Asymptotic Behavior of a Higher-Order Recursive Sequence

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1 International Journal of Difference Equations ISSN , Volume 7, Number 2, pp (202) Asymptotic Behavior of a Higher-Order Recursive Sequence Özkan Öcalan Afyon Kocatepe University Faculty of Science Arts Department of Mathematics Afyonkarahisar, Turkey ozkan@aku.edu.tr Abstract In this paper, we investigate the boundedness character the global behavior of positive solutions of the difference equation + = p n + k, n = 0,,, where k N {p n } is a sequence of nonnegative real numbers which converges to p the initial conditions x k,, x 0 are arbitrary positive real numbers. AMS Subject Classifications: 39A0. Keywords: Asymptotic behavior, boundedness, global behavior, recursive sequence. Introduction Our aim in this paper is to study the boundedness character the global asymptotic behavior of positive solutions of the difference equation + = p n + k, n = 0,,, (.) where k N {p n } is a sequence of nonnegative real numbers which converges to p the initial conditions x k,, x 0 are arbitrary positive real numbers. Eq. (.) was studied by many authors with k =. Received June 29, 202; Accepted November 4, 202 Communicated by Martin Bohner

2 76 Özkan Öcalan When k =, in [6] the author studied asymptotic behavior of the positive solutions of the difference equation + = p n +, n = 0,,, (.2) where {p n } is a sequence of nonnegative real numbers which converges to p the initial conditions x, x 0 are arbitrary positive real numbers. It was prove in [6] that if lim p n = p >, then every positive solution of Eq. (.2) is bounded converges to p +, if 0 p <, then there exist solutions of Eq. (.2) that are unbounded. In [3 5] the authors studied the Eq. (.2) where {p n } is a positive two periodic sequence. In [2], when {p n } is a positive bounded sequence, the authors obtained conditions for the boundedness persistence of solutions for the existence of unbounded solutions, they also obtained global attractivity results for Eq. (.2). For the autonomous case of Eq. (.) we can refer to []. The paper is organized as follows. In Section 2 we investigate the boundedness character of positive solutions of Eq. (.). We prove that if k is odd 0 p <, then there exist unbounded solutions of Eq. (.) when the case k N p, then every positive solution of Eq. (.) is bounded. Section 3 is devoted to the global attractivity results of the positive solutions of Eq. (.). We show that when the case k N p >, then every positive solution of Eq. (.) converges to (p + ). 2 Boundedness Character of Eq. (.) In this section, we investigate the boundedness character of Eq. (.). We show that if p, then every positive solution of Eq. (.) is bounded that when k is odd 0 p <, then there exist unbounded solutions of Eq. (.). Theorem 2.. Suppose that lim p n = p, then every positive solution of Eq. (.) is bounded. Proof. First, we assume that p >. et ε (0, p ), from (.) one can see that p ε for n. Now, we shall prove that { } is bounded. Assume without loss of generality that > p ε for n = k, k +,, 0. Then one can find (p ε, p ε + ) such that p + ε for n = k, k +,, 0.

3 Asymptotic Behavior of a Recursive Sequence 77 Set f(u, v) = p ε + v u. Note that since p > ( ) f, p + ε p + ε ( ) f p + ε, =. Now x = f(x 0, x k ) f (, ) p + ε p + ε. In a similar way it is true that x. By induction we obtain that p + ε for n = k, k +,. The proof is complete. Now, we assume that p =. et ε (0, δ) δ (0, ), since lim p n =, from (.) one can see that ε + δ for n. Then one can find ( ε + δ, 2 ε + δ) such that + ε + δ for n = k, k +,, 0. Therefore, the rest of the proof is similar the above it is omitted. Now we study the boundedness character of Eq. (.) for the case k is odd 0 p <. We prove that in this case, there exist unbounded solutions of Eq. (.) Theorem 2.2. Consider Eq. (.) when the case k is odd. If 0 p <, then there exist solutions of Eq. (.) that are unbounded. Proof. It is clear that since lim p n = p, we may assume p ε < p n < p + ε, where ε is a sufficiently small arbitrary positive number. Now, let 0 < p <, δ (0, p), ε (0, δ). Choose the initial conditions such that p < x k+, x k+3,, x 0 < p+δ x k, x k+2,, x > p δ > +p+δ. Then, x = p 0 + x k x 0 x 2 = p + x k+ x > p ε + x k p + δ > p ε + x k < p + ε + p + δ x < p + ε + x < p + ε <.

4 78 Özkan Öcalan Further we have x 3 = p 2 + x k+2 x 2 > p ε + x k+2 p + δ > p ε + x k+2 x 4 = p 3 + x k+3 x 3 < p + ε + p + δ x 3 < p + ε + x 3 < p + ε <. Therefore, we obtain p ε < x k+ < x k+2 > 2 (p ε) + x k. By induction, for i =, 2,, we have p ε < x (k+)i < x (k+)i+ > (i + ) (p ε) + x k. Thus, lim x (k+)i+ = lim x (k+)i = p. i i Now, let p = 0, δ (0, ), ε (0, δ) choose the initial conditions such that x k, x k+2,, x > ( δ ε) 0 < x k+, x k+3,, x 0 <. So, we have Further we have x > x k x 0 > ( δ ε) x 2 < ε + x k+ x < ε + x < ε + δ ε = δ. x 3 > x k+2 x 2 > ( δ)( δ ε) > ( δ ε) Therefore, we obtain x 4 < ε + x k+3 x 3 < ε + x 3 < ε + δ ε = δ Thus, we obtain The proof is complete. 0 < x 2n < δ x 2n+ > ( δ) n ( δ ε). lim x 2n+ = lim x 2n = 0.

5 Asymptotic Behavior of a Recursive Sequence 79 3 Global Attractivity of Eq. (.) In this section, when k N, we show that if p >, then every positive solution of Eq. (.) converges to (p + ). Theorem 3.. Consider Eq. (.) when the case k N. Assume that p >. Then every positive solution of Eq. (.) converges to (p + ). Proof. Since p >, by Theorem 2. every positive solution of Eq. (.) is bounded, then we have the following Then it is easy to see from Eq. (.) that Thus, we have This implies that Then, we get s = lim inf S = lim sup. s p + s S S p + S s. ss ps + s Ss ps + S. ps + s Ss ps + S. p(s s) (S s). Since p >, if s < S, then we arrive a contradiction. Thus, the proof is complete. References [] R. Devault, C. Kent W. Kosmala, On the recursive sequence + = p + ( k / ), J. Difference Equ. Appl., 9(8):72 730, [2] R. Devault, V.. Kocić D. Stutson, Global behavior of solutions of the nonlinear difference equation + = p n +, J. Difference Equ. Appl., (8):707 79, [3] M. R. S. Kulenović, G. adas C. B. Overdeep, On the dynamics of + = p n + ( / ), J. Difference Equ. Appl., 9(): , [4] M. R. S. Kulenović, G. adas C. B. Overdeep, On the dynamics of + = p n +( / ) with a period-two coefficient, J. Difference Equ. Appl., 0(0):905 94, 2004.

6 80 Özkan Öcalan [5] S. Stević, On the recursive sequence + = α n + ( / ) II, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 0:9 96, [6] S. Stević, On the recursive sequence + = α n + ( / ), Int. J. Math. Sci., 2(2): , 2003.