Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 463169, 11 pages doi:10.1155/2009/463169 Research Article On Boundedness of Solutions of the Difference Equation xn pxn qxn−1 / xn for q > p > Hongjian Xi,1, Taixiang Sun,1 Weiyong Yu,1 and Jinfeng Zhao1 Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, China Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China Correspondence should be addressed to Taixiang Sun, stx1963@163.com Received February 2009; Revised 19 April 2009; Accepted June 2009 Recommended by Agacik Zafer We study the boundedness of the difference equation xn pxn qxn−1 / xn , n 0, 1, , where q > p > and the initial values x−1 , x0 ∈ 0, ∞ We show that the solution {xn }∞ −1 of n this equation converges to x q p − if xn ≥ x or xn ≤ x for all n ≥ −1; otherwise {xn }∞ −1 is n unbounded Besides, we obtain the set of all initial values x−1 , x0 ∈ 0, ∞ × 0, ∞ such that the positive solutions {xn }∞ −1 of this equation are bounded, which answers the open problem 6.10.12 n proposed by Kulenovi´ and Ladas 2002 c Copyright q 2009 Hongjian Xi et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction In this paper, we study the following difference equation: xn pxn qxn−1 , xn n 0, 1, , 1.1 where p, q ∈ 0, ∞ with q > p and the initial values x−1 , x0 ∈ 0, ∞ The global behavior of 1.1 for the case p q < is certainly folklore It can be found, for example, in see also a precise result in The global stability of 1.1 for the case p q follows from the main result in see also Lemma in Stevi´ ’s paper Some generalizations of Copson’s result can be found, c for example, in papers 5–8 Some more sophisticated results, such as finding the asymptotic behavior of solutions of 1.1 for the case p q even when p can be found, for Advances in Difference Equations example, in papers see also 8–11 Some other properties of 1.1 have been also treated in The case q p was treated for the first time by Stevi´ ’s in paper 12 The main trick c from 12 has been later used with a success for many times; see, for example, 13–15 Some existing results for 1.1 are summarized as follows 16 Theorem A If p q ≤ 1, then the zero equilibrium of 1.1 is globally asymptotically stable If q 1, then the equilibrium x p of 1.1 is globally asymptotically stable If < q < p, then every positive solution of 1.1 converges to the positive equilibrium x p q − If q p, then every positive solution of 1.1 converges to a period-two solution If q > p, then 1.1 has unbounded solutions In 16 , Kulenovi´ and Ladas proposed the following open problem c Open problem B (see Open problem 6.10.12of [16]) Assume that q ∈ 1, ∞ a Find the set B of all initial conditions x−1 , x0 ∈ 0, ∞ × 0, ∞ such that the solutions {xn }∞ −1 of 1.1 are bounded n b Let x−1 , x0 ∈ B Investigate the asymptotic behavior of {xn }∞ −1 n In this paper, we will obtain the following results: let p, q ∈ 0, ∞ with q > p, and let {xn }∞ −1 be a positive solution of 1.1 with the initial values x−1 , x0 ∈ 0, ∞ × 0, ∞ n If xn ≥ x for all n ≥ −1 or xn ≤ x for all n ≥ −1 , then {xn }∞ −1 converges to x q p − n