Anna Andruch-Sobi lo, Ma lgorzata Migda. FURTHER PROPERTIES OF THE RATIONAL RECURSIVE SEQUENCE x n+1 =

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1 Ouscula Mathematica Vol. 26 No Anna Andruch-Sobi lo, Ma lgorzata Migda FURTHER PROPERTIES OF THE RATIONAL RECURSIVE SEQUENCE x n+1 = Abstract. In this aer we consider the difference equation x n+1 = b + cx nx n 1, n = 0, 1,... (E) with ositive arameters a and c, negative arameter b and nonnegative initial conditions. We investigate the asymtotic behavior of solutions of equation (E). Keywords: difference equation, exlicit formula, ositive solutions, asymtotic stability. Mathematics Subject Classification: 39A10 1. INTRODUCTION In this aer we consider the following rational difference equation x n+1 = b + cx n x n 1, n = 0, 1,... (E) where b is a negative real number and a and c are ositive real numbers and the initial conditions x 1, x 0 are nonnegative real numbers such that at least one of them is ositive. Eq. (E) in the case of ositive b was considered in [1].We use the exlicit formula for solutions of Eq. (E) in investigating their behavior. There has been a lot of work concerning the asymtotic behavior of solutions of rational difference equations. Second order rational difference equations were investigated, for examle, in [1 13]. This aer is motivated by the short notes [4], where the author studied the rational difference equation x n+1 = x n x n x n 1, n = 0, 1,

2 388 Anna Andruch-Sobi lo, Ma lgorzata Migda 2. MAIN RESULTS Let = b a, q = c a. Then Eq. (E) can be rewritten as x n+1 = x n 1 + qx n x n 1, n = 0, 1,.... (E1) The change of variables x n = 1 q y n reduces the above equation to y n+1 = y n 1 + y n y n 1, n = 0, 1,... (E2) where is a negative real number, the initial conditions y 1, y 0 are nonnegative real numbers such that at least one of them is ositive. We will also assume y 0 y 1 n (1 ) n 1 for n = 1, 2,..., 1 and y 0 y 1 1 for = 1 (which ensures that the denominator in Eq. (E2) is not equal to zero). Hereafter, we focus our attention on Eq. (E2) instead of Eq. (E). Note, that the solution {y n } of Eq. (E2) with y 1 = 0 or y 0 = 0 is oscillatory. In fact, in this case there is { {y n } = 0, y 0, 0, y0, 0, y0 2,... } { or {y n }= y 1, 0, y 1 } y 1, 0,, 0, Obviously, if = 1, these solutions are 4-eriodic. Here, we review some results which will be useful in our investigation of the behavior of solutions of Eq. (E2). Let I be some interval on the real line and let f : I I I be a continuous function. Definition 1. ([8]) For every air of initial conditions (x 1, x 0 ) I I, the difference equation x n+1 = f(x n, x n 1 ), n = 0, 1,... (E3) has the unique solution {x n } n= 1, which is called a recursive sequence. An equilibrium oint of (E3) is a oint α I with f(α, α) = α; it is also called a trivial solution of Eq. (E3). Definition 2. ([13]) Let α be an equilibrium oint of Eq.(E3): (i) α is stable if for every ε > 0 there exists δ > 0 such that for any initial conditions (x 1, x 0 ) I I with x 1 α + x 0 α < δ, the inequality x n α < ε holds for n = 1, 2,...; (ii) α is a local attractor if there exists γ > 0 such that x n α holds for any initial conditions (x 1, x 0 ) I I with x 1 α + x 0 α < γ; (iii) α is locally asymtotically stable if it is stable and is a local attractor; (iv) α is a reeller if there exists γ > 0 such that for each (x 1, x 0 ) I I with x 1 α + x 0 α < γ, there exists N such that x N α γ. Assume α is an equilibrium oint of Eq. (E3). Let r = f(α,α) x n, s = f(α,α) x n 1. Then the linearized equation associated with Eq. (E3) about the equilibrium α is z n+1 + rz n + sz n 1 = 0. (E4)

