a 4i 4 + a4i 5 a a 4i + ρ 2a 4i + a4i 2 a a 4i 2 + ρ 2a 4i 3 + a4i 4 a } } } } x 4 k i2 n+ i2 a 4i 3 + ρ 2a 4i 4 + a4i 5 a } a 4i 5 + ρ } 2a 4i 6 + a4i 7 a x 4n h } a4i 2 + ρ 2a 4i 3 + a4i 4 a a 4i + ρ } 2a 4i + a4i 2 a where n 2, 3,.... Proof. By using the mathematical induction as in theorem. Lemma 5 The solutions x n } of the difference eqs () and (*) have no prime period two solution

10 330 T. F. Irahim Now we recall some notations and results which will e useful in our study. Definition The difference equation F (x n,x,..., x n k ), n 0,,... (0) is said to e persistence if there exist numers m and M with 0 <m M< such that for any initial conditions x k,x k+,...,, (0, ) there exists a positive integer N which depends on the initial conditions such that m x n M for all n N. Definition 2 (i) The equilirium point of Eq.(0) is locally stale if for every ɛ>0, there exists δ>0 such that for all x k,x k+,...,, I with we have x k x + x k+ x x <δ, x n x <ɛ for all n k. (ii) The equilirium point x of Eq.(0) is locally asymptotically stale if x is locally stale solution of Eq.(0) and there exists γ>0, such that for all x k,x k+,...,, I with we have x k x + x k+ x x <γ, lim n x n x. The linearized equation of Eq.(0) aout the equilirium x is the linear difference equation y n+ k i0 F(x, x,..., x) x n i y n i. Theorem A 9 Assume that p, q R. Then p + q < is a sufficient condition for the asymptotic staility of the difference equation x n+ px n qx 0, n 0,,....

11 Rational difference equation 33 5 Local Staility of the Equilirium Points In this section we study the local staility of the solutions of Eq.(). We first give the equilirium points of Eq(). Lemma 6 The equilirium points of the difference eq () are 0 and ± Proof. a x x 2 x (a + x 2 ) x 2 ( a + x 2) x 2 x ( 2 +a + x 2) 0 a thus the equilirium points of the difference eq () are 0 and ±. Remark 4 When a, then the only equilirium point of the difference eq (*) is 0. a Theorem 7 The equilirium points x ± are locally asymptotically stale if a and in this case a 2a 2 4a 2 6a +. Proof. We will prove the theorem at the equilirium point x + a proof at the equilirium point x y the same way. let f :(0, ) 3 (0, ) e a continuous function defined y f(u, v, w) uw v (a + uw) a and the Therefore it follows that f(u, v) u v (a + uw) w uw (vw) v 2 (a + uw) 2 aw v (a + uw) 2 f(u, v) v uw v 2 (a + uw)

12 332 T. F. Irahim At the equilirium point x f(u, v) w a au v (a + uw) 2 we have f( x, x, x) u a ( ( a + a )) 2 a (2a ) 2 p f( x, x, x) v a (2a ) p 2 f( x, x, x) w a (2a ) 2 p 3 Then the linearized equation of Eq.() aout x a is y n+ p y n p 2 y p 3 y n 2 0. i. e. y n+ a (2a ) 2 y n + a (2a ) y a (2a ) 2 y n 2 0 Whose characteristic equation is λ 3 a a (2a ) 2 λ2 + (2a ) λ a (2a ) 2 0 By the generalization of theorem A we have p + p 2 + p 3 a (2a ) 2 + a (2a ) + a (2a ) 2 a + a (2a ) + a (2a ) 2 a 2a 2 4a 2 6a +

13 Rational difference equation Boundedness of solutions Here we study the oundedness of Eq.(*). Theorem 8 Every solution of Eq.(*) is ounded from aove. Proof: Let x n } n k e a solution of Eq.(*). It follows from Eq.(*) that x n x n 2 x ( + x n x n 2 ) x n x n 2 x + x x n x n 2 x Then x n x n 2 for all n 0. This means that every solution of eq(*) is ouneded from aove y M max(,, ). References R.P. Agarwal, Difference Equations and Inequalities, Dekker, New York, M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comp., 76(2), (2006), C. Cinar, On the positive solutions of the difference equation ax, Appl. Math. Comp.(in press), x n x 4 C. Cinar, On the positive solutions of the difference equation x, Appl. Math. Comp., 50, (2004), x n x 5 C. Cinar, On the difference equation Appl. Math. Comp., 58, (2004), x +x n x, 6 S. Elaydi, An Introduction to Difference Equations, Springer, New York, C. H. Gions,M. R.S. Kulenovvic,G. Ladas,H. D. Voulov, On the trichotomy character of (α + βx n + γx ) / (A + x n ), J. of Difference Eqs and App. 8(), (2002),

14 334 T. F. Irahim 8 R. Karatas, C. Cinar and D. Simsek, On positive solutions of x n 5 the difference equation, Int. J. Contemp. +x n 2 x n 5 Math. Sci., Vol., no. 0, (2006), V.L. Kocic and G. Ladas, Gloal Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic 0 V. Lakshmikantham, D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Academic Press, New York, 998. D. Simsek, C. Cinar and I. Yalcinkaya, On the Recursive Sequence x n 3, Int. J. Contemp. Math. Sci., Vol., no. +x 0,( 2006), Received: March, 2009

Dynamics of a Rational Recursive Sequence

International Journal of Difference Equations ISSN 0973-6069, Volume 4, Number 2, pp 185 200 (2009) http://campusmstedu/ijde Dynamics of a Rational Recursive Sequence E M Elsayed Mansoura University Department