Otherwise {xn }∞ −1 is unbounded n For closely related results see 17–34 Some Definitions and Lemmas In this section, let q > p > and x q p − be the positive equilibrium of 1.1 Write D 0, ∞ × 0, ∞ and define f : D → D by, for all x, y ∈ D, f x, y y, py qx y It is easy to see that if {xn }∞ −1 is a solution of 1.1 , then f n x−1 , x0 n Let A1 0, x × 0, x , A3 0, x × x, ∞ , A2 A4 2.1 xn−1 , xn for any n ≥ x, ∞ × x, ∞ , x, ∞ × 0, x , R0 {x} × 0, x , L0 {x} × x, ∞ , R1 0, x × {x}, L1 x, ∞ × {x} 2.2 Advances in Difference Equations Then D ∪4 Ai ∪ L0 ∪ L1 ∪ R0 ∪ R1 ∪ { x, x } The proof of Lemma 2.1 is quite similar to i that of Lemma in 35 and hence is omitted Lemma 2.1 The following statements are true f : D → f D is a homeomorphism f L1 L0 and f L0 ⊂ A4 f R1 {x} × p, x and f R0 ⊂ A3 f A3 ⊂ A4 and f A4 ⊂ A3 A2 ∪ L1 ⊂ f A2 ⊂ A2 ∪ L1 ∪ A4 and A1 ∪ R1 ⊂ f A1 ⊂ A1 ∪ R1 ∪ A3 p > 1, and let {xn }∞ −1 be a positive solution of 1.1 n Lemma 2.2 Let q > 1 If limn → ∞ x2n a ∈ 0, ∞ and a / p, then limn → If limn → ∞ x2n−1 ∞ x2n b ∈ 0, ∞ and b / p, then limn → a b ∞ x2n x x Proof We show only because the proof of follows from by using the change yn xn−1 and the fact that is autonomous Since limn → ∞ x2n a ∈ 0, ∞ and a / p, by 1.1 we have lim x2n n→ ∞ lim n→ ∞ q−1 a a−p qx2n − x2n x2n − p 2.3 Also it follows from 1.1 that a lim x2n n→ ∞ from which we have a qx2n−1 − x2n ∞ x2n − p lim n→ x and limn → ∞ x2n a q−1 a q−1 a−p a−p , 2.4 x This completes the proof Lemma 2.3 Let q > p > 1, and let {xn }∞ −1 be a positive solution of 1.1 with the initial values n x−1 , x0 ∈ A4 If there exists some n ≥ such that x2n−1 ≥ x2n , then x2n ≥ x2n Proof Since x−1 , x0 ∈ A4 , it follows from Lemma 2.1 that x2n−1 , x2n ∈ A4 for any n ≥ Without loss of generality we may assume that n 0, that is, x−1 ≥ x1 Now we show x0 ≥ x2 Suppose for the sake of contradiction that x0 < x2 , then x−1 ≥ x1 x0 < x2 px0 qx−1 , x0 px1 qx0 x1 2.5 2.6 By 2.5 we have x0 ≥ x−1 q − , x−1 − p 2.7 Advances in Difference Equations and by 2.6 we get q − − qx−1 x0 p2 q − − p x0 pqx−1 > 2.8 Claim If x−1 ≥ x, then p2 q − − qx−1 − q − − p pqx−1 ≥ 2.9 Proof of Claim Let g x p2 q − − qx − q − − p pqx x ≥ x , then we have g x qx − p2 − q − 4pq q − − p 2q ≥ 2q q − p2 p 1−q 2q q − q − p − p p 2.10 p > Since x−1 ≥ x, it follows p2 q − − qx−1 ≥ q2 qp − 2q q2 − 2q qp 2 − q − − p pqx−1 − p2 − p2 − q − − p qp q 2qp q2 − 2q − q − 2q q2 − 2q 2 − p2 − pq 1−p p−1 − p2 2.11 pq ≥ This completes the proof of Claim By 2.8 , we have x0 > λ1 qx−1 − p2 − q qx−1 − p2 − q q−1−p − 4pq q − − p x−1 2.12 Advances in Difference Equations or x0 < λ2 qx−1 − p2 − q − qx−1 − p2 − q − 4pq q − − p x−1 q−1−p 2.13 Claim We have λ1 ≥ x, 2.14 x−1 q − x−1 − p 2.15 λ2 ≤ Proof of Claim Since q q p − − p2 − q q2 − p2 − 2q q − 4pq q − − p p q−1 − qp 2.16 p−1 q−1−p − q q p − − p2 − q , we have λ1 qx−1 − p2 − q qx−1 − p2 − q − 4pq q − − p x−1 q−1−p ≥ qx − p2 − q qx − p2 − q − 4pq q − − p x q−1−p q q p − − p2 − q q q p − − p2 − q − 4pq q − − p p q−1 q−1−p ≥ q p−1 x 2.17 The proof of 2.14 is completed Now we show 2.15 Let h x pq x − p − x−p q−1 