3 Further roerties of the rational recursive sequence x n+1 = Theorem A ([7])(Linearized stability theorem). (i) If r < 1 + s and s < 1, then α is locally asymtotically stable. (ii) If r < 1 + s and s > 1 then α is a reeller. The equilibria of Eq. (E2) are the solutions of the equation ȳ = ȳ + ȳ So, equilibrium oints of Eq. (E2) are ȳ = 0 and ȳ = ± 1. The local asymtotic behavior of the zero equilibrium of Eq. (E2) is characterized by the following result. Theorem 1. The following statements are true: (i) if (, 1), then ȳ = 0 is locally asymtotically stable; (ii) if ( 1, 0), then ȳ = 0 is a reeller. Proof. For Eq. (E2), there is y 2 n 1 f = y n ( + y n y n 1 ) 2, f = y n 1 ( + y n y n 1 ) 2. Therefore, for ȳ = 0 we get r = 0, s = 1 and the linearized equation associated with Eq. (E2) about the equilibrium ȳ = 0 is z n+1 1 z n 1 = 0. (i) The result follows from Theorem A(i) and the following relations r (1 + s) = < 0, and s = 1 < 1. (ii) The result follows from Theorem A(ii) and the following relations r 1 + s = 1 = 1 < 0 and 1 > 1. This comletes the roof.

4 390 Anna Andruch-Sobi lo, Ma lgorzata Migda It is easy to see that the method use in the roof of Theorem 1 in [1] can be use in our case too. Thus the following formula y n = n+1 2Q 1 [ 2i 2i 1 P +y 0y 1 k ] y 1 n+1 2Q 1 P [ 2i+1 2i +y 0y 1 k ] n 1 2 Q [ 2i+1 P2i +y 0y 1 k ] y 0 n 1 2 Q [ 2i+2 2i+1 P +y 0y 1 k ] for n odd, for n even, holds for all solutions of Eq. (E2) with ositive initial conditions y 1, y 0 such that y 0 y 1 n (1 ) n 1 for n = 1, 2,..., 1 and y 0 y 1 1 for = 1. If all arameters and initial conditions in Eq. (E) are ositive, then all solutions of Eq. (E) are ositive, too. It is not true in the case of negative b. In the next theorem we give sufficient conditions for every solution of Eq. (E2) to be ositive. Theorem 2. Assume that ( 1, 0). Let {y n } be a solution of Eq. (E2) with ositive initial conditions y 1, y 0 such that y 0 y 1 n (1 ) n 1 for n = 1, 2,.... If y 0 y 1 > then {y n } is ositive. Proof. Let {y n } be a solution of Eq. (E2). From (1), for the subsequence {y 2n 1 } there follows n 1 [ 2i 2i 1 y 2n 1 = y 1. n 1 [ 2i+1 2i Obviously, for ( 1, 0), 2i 1 2i + y 0 y 1 for all i = 0, 1,.... On the other hand, if y 0 y 1 >, then k > 0 (1) 2i 2i+1 + y 0 y 1 k > 0 (2) for all i = 0, 1,.... Therefore, all terms of the sequence {y 2n 1 } are ositive. For n even the roof is similar. Remark 1. If y 0 y 1 = 1 then from (E2) we get y n+1 = yn 1 +y ny n 1 = y n 1. Hence {y 2n } = {y 0, y 0, y 0,...} and {y 2n 1 } = {y 1, y 1, y 1,...}.

5 Further roerties of the rational recursive sequence x n+1 = 391 Theorem 3. Assume that ( 1, 0). Let {y n } be a solution of Eq. (E2) with ositive initial conditions y 1, y 0 such that y 0 y 1 n (1 ) n 1 for n = 1, 2,.... If < y 0 y 1 < 1 then the subsequence {y 2n } is decreasing and subsequence {y 2n 1 } is increasing. Proof. Let {y n } be a solution of Eq. (E2). From (1), for the subsequence {y 2n } there follows n 1 [ 2i+1 2i y 2n = y 0. n 1 [ 2i+2 2i+1 Thus for n 1 y 2n+2 y 2n = n [ 2i+1 2i n 1 [ 2i+2 2i+1 = n [ 2i+2 2i+1 n 1 [ 2i+1 2i 2n 2n+1 + y 0 y 1 k =. 2n+2 2n+1 + y 0 y 1 k (3) Since y 0 y 1 < 1, there is y 0 y 1 2n+1 > 2n+1 2n+2. Hence and therefore 2n+1 y 0 y 1 ( k 2n+1 + y 0 y 1 2n 2n k ) > 2n+1 2n+2, 2n+1 k < 2n+2 + y 0 y 1 k. From the above inequality, by (2) and (3) it follows that the subsequence {y 2n } is decreasing. Similarly we rove that the subsequence {y 2n 1 } is increasing. This comletes the roof. Theorem 4. Assume that 2. Let {y n } be a solution of Eq. (E2) with ositive initial conditions y 1, y 0 (0, 1). Then the subsequences {y 4n 1 } and {y 4n } are both ositive and decreasing, while subsequences {y 4n+1 } and {y 4n+2 } are both negative and increasing.

