International Journal of Difference Equations ISSN 0973-532, Volume 4, Number, pp. 43 48 (2009) http://campus.mst.edu/ijde Symmetric Functions Difference Equations with Asymptotically Period-two Solutions Kenneth S. Berenhaut Richard T. Guy Wake Forest Uniersity Department of Mathematics Winston-Salem, NC 2709 berenhks@wfu.edu guyrt7@wfu.edu Abstract This paper introduces easily erified conditions which guarantee that all solutions to the equation y n f(y n k, y n m ), with k, m gcd(k, m) are asymptotically periodic with period two. A recent result of Sun Xi is employed. Seeral examples are included. AMS Subject Classifications: 39A0, 39A. Keywords: Difference equations, periodicity, symmetric functions, ratios, recursie equation. Introduction In this paper, we consider recursie equations of the form y n f(y n k, y n m ), (.) for n 0, where k, m, gcd(k, m), s max{k, m} y s, y s+,..., y (0, ). Recently, Sun Xi [6] Steić [4] proed the following interesting result regarding criteria for asymptotically two-periodic behaior of solutions to (.). Theorem.. Suppose that {y i } satisfies (.), in addition (i) f C((0, ) 2, (a, )) with a inf 0, (.2) (u,) (0, ) 2 Receied Noember 27, 2008; Accepted December 0, 2008 Communicated by Gerasimos Ladas
44 K. S. Berenhaut R. T. Guy (ii) is increasing in u decreasing in, (iii) there exists a decreasing function g C((a, ), (a, )) such that g() x, for x > a, (.3) x f(x, ), for x > a, (.4) a. (.5) x a + x Then, if k is een m is odd, eery positie solution to (.) conerges to a (not necessarily prime) two-periodic solution. Otherwise, eery solution conerges to the unique equilibrium of (.). For earlier related results motiation, see [, 2, 5]. Remark.2. Note that the self-inerse property in (.3) gies that the range of g must comprise all of (a, ). Hence under the assumption that g is decreasing, (.5) is immediately satisfied may be remoed from the statement of Theorem.. 2 Main Result Examples In this short note, we will gie easily erified conditions for existence of a function g satisfying the requirements of Theorem.. In particular, we will proe the following. Theorem 2.. Suppose that {y i } satisfies (.) with h(u, )/ for some function h where (i) h C((0, ) 2, (0, )) is such that h(u, ) is symmetric in u increasing in u, (ii) the function f is decreasing in, (iii) with a as in (.2), for all > a, there exist C D (possibly infinite) such that /u C > /u D <. (2.) u a + u Then, there exists a continuous function g satisfying (.3), (.4) (.5). Hence by Theorem., if k is een m is odd, eery positie solution to (.) conerges to a (not necessarily prime) two-periodic solution, otherwise, eery positie solution conerges to the unique equilibrium of (.).