qx − p2 − q q−1 q − − p x 2.18 Advances in Difference Equations Note that 2pq − 2q q − < 0; it follows that if x ≥ x, then h x qx − p2 − q 2pq x − p − q−1 ≤ 2pq q − − q−1 2pq − q − p2 q−1 q p p q q−1 x−p − q−1 q−1−p q2 − p − q < 0, 2.19 which implies that h x is decreasing for x ≥ x Since x−1 ≥ x and pq q − − q−1 q−1 q−1 hx q−1−p q q q p−1 p − − p2 − q 2.20 0, it follows that pq x−1 − p h x−1 q−1 − x−1 − p q − qx−1 − p2 − q q − − p x−1 ≤ h x 2.21 Thus q−1 qx−1 − p2 − q ≥ 4p2 q2 x−1 − p q−1 2 − 4pq q − − p x−1 − 4pq x−1 − p q − qx−1 − p2 − q qx−1 − p2 − q 2.22 This implies that q−1 qx−1 − p2 − q ≥ 2pq x−1 − p − q − − 4pq q − − p x−1 2.23 qx−1 − p2 − q Finally we have x−1 q − ≥ x−1 − p q − − p pqx−1 q−1−p 1 qx−1 − p2 − q qx−1 − p2 − q − 1 qx−1 − p2 − q q−1−p qx−1 − p2 − q 2 − 4pq q − − p x−1 − 4pq q − − p x−1 λ2 2.24 The proof of 2.15 is completed Advances in Difference Equations Note that x0 < x since x−1 , x0 ∈ A4 By 2.12 , 2.13 , 2.14 , and 2.15 , we see x0 < x−1 q − / x−1 − p , which contradicts to 2.7 The proof of Lemma 2.3 is completed Main Results In this section, we investigate the boundedness of solutions of 1.1 Let q > p > 1, and let {xn }∞ −1 be a positive solution of 1.1 with the initial values x−1 , x0 ∈ 0, ∞ × 0, ∞ , n then we see that xn − x xn − x < for some n ≥ −1 or xn ≥ x for all n ≥ −1 or xn ≤ x for all n ≥ −1 Theorem 3.1 Let q > p > 1, and let {xn }∞ −1 be a positive solution of 1.1 such that xn ≥ x for n all n ≥ −1 or xn ≤ x for all n ≥ −1, then {xn }∞ −1 converges to x q p − n Proof Case < xn ≤ x for any n ≥ −1 If < x2n ≤ q − for some n, then x2n − x2n−1 px2n qx2n−1 − x2n−1 − x2n−1 x2n > x2n 3.1 If q − < x2n ≤ x for some n, then px2n px ≥ x2n − q x − q which implies that px2n ≥ x2n−1 x2n − q x2n − x2n−1 px2n x ≥ x2n−1 , and qx2n−1 − x2n−1 − x2n−1 x2n ≥ x2n Thus x ≥ x2n ≥ x2n−1 for any n ≥ In similar fashion, we can show x ≥ x2n n ≥ Let limn → ∞ x2n a and limn → ∞ x2n b, then a which implies a b x pb 3.2 qa , b b pa qb , a 3.3 ≥ x2n for any 3.4 Advances in Difference Equations Case xn ≥ x p q − for any n ≥ −1 Since f x, y decreasing in y, it follows that for any n ≥ −1, xn py qx / y x > p/q is pxn qxn xn 3.5 px qxn ≤ ≤ xn x In similar fashion, we can show that limn → proof ∞ x2n limn → ∞ x2n x This completes the Lemma 3.2 see 20, Theorem Let I be a set, and let F : I × I → I be a function F u, v which decreases in u and increases in v, then for every positive solution {xn }n∞ of equation xn −1 F xn , xn−1 , {x2n }∞ and {x2n−1 }∞ exactly one of the following n n They are both monotonically increasing They are both monotonically decreasing Eventually, one of them is monotonically increasing, and the other is monotonically decreasing Remark 3.3 Using arguments similar to ones in the proof of Lemma 3.2, Stevi´ proved c Theorem in 25 Beside this, this trick have been used by Stevi´ in 18, 28, 29 c Theorem 3.4 Let q > p > 1, and let {xn }∞ −1 be a positive solution of 1.1 such that xn n x xn − x < for some n ≥ −1, then {xn }∞ −1 is unbounded n − Proof We may assume without loss of generality that x0 − x x−1 − x < and