Invariants for Some Rational Recursive Sequences with Periodic Coefficients By J.FEUER, E.J.JANOWSKI, and G.LADAS Department of Mathematics University of Rhode Island Kingston, R.I. 02881-0816, USA Abstract We present some invariants for the rational recursive sequence 1 = a n b n (c n d n ) 1, n = 0, 1,... where {a n }, {b n }, {c n }, and {d n } are positive periodic sequences of certain periods and the initial conditions x 1 and x 0 are arbitrary positive numbers. The case where some of the coefficients are identically zero is also investigated. Key Words: invariant, periodic sequence, recursive sequence AMS 1990 Mathematics Subject Classification: 39A12
1 Introduction Consider the rational recursive sequence 1 = a n b n (c n d n ) 1, n = 0, 1,... (1.1) where {a n }, {b n }, {c n }, and {d n } are periodic sequences of positive numbers and where the initial conditions x 1 and x 0 are arbitrary positive numbers. Our aim is to exhibit the invariants which we have discovered for this equation when the coefficients have certain periods. This paper was motivated by a desire to extend to equations with periodic coefficients the result on invariants obtained in [1] for the rational recursive sequence with constant coefficients, 1 = α β (γ δ) 1, n = 0, 1,.... (1.2) For other references dealing with invariants see [2] [5]. 2 Motivation In this section we explain our motivation for assuming the form of the invariant for Eq.(1.1), by first presenting an invariant for Eq.(1.2) in a form which is easy to generalize to invariants with periodic coefficients. We adapted this invariant from [1], and present it in the following theorem. Theorem A Eq.(1.2) possesses the following invariant: I n = βα βδ α2 αδ 1 (αγ δ 2 ) 1 γδ 1 αδ (αγ δ 2 ) 1 1 βδ α2 1 1
Theorem A suggests that we look for invariants for Eq.(1.1) of the form, I n = P n 1 R n S n 1 U n V n 1 W n 1 T n 1 Z n 1 (2.1) Then one can see that the coefficients of Eq.(2.1) satisfy the following system of equations: T n1 = U n Z n1 = R n P n1 d n = S n b n P n1 c n R n1 d n = S n a n T n b n R n1 c n S n1 d n = T n a n V n b n (2.2) S n1 c n = V n a n W n1 b n = P n d n U n1 b n W n1 a n = P n c n Z n d n U n1 a n V n1 b n = Z n c n W n d n V n1 a n = W n c n. In fact, all the invariants which we will display in this paper are solutions of this system. 3 Invariants When {b n } and {c n } are equal to constants b and c respectively, Eq.(1.1) becomes, a n b 1 =, n = 0, 1,.... (3.1) (c d n ) 1 2
Theorem 3.1 Assume that the coefficients {a n } and {d n } of Eq.(3.1) are periodic of period 3 and that b and c are positive constants. Then Eq.(3.1) possesses the following invariant: I n = a n1b a n1a n2 bd n 1 a n2 d n 1 (d n1 d n2 a n c) d n1 c 1 a n d n2 1 (d n d n1 a n2 c) 1 a na n1 bd n2 1 Proof. For all n 0, we have I n1 = a n2b 1 a n2a n bd n1 1 a n d n1 1 (d n2 d n a n1 c)1 d n2 c1 a n1 d n 1 (d n1 d n2 a n c) a n1a n2 bd n. By substituting Eq.(3.1) we obtain I n1 = (a n2 b (a n2 a n bd n1 ) a n d n1 x 2 n) (c d n ) 1 (a n b) (a n1 d n (d n2 d n a n1 c) d n2 cx 2 (a n b) n) (c d n ) 1 and so (d n1 d n2 a n c) a n1a n2 bd n I n1 = (a n2 d n1 ) (c d n ) 1 (a n1 d n2 ) (a n b) 1 (d n1 d n2 a n c) a n1a n2 bd n = I n. 3
The proofs of the remaining theorems in this paper are similar to that of Theorem 3.1 and will be omitted. When only {c n } is constant and the other three coefficients are periodic, Eq.(1.1) becomes, 1 = a n b n (c d n ) 1, n = 0, 1,.... (3.2) Theorem 3.2 Assume that {b n } is periodic of period 5, {d n } is periodic of period 3, c is a positive constant, and ca n = d n1 d n2 for all n. Then Eq.(3.2) possesses the following invariant: I n = a n 2b n b n 1 d nb n 1 b n1 a n 1 a n 2 b n 1 a n 1 d n b n1 1 (d n1 d n2 b n3 ca n b n2 ) cd n1 b n2 1 a n d n2 b n3 1 (ca n 1 b n1 d n d n1 b n2 ) 1 a na n1 b n 1 d n2 b n b n3 1 Remark 3.1 When a change of variables converts an equation into a form for which an invariant is known, then applying the reverse change of variables to the known invariant yields an invariant for the original equation. We now demonstrate this technique on Eq.(1.1) when {b n } is the only constant and the other three coefficients are periodic, namely 1 = a n b (c n d n ) 1, n = 0, 1,.... (3.3) By the change of variables = 1 y n Eq.(3.3) becomes, y n1 = d ny n c n (by n a n )y n 1, n = 0, 1,... (3.4) which is of the form of Eq.(3.2). From the application of the reverse change of variables y n = 1 to the invariant of Eq.(3.4), we get the following 4