Difference Equations with Period-two Behaior 45 Now, note that under assumptions () (2) in Theorem 2., we hae that, for fixed > a, u h(u, ) u h(, u) u f(, u) (2.2) is decreasing in u hence the (possibly infinite) its in (2.) exist. A key point here is that there is no need to hae a closed form for g to erify the hypotheses of Theorem.. In fact, finding such a closed form may not be ery practical in practice (see for instance Example 2.6 below). Before turning to a proof of Theorem 2., we gie seeral examples of difference equations satisfying the requirements of the theorem. Example 2.2. Consider the equation x n + x n k. (2.3) Here we hae h(u, ) u +, + u/, a, for >, u + u u + u + + > (2.4) + <. (2.5) Hence Theorem 2. is applicable all positie solutions are asymptotically twoperiodic wheneer k is een m is odd, otherwise all positie solutions conerge to the unique equilibrium. See [ 3] the references therein for further discussion of Eq. (2.3). Example 2.3. Consider the equation x n + x n k + xn k. (2.6) Here we hae h(u, ) u+ + u, +u/ + u/, a, for >, u + u u + u + + + u + > (2.7) + + u By Theorem 2., we hae the required asymptotic two-periodic behaior. <. (2.8)
46 K. S. Berenhaut R. T. Guy Example 2.4. Consider the equation x n ( xn k x n k + + x ) n m. (2.9) + Here we hae h(u, ) u/(u + ) + /( + ), (u/(u + ) + /( + ))/, a 0, for > 0, u 0 + u u 0 + (u + ) + u( + ) (2.0) u (u + ) + u( + ) 0 <. (2.) By Theorem 2., we hae the required asymptotic two-periodic behaior. Example 2.5. Consider the equation x n + x n k + log(x n k ). (2.2) Here we hae h(u, ) u + + log(u), + u/ + log(u)/, a, for >, u + u u + u + + log(u) u + + log(u) u + + log() By Theorem 2., we hae the required asymptotic two-periodic behaior. Example 2.6. Consider the equation > (2.3) <. (2.4) x n xα n k + xβ n k + xα n m + x β n m. (2.5) Here, for 0 < α, β <, we hae h(u, ) u α +u β + α + β, u α / +u β / + ( α) + ( β), a 0, for > 0, u 0 + u u 0 + u + α u + β u + (2.6) α u β u u + α u + β u + 0 <. (2.7) α u β By Theorem 2., we hae the required asymptotic two-periodic behaior.
Difference Equations with Period-two Behaior 47 Remark 2.7. To see that the requirement in (3) cannot be remoed, consider the equation x n + x n k + x n k. (2.8) Here, we hae h(u, ) u + + u, + u/ + u a, yet the equation possesses no equilibrium. Indeed, for >, u + u u + + u + 2 + >, (2.9) but + + + >. (2.20) Condition (3) is not satisfied Theorem 2. is not applicable. 3 Proof of the Main Result We now turn to a proof of Theorem 2.. Proof of Theorem 2.. By Theorem. Remark.2, we need only show that there exists a decreasing continuous function g which satisfies (.3) (.4). To that end, note that by (2.2), /u is decreasing in u hence for fixed > a, by (3), there exists a unique u g(), say, which satisfies f(g(), )/g(). (3.) To see that the function g is decreasing, note that for x > a ɛ > 0, by (2) the definition of g, f(, x + ɛ) < f(, x) f(g(x + ɛ), x + ɛ), (3.2) g(x + ɛ) hence since /u is decreasing in u, g(x + ɛ) <, as required. Now, note that for x > a, by the definition of g the assumptions on f h, we hae f(x, ) h(x, ) h(, x) f(, x)x x, (3.3) (.4) is satisfied. Since u g() is the unique alue satisfying f(u, ) u, (3.3) gies that (.3) also holds. The continuity of g follows from its monotonicity the fact that the range of g is (a, ) (see Remark.2), the theorem is proen. Remark 3.. Exping on ideas in [], Steić [4, 5] showed that similar asymptotically two-periodic behaior can occur for multiariable functions f in (.), with arying delays. The interested reader may erify that the ideas introduced here are applicable in that case as well.
48 K. S. Berenhaut R. T. Guy References [] K. S. Berenhaut, J. D. Foley S. Steić, The global attractiity of the rational difference equation y n + y n k. Proc. Amer. Math. Soc. 35(2007), no. 4, y n m 33 40. [2] E. A. Groe G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC Press, Boca Raton (2004). [3] W. T. Patula H. D. Voulo, On the oscillation periodic character of a third order rational difference equation. Proc. Amer. Math. Soc. 3(2003), no. 3, 905 909. [4] S. Steić, Asymptotic periodicity of a higher-order difference equation. Discrete Dynamics in Nature Society Volume 2007 (2007), Article ID 3737, 9 pages. [5] S. Steić, On the recursie sequence x n + k α i x n pi / i m β j x n qj. Discrete Dynamics in Nature Society Volume 2007 (2007), Article ID 39404, 7 pages, 2007. [6] T. Sun H. Xi, The periodic character of the difference equation x n+ f(x n l+, x n 2k+ ), Adances in Difference Equations Volume 2008 (2008), Article ID 43723, 6 pages. j