x−1 , x0 ∈ A4 the proof for x−1 , x0 ∈ A3 is similar From Lemma 2.1 we see x2n−1 , x2n ∈ A4 for all n ≥ 0.If {x2n }∞ is eventually increasing, then it follows from Lemma 2.3 that {x2n−1 }∞ is n n eventually increasing Thus limn → ∞ x2n−1 b > x and limn → ∞ x2n a ≤ x, it follows from Lemma 2.2 that b ∞ If {x2n }∞ is not eventually increasing, then there exists some N ≥ such that n x2N ≥ x2N px2N qx2N , x2N 3.6 from which we obtain x2N ≥ px2N / x2N − q ≥ p, since x2N ≥ x p q − and q > Since f y, x py qx / y p qx − p / y x ≥ p, y ≥ p is increasing in x and is decreasing in y, we have that x2n ≥ p for any n ≥ N It follows from Lemma 3.2 that {x2n }∞ is eventually decreasing Thus limn → ∞ x2n a < x and limn → ∞ x2n−1 b ≥ x n It follows from Lemma 2.2 that b ∞ This completes the proof By Theorems 3.1 and 3.4 we have the following Corollary 3.5 Let q > p > 1, and let {xn }∞ −1 be a positive bounded solution of 1.1 , then n xn−1 ≥ xn ≥ x for all n ≥ or x ≥ xn ≥ xn−1 for all n ≥ Advances in Difference Equations Now one can find out the set of all initial values x−1 , x0 ∈ 0, ∞ × 0, ∞ such that the positive solutions of 1.1 are bounded Let P0 A2 , Q0 A1 For any n ≥ 1, let Pn f −1 Pn−1 , It follows from Lemma 2.1 that P1 Qn f −1 Qn−1 f −1 P0 ⊂ P0 , Q1 Pn ⊂ Pn−1 , Qn ⊂ Qn−1 3.7 f −1 Q0 ⊂ Q0 , which implies 3.8 for any n ≥ Let S be the set of all initial values x−1 , x0 ∈ 0, ∞ × 0, ∞ such that the positive solutions {xn }∞ −1 of 1.1 are bounded Then we have the following theorem n Theorem 3.6 S ∞ n Qn ∪ ∞ n Pn ⊂ A1 ∪ A2 ∪ { x, x } Proof Let {xn }∞ −1 be a positive solution of 1.1 with the initial values x−1 , x0 ∈ S n xn−1 , xn ∈ A1 for any n ≥ 0, which implies If x−1 , x0 ∈ ∞ Qn , then f n x−1 , x0 n xn ≤ x for any n ≥ −1 It follows from Theorem 3.1 that limn → ∞ xn x xn−1 , xn ∈ A2 , which implies xn ≥ x for any If x−1 , x0 ∈ ∞ Pn , then f n x−1 , x0 n n ≥ −1 It follows from Theorem 3.1 that limn → ∞ xn x Now assume that {xn }∞ −1 is a positive solution of 1.1 with the initial values n x−1 , x0 ∈ D − S If x−1 , x0 ∈ A3 A4 L0 L1 R0 R1 , then it follows from Lemma 2.1 that x1 , x2 ∈ { x, y : x − x y − x < 0}, which along with Theorem 3.4 implies f x−1 , x0 that {xn } is unbounded If x−1 , x0 ∈ A2 − ∞ Pn , then there exists n ≥ such that x−1 , x0 ∈ Pn − Pn n xn−1 , xn ∈ A2 − f −1 A2 By Lemma 2.1, we obtain f −n A2 − f −n−1 A2 Thus f n x−1 , x0 n n x−1 , x0 ∈ L1 A4 and f x−1 , x0 xn , xn ∈ A4 , which along with Theorem 3.4 f implies that {xn } is unbounded If x−1 , x0 ∈ A1 − ∞ Qn , then there exists n ≥ such that x−1 , x0 ∈ Qn − Qn Qn − n xn−1 , xn ∈ A1 − f −1 A1 Again by Lemma 2.1 and Theorem 3.4, f −1 Qn and f n x−1 , x0 we have that {xn } is unbounded This completes the proof Acknowledgment Project Supported by NNSF of China 10861002 and NSF of Guangxi 0640205, 0728002 References M R Taskovi´ , Nonlinear Functional Analysis Vol I: Fundamental Elements of Theory, Zavod, za c udˇ benike i nastavna sredstva, Beograd, Serbia, 1993 z S Stevi´ , “Behavior of the positive solutions of the generalized Beddington-Holt equation,” c PanAmerican