6 392 Anna Andruch-Sobi lo, Ma lgorzata Migda Proof. Let y 1, y 0 (0, 1). Then y 1, y 2 (0, 1) and y 3, y 4 ( 1, 0). By induction we can rove that {y 4n 1 }, {y 4n } (0, 1) and {y 4n+1 }, {y 4n+2 } ( 1, 0), n = 0, 1,.... Since, by (1), we have y 4n+4 y 4n = (4n y 0 y 4n )( 4n y 0 y 4n ) ( 4n y 0 y 4n )( 4n y 0 y 4n ) < 1, y 4n+4 < y 4n, n = 0, 1,... Similarly we can see that y 4n+3 < y 4n 1, and y 4n+5 > y 4n+1, y 4n+6 > y 4n+2 for n = 0, 1,... and the result follows. 3. NUMERICAL RESULTS Examle 1. Let y 1 = 3 4, y 0 = 1 be the initial conditions of Eq. (E2) with = 1 2. Then, by Theorem 2, the solution is ositive. Table 1 sets forth the values of y n for selected small n s. Table 1 n y(n) n y(n) Table 2 n y(n) n y(n) Examle 2. Let = 2/3, y( 1) = 4/3, y(0) = 1. Thus the condition < y(0)y( 1) < 1 holds and by Theorem 3, the subsequence {y 2n } is decreasing and subsequence {y 2n 1 } is increasing. Table 2 sets forth the values of y n for selected small n s. Examle 3. Let = 11, y( 1) = 0.2, y(0) = 0.5. Then, by Theorem 4, the subsequences {y 4n 1 } and {y 4n } are both ositive and decreasing, while the subsequences {y 4n+1 } and {y 4n+2 } are both negative and increasing.

7 Further roerties of the rational recursive sequence x n+1 = Table 3 sets forth the values of y n for selected small n s. Table 3 n y(n) n y(n) E E E E E E E E E E E E E E E E E E E E E E E E REFERENCES [1] A. Andruch-Sobi lo, M. Migda, On the rational recursive sequence x n+1 = (submitted). [2] C. Cinar, On the ositive solutions of the difference equation x n+1 = x n 1 1+x nx n 1, Al. Math. and Com. 150 (2004), [3] C. Cinar, On the ositive solutions of the difference equation x n+1 = 1+bx nx n 1, Al. Math. and Com. 156 (2004), x [4] C. Cinar, On the ositive solutions of the difference equation x n+1 = n 1 1+x nx n 1, Al. Math. and Com. 158 (2004), [5] D.C. Chang, S. Stević, On the recursive sequence x n+1 = α + βx n 1 1+g(x n), Al. Anal. 82 (2003), [6] H.M. El-Owaidy, A.M. Ahmed, M.S. Moousa, On the recursive sequences x n+1 = αx n 1 β±x n, Al. Math. Com., 145 (2003), [7] V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Alications, Kluwer-Academic Publishers, Dordrecht, [8] W.A. Kosmala, M.R.S. Kulenovic, G. Ladas, C.T. Teixeira, On the recursive sequence y n+1 = +y n 1 qy n+y n 1, J. Math. Anal. Al. 251 (2000), [9] W.T. Li, H.R. Sun, Dynamics of rational difference equation, Al. Math. Comut. 163 (2005),

8 394 Anna Andruch-Sobi lo, Ma lgorzata Migda [10] Y.S. Huang, P.M. Knof, Bondedness of ositive solutions of second-order rational difference equations, J. Difference Equ. Al. 10 (2004), [11] S. Stević, More on a rational recurrence relation, Al. Math. E-Notes 4 (2004), [12] X. Yan, W. Li, Global attractivity in the recursive sequence x n+1 = α βxn γ x n 1, Al. Math. Comut. 138 (2003), [13] X. Yang, W. Su, B. Chen, G. Megson, D. Evans, On the reecursive sequences x n = +bx n 2 c+dx n 1 x n 2, Al. Math. Com. 162 (2005), Anna Andruch-Sobi lo andruch@math.ut.oznan.l Poznań University of Technology Institute of Mathematics Piotrowo 3A, Poznań, Poland Ma lgorzata Migda mmigda@math.ut.oznan.l Poznań University of Technology Institute of Mathematics Piotrowo 3A, Poznań, Poland Received: Setember 23, 2005.