invariant for Eq.(3.3) when {c n } is periodic of period 5, {a n } is periodic of period 3, b is a positive constant, and bd n = a n1 a n2 for all n: I n = ba n 2c n2 a n 1a n 2 c n2 bd n c n 2 1 a n 1 d n c n3 1 (a n c n 1 c n1 d n1 d n2 c n ) d n1 c n c n 1 1 a n d n2 c n1 1 (a n 1 c n c n3 d n d n1 c n 1 ) 1 bd n2c n1 a n a n1 c n2 1 When the coefficients {a n } and {b n } are equal periodic sequences of period 2, and {c n } and {d n } are constants c and d respectively, then Eq.(1.1) becomes, 1 = a n a n, n = 0, 1,.... (3.5) (c d) 1 Theorem 3.3 Assume that {a n } is periodic of period 2 and that c and d are positive constants. Then Eq.(3.5) possesses the following invariant: I n = a n 1a n 2 a n 1(a n 2 d) a n 1d 1 (a n c d 2 ) 1 cd 1 da n 1 (a n 1 c d 2 ) 1 a n(a n 1 d) 1 When {c n } and {d n } are equal periodic sequences of period 2, and {a n } and {b n } are constants a and b respectively, Eq.(1.1) becomes, 1 = a b (c n c n ) 1, n = 0, 1,... and by Remark 3.1 it has an invariant which can easily be found. 4 Special Cases In this section we present several special cases of Eq.(1.1) which possess interesting new invariants when some of the coefficients are identically zero. 5
Consider Eq.(1.1) when {a n } is identically zero, namely 1 = b n (c n d n ) 1, n = 0, 1,.... (4.1) Theorem 4.1 Assume that {b n } and {d n } are periodic of period 2, and {c n } is periodic of period 4. Then Eq.(4.1) possesses the following invariant: I n = c n2b n 1 c n d n 1 c n c n 1 1 c n 1 d n 1 b nc n1 1 Eq.(4.1) also possesses an invariant when {d n } is periodic of period 3. Theorem 4.2 Assume that {b n } is periodic of period 2, {c n } is periodic of period 4, and {d n } is periodic of period 3. Then Eq.(4.1) possesses the following invariant: I n = c n2b n 1 d n c n d n 1 d n1 c n c n 1 d n1 1 c n 1 d n d n1 1 b nc n1 d n 1 1 Consider Eq.(1.1) when {b n } is identically zero, that is a n 1 =, n = 0, 1,.... (4.2) (c n d n ) 1 Theorem 4.3 Assume that {a n } and {d n } are both periodic of period 3 and {c n } is periodic of period 6. Then Eq.(4.2) possesses the following invariant: I n = a n 1a n1 c n3 1 a n 1 c n4 d n (a n c n 1 c n1 c n d n 1 d n1 ) c n 1 c n d n1 1 a n c n1 d n 1 1 (a n 1 c n c n4 c n 1 d n d n1 ) 1 a na n1 c n2 1 (4.3) 6
We can also get an invariant for Eq.(4.2) when {d n } is periodic of period 4 and {a n } is periodic of period 2. Theorem 4.4 Assume that {a n } is periodic of period 2, {c n } is periodic of period 6, and {d n } is periodic of period 4. Then Eq.(4.2) possesses the following invariant: I n = a n 1a n c n3 d n 1 a n 1 c n4 d n d n1 (a n c n 1 c n1 d n2 c n d n 1 d n1 d n2 ) c n 1 c n d n1 d n2 1 a n c n1 d n 1 d n 2 1 (a n 1 c n c n4 d n1 c n 1 d n d n1 d n2 ) 1 a na n 1 c n2 d n 1 1 (4.4) When {c n } is identically zero, we get the following equation, 1 = a n b n d n 1, n = 0, 1,.... (4.5) By applying the change of variables = 1 y n y n1 = we have, d n y n (b n y n a n )y n 1, n = 0, 1,.... From Remark 3.1 we can show that Eq.(4.5) possesses an invariant when {a n } is periodic of period 2, 3, or 4, {b n } is periodic of period 2, 3, or 6, and {d n } is periodic of period 2 or 3. These invariants can be obtained from Eq.(4.3) or Eq.(4.4). In a similar way, one can derive an invariant for the equation, 1 = a n b n c n 1, n = 0, 1,... when {a n } is periodic of period 2 or 3, {b n } is periodic of period 2 or 4, and {c n } is periodic of period 2. Remark 4.1 An interesting problem for further study is to obtain new invariants for Eq.(1.1) with periodic coefficients either by a systematic investigation of System (2.2) or by using other methods. 7
5 References 1. E.A.Grove, E.J.Janowski, C.M.Kent, and G.Ladas, On the rational recursive sequence 1 = αxnβ (γδ) 1, Communications on Applied Nonlinear Analysis 1(1994), 61-72. 2. V.L.Kocic and G.Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. 3. V.L.Kocic, G.Ladas, and I.W.Rodrigues, On rational recursive sequences, J. Math. Anal. Appl. 173(1993), 127-157. 4. E.J.Janowski, V.L.Kocic, G.Ladas, and S.W. Schultz, Global behavior of solutions of 1 = max{xn,a} 1, Proceedings of the First International Conference on Difference Equations, May 25-28, 1994, San Antonio, Texas, USA, Gordon and Breach Science Publishers, (to appear). 5. G.Ladas, Invariants for generalized Lyness equations, Journal of Difference Equations and Applications 1(1995), (to appear). 8