Mathematical Journal, vol 10, no 4, pp 77–85, 2000 E T Copson, “On a generalisation of monotonic sequences,” Proceedings of the Edinburgh Mathematical Society Series 2, vol 17, pp 159–164, 1970 S Stevi´ , “Asymptotic behavior of a sequence defined by iteration with applications,” Colloquium c Mathematicum, vol 93, no 2, pp 267–276, 2002 10 Advances in Difference Equations S Stevi´ , “A note on bounded sequences satisfying linear inequalities,” Indian Journal of Mathematics, c vol 43, no 2, pp 223–230, 2001 S Stevi´ , “A generalization of the Copson’s theorem concerning sequences which satisfy a linear c inequality,” Indian Journal of Mathematics, vol 43, no 3, pp 277–282, 2001 S Stevi´ , “A global convergence result,” Indian Journal of Mathematics, vol 44, no 3, pp 361–368, 2002 c S Stevi´ , “A note on the recursive sequence xn c pk xn pk−1 xn−1 · · · p1 xn−k ,” Ukrainian Mathematical Journal, vol 55, no 4, pp 691–697, 2003 S Stevi´ , “On the recursive sequence xn xn−1 /g xn ,” Taiwanese Journal of Mathematics, vol 6, no c 3, pp 405–414, 2002 10 S Stevi´ , “Asymptotics of some classes of higher-order difference equations,” Discrete Dynamics in c Nature and Society, vol 2007, Article ID 56813, 20 pages, 2007 11 S Stevi´ , “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics c Letters, vol 20, no 1, pp 28–31, 2007 12 S Stevi´ , “On the recursive sequence xn g xn , xn−1 / A xn ,” Applied Mathematics Letters, vol 15, c pp 305–308, 2002 13 K S Berenhaut, J E Dice, J D Foley, B Iriˇ anin, and S Stevi´ , “Periodic solutions of the rational c c difference equation yn yn−3 yn−4 /yn−1 ,” Journal of Difference Equations and Applications, vol 12, no 2, pp 183–189, 2006 14 S Stevi´ , “Periodic character of a class of difference equation,” Journal of Difference Equations and c Applications, vol 10, no 6, pp 615–619, 2004 15 S Stevi´ , “On the difference equation xn c α βxn−1 γxn−2 f xn−1 , xn−2 /xn ,” Dynamics of Continuous, Discrete & Impulsive Systems, vol 14, no 3, pp 459–463, 2007 16 M R S Kulenovi´ and G Ladas, Dynamics of Second Order Rational Difference Equations, with Open c Problems and Conjectures, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2002 17 A M Amleh, E A Grove, G Ladas, and D A Georgiou, “On the recursive sequence xn α xn−1 /xn ,” Journal of Mathematical Analysis and Applications, vol 233, no 2, pp 790–798, 1999 18 K S Berenhaut and S Stevi´ , “The behaviour of the positive solutions of the difference equation c xn A xn−2 /xn−1 p ,” Journal of Difference Equations and Applications, vol 12, no 9, pp 909–918, 2006 19 E Camouzis and G Ladas, “When does periodicity destroy boundedness in rational equations?” Journal of Difference Equations and Applications, vol 12, no 9, pp 961–979, 2006 20 E Camouzis and G Ladas, “When does local asymptotic stability imply global attractivity in rational equations?” Journal of Difference Equations and Applications, vol 12, no 8, pp 863–885, 2006 21 R Devault, V L Kocic, and D Stutson, “Global behavior of solutions of the nonlinear difference equation xn pn xn−1 /xn ,” Journal of