ON THE RECURSIVE SEQUENCE x n+1 = A x n. 1. Introduction Our aim in this paper is to establish that every positive solution of the equation

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1 PROCEEDINGS OF THE MERICN MTHEMTICL SOCIETY Volume 26, Number, November 998, Pages S (98) ON THE RECURSIVE SEQUENCE x n+ = x n R. DEVULT, G. LDS, ND S. W. SCHULTZ (Communicated by Hal L. Smith) bstract. We show that every positive solution of the equation x n+ =, n =0,,..., x n where (0, ), converges to a period two solution.. Introduction Our aim in this paper is to establish that every positive solution of the equation () where x n+ =, x n n =0,,..., (2) x 2,x,x 0, (0, ), converges to a period two solution. This confirms conjecture in []. More precisely, we show that each of the two subsequences { } {+ } of every positive solution of Eq. () converges to a finite it. One of the main ingredients in our proof, an interesting result in its own right, is to show that every positive solution of the equation y n+ = + y n (3), n =0,,..., y n where (4) y,y 0, (0, ), converges to its equilibrium ȳ =+. From this it follows easily that the equilibrium + of Eq. (3) is globally asymptotically stable. It follows from the work in [2] that every positive solution of Eq. () is bounded persists. It was also observed in [2] that for every real number q, Eq.() possesses the period two solution q, +,q, q +,.... q Received by the editors March 8, Mathematics Subject Classification. Primary 390. Key words phrases. Recursive sequence, global asymptotic stability, period two solution c 998 merican Mathematical Society

2 3258 R. DEVULT, G. LDS, ND S. W. SCHULTZ We say that a solution {x n } of a difference equation is bounded persists if there exist positive constants P Q such that P x n Q for n =, 0,.... positive semicycle of a solution {y n } of Eq. (3) consists of a string of terms {y l,y l+,...,y m }, all greater than or equal to the equilibrium ȳ, with l m,suchthat either l =, or l> y l < ȳ either m =, or m< y m+ < ȳ. negative semicycle of a solution {y n } of Eq. (3) consists of a string of terms {y l,y l+,...,y m }, all less than the equilibrium ȳ, with l m, such that either l =, or l> y l ȳ either m =, or m< y m+ ȳ. The first semicycle of a solution starts with the term y, is positive if y ȳ negative if y < ȳ. 2. Global asymptotic stability of Eq. (3) Let {x n } be a positive solution of Eq. (), set y n = x n x n. Then clearly, {y n } satisfies Eq. (3), condition (4) holds. The main result in this section is the following. Theorem. ssume that (4) holds. Then the positive equilibrium ȳ =+ of Eq. (3) is globally aymptotically stable. Before we establish the above theorem we need the following result. Lemma. Let {y n } be a nontrivial positive solution of Eq. (3). Then the following statements are true. (a) {y n } oscillates about the equilibrium ȳ =+with semicycles of length two or three. (b) The extreme in a semicycle occurs in the first or second term. (c) For n>2, <y n <+ +. Proof. We will first show that every positive semicycle, except possibly the first, has two or three terms. The case for the negative semicycles is similar is omitted. Let ȳ be the first term in a positive semicycle, other than the first positive semicycle. Then < ȳ, + = + >+ ȳ =+ + ȳ. Now if + >,then +2 = + + >+=ȳ.

3 ON THE RECURSIVE SEQUENCE x n+ = xn 3259 lso +2 = = + + ȳ + <+. Thus ȳ<+2 <+, so +3 = + +2 <+=ȳ, + the positive semicycle has length three. If +, then +2 = + + +=ȳ, the positive semicycle has length at most three, with length equal to two unless + =. From this it is clear that every solution {y n } of Eq. (3) oscillates about ȳ =+. It is also clear from the above that the extreme in a semicycle occurs in the first or second term. From Eq. (3) one can see that y n >for n>0.let,wheren>2, be the first term in a positive semicycle. Then < <ȳ < 2 <ȳ, giving = + <+ ȳ 2 =++. Now + = + = = + <+ 2. The proof is complete. Proof of Theorem. It is easy to see (by linearized stability analysis) that ȳ =+ is locally asymptotically stable. So it remains to show that if {y n } is a nontrivial solution of Eq. (3), then y n =+. To this end, define the sequences {L n } {U n } as follows: L =, U = +, for n =,2,... L n+ = + U n+ = +. U n L n+ Now, it can be seen that {U n } {L n } are sequences of upper lower bounds for the semicycles of the solutions {y n } of Eq. (3). lso L n+ = + + L n U n+ = + +. U n From this it follows that L <L 2 <... < L n <L n+ <... < ȳ <... < U n+ <U n <... < U 2 <U. Thus L n = L ȳ U n = U ȳ.

4 3260 R. DEVULT, G. LDS, ND S. W. SCHULTZ But since the only solution to y = + + y is y =ȳ, it must be true that U = L =ȳ. The proof is complete. (5) (6) 3. Periodic character of the solutions of Eq. () It follows from Eq. () that + = +2 =, n =0,,..., 2, n =0,,..., + so (7) +2 = + 2 2, n =,2, The following result summarizes some of the properties of the solutions of Eq. () Lemma 2. Let {x n } be a positive solution of Eq. (). Then the following statements are true. (i) For N 0, let m N =min{x 2N 2,x 2N,x 2N+2 } =max{x 2N 2,x 2N,x 2N+2 }. Then m N for n N. (ii) There exist positive numbers m M such that m x n M for n = 0,,... (iii) Proof. To prove (i), note that the function f(x, y, z) = is increasing in x, y, z. Thus x 2N+4 = x 2N +2 + x 2N 2 =. x y x 2N x 2N 2 y + z + = by induction, for n N. Similarly m N for n N. The proof of (ii) follows directly from (i), since (i) implies that the sequences of even odd terms of Eq. (3) are bounded. To prove (iii), note that y 2n = converges to +, so converges to. Thus y 2n y 2n = 2 = 2 2 =.

5 ON THE RECURSIVE SEQUENCE x n+ = xn 326 The main result in this paper is the following: Theorem 2. Let {x n } be a positive solution of Eq. (). Then there exist positive constants L E L O such that L E L O =+ = L E + = L O. Proof. It follows from Eq. (5) Lemma 2 that 2. 2 M Hence 2 =0. It is now clear from Lemma 2 (i) (iii) that exists is a positive number L E. From Eq. (5) it then follows that + also exists is a positive number L O. Finally, from Eq. (5) we see that L E L O =+, the proof is complete. References [] G. Ladas, Open Problems Conjectures, Journal of Difference Equations pplications 2 (996), [2] Ch. G. Philos, I. K. Purnaras Y. G. Sficas, Global attractivity in a nonlinear difference equation, pplied Mathematics Computers 62, (994), MR 95h:39008 Division of Mathematics Sciences, Northwestern State University, Natchitoches, Louisiana address: rich@alpha.nsula.edu Department of Mathematics, University of Rhode Isl, Kingston, Rhode Isl address: gladas@math.uri.edu Department of Mathematics Computer Science, Providence College, Providence, Rhode Isl address: sschultz@providence.edu