Difference Equations and Applications, vol 11, no 8, pp 707– 719, 2005 22 J Feuer, “On the behavior of solutions of xn p xn−1 /xn ,” Applicable Analysis, vol 83, no 6, pp 599–606, 2004 23 M R S Kulenovi´ , G Ladas, and N R Prokup, “On the recursive sequence xn c αxn βxn−1 / xn ,” Journal of Difference Equations and Applications, vol 6, no 5, pp 563–576, 2000 24 S Stevi´ , “Asymptotic behavior of a nonlinear difference equation,” Indian Journal of Pure and Applied c Mathematics, vol 34, no 12, pp 1681–1687, 2003 25 S Stevi´ , “On the recursive sequence xn c αn xn−1 /xn II,” Dynamics of Continuous, Discrete & Impulsive Systems Series A, vol 10, no 6, pp 911–916, 2003 p p 26 S Stevi´ , “On the recursive sequence xn α xn−1 /xn ,” Journal of Applied Mathematics & Computing, c vol 18, no 1-2, pp 229–234, 2005 k m α 27 S Stevi´ , “On the recursive sequence xn c i αi xn−pi / j βj xn−qj ,” Journal of Difference Equations and Applications, vol 13, no 1, pp 41–46, 2007 28 S Stevi´ , “On the difference equation xn αn xn−1 /xn ,” Computers & Mathematics with Applications, c vol 56, no 5, pp 1159–1171, 2008 29 S Stevi´ and K S Berenhaut, “The behavior of positive solutions of a nonlinear second-order c difference equation,” Abstract and Applied Analysis, vol 2008, Article ID 653243, pages, 2008 30 T Sun and H Xi, “Global asymptotic stability of a family of difference equations,” Journal of Mathematical Analysis and Applications, vol 309, no 2, pp 724–728, 2005 31 T Sun, H Xi, and Z Chen, “Global asymptotic stability of a family of nonlinear recursive sequences,” Journal of Difference Equations and Applications, vol 11, no 13, pp 1165–1168, 2005 Advances in Difference Equations 11 32 T Sun and H Xi, “On the system of rational difference equations xn f yn−q , xn−s , yn g xn−t , yn−p ,” Advances in Difference Equations, vol 2006, Article ID 51520, pages, 2006 33 T Sun, H Xi, and L Hong, “On the system of rational difference equations xn f yn , xn−k , yn f xn , yn−k ,” Advances in Difference Equations, vol 2006, Article ID 16949, pages, 2006 34 H Xi and T Sun, “Global behavior of a higher-order rational difference equation,” Advances in Difference Equations, vol 2006, Article ID 27637, pages, 2006 35 T Sun and H Xi, “On the basin of attraction of the two cycle of the difference equation xn xn−1 / p qxn xn−1 ,” Journal of Difference Equations and Applications, vol 13, no 10, pp 945–952, 2007  ... 4pq q − − p p q? ? ?1 − qp 2 .16 p? ? ?1 q? ? ?1? ? ?p − q q p − − p2 − q , we have ? ?1 qx? ?1 − p2 − q qx? ?1 − p2 − q − 4pq q − − p x? ?1 q? ? ?1? ? ?p ≥ qx − p2 − q qx − p2 − q − 4pq q − − p x q? ? ?1? ? ?p q q p − − p2 − q q q p. .. x qx − p2 − q − 4pq q − − p 2q ≥ 2q q − p2 p 1? ? ?q 2q q − q − p − p p 2 .10 p > Since x? ?1 ≥ x, it follows p2 q − − qx? ?1 ≥ q2 qp − 2q q2 − 2q qp 2 − q − − p pqx? ?1 − p2 − p2 − q − − p qp q 2qp q2 ... − q Finally we have x? ?1 q − ≥ x? ?1 − p q − − p pqx? ?1 q? ? ?1? ? ?p 1 qx? ?1 − p2 − q qx? ?1 − p2 − q − 1 qx? ?1 − p2 − q q? ?1? ? ?p qx? ?1 − p2 − q 2 − 4pq q − − p x? ?1 − 4pq q − − p x? ?1 λ2 2.24 The proof of 2 .15