Global attractivity in a rational delay difference equation with quadratic terms

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1 Journal of Difference Equations and Applications ISSN: (Print) (Online) Journal homepage: Global attractivity in a rational delay difference equation with quadratic terms C.M. Kent & H. Sedaghat To cite this article: C.M. Kent & H. Sedaghat (2011) Global attractivity in a rational delay difference equation with quadratic terms, Journal of Difference Equations and Applications, 17:04, To link to this article: Published online: 28 Mar Submit your article to this journal Article views: 100 View related articles Citing articles: 9 View citing articles Full Terms & Conditions of access and use can be found at Download by: [VCU Libraries Serials] Date: 20 February 2017, At: 13:02

2 Journal of Difference Equations and Applications Vol. 17, No. 4, April 2011, Global attractivity in a rational delay difference equation with quadratic terms C.M. Kent and H. Sedaghat* Department of Mathematics, Virginia Commonwealth University, Richmond, VA , USA (Received 8 December 2008; final version received 16 May 2009) For the following rational difference equation with arbitrary delay and quadratic terms: x nþ1 ¼ Ax2 n þ Bx nx n2k þ Cx 2 n2k þ Dx n þ Ex n2k þ F ; ax n þ bx n2k þ g we determine sufficient conditions on the parameter values which guarantee that the unique non-negative fixed point attracts all positive solutions. When the fixed point is the origin (F ¼ 0), we show that it attracts all non-negative solutions of the more general equation x nþ1 ¼ Ax2 n þ Bx nx n2k þ Cx 2 n2k þ D 1x n þ D 2 x n21 þ þ D m x n2mþ1 ; ax n þ bx n2k þ g where m [ {1; 2;...}. We also show that altering some of the above conditions on parameters causes the origin to not be globally attracting. Keywords: rational; delay; quadratic terms; global stability 2000 Mathematics Subject Classification: 39A10; 39A11 1. Introduction Consider the rational difference equation x nþ1 ¼ Ax2 n þ Bx nx n2k þ Cx 2 n2k þ Dx n þ Ex n2k þ F ; ð1þ ax n þ bx n2k þ g where k [ {1; 2;...}, all parameters (coefficients and initial values) are non-negative and A þ B þ C. 0; a þ b. 0: ð2þ Note that if all initial values x 0 ; x 21 ;...; x 2k are positive, then all solutions of (1) are positive. For over a decade rational equations with linear expressions in both the numerator and the denominator have been studied methodically and extensively; see, for example Refs. [2,5,9]. However, rational equations with quadratic terms in the numerator or the denominator have not been studied systematically. These equations exhibit a rich variety of dynamic behaviours and offer substantial insights into rational difference equations. *Corresponding author. hsedagha@vcu.edu ISSN print/issn online q 2011 Taylor & Francis DOI: /

3 458 C.M. Kent and H. Sedaghat For k ¼ 1, the dynamics of equation (1) were studied in Refs. [3,4]. In this paper, we first determine sufficient conditions on the parameter values that for arbitrary k imply the existence of a unique, positive fixed point for (1) that attracts all of its positive solutions. When the fixed point is the origin, i.e. F ¼ 0, the results in Ref. [7] extend the conditions in Ref. [3] for the global attractivity of the origin to arbitrary k. In this paper, we extend the conditions in Ref. [3] in a different direction to the more general equation x nþ1 ¼ Ax2 n þ Bx nx n2k þ Cx 2 n2k þ D 1x n þ D 2 x n21 þ þ D m x n2mþ1 ; ð3þ ax n þ bx n2k þ g where m [ {1; 2;...}, all the parameters are non-negative and (2) holds. We obtain sufficient conditions on the parameters that imply the origin is the unique non-negative fixed point of (3) which is globally asymptotically stable relative to ½0; 1Þ m. Here, we may assume without loss of generality that m. k in (3) since we may set D j ¼ 0 for various values of j and any value of m. For general background and definitions of basic concepts, we refer to texts such as Refs. [5,8,9,10]. 2. Global attractivity of the positive fixed point First, we need the following coordinate-wise monotonicity result from Ref. [6]. Lemma 1. Let I be an open interval of real numbers and suppose that f [ CðI m ; RÞ is nondecreasing in each coordinate. Let x [ I be a fixed point of the difference equation x nþ1 ¼ f ðx n ; x n21 ;...; x n2mþ1 Þ; ð4þ and assume that the function hðtþ ¼f ðt;...; tþ satisfies the conditions hðtþ. t if t, x and hðtþ, t if t. x; t [ I: ð5þ Then I is an invariant interval of (4) and x attracts all solutions with initial values in I. With m ¼ k þ 1 and I ¼ð0; 1Þ, we can apply Lemma 1 to equation (1) as in the next result. We note that the nontrivial case without quadratic terms, i.e. is included in the hypotheses of this result. A ¼ B ¼ C ¼ 0; a þ b. 0 Theorem 1. Assume that all parameters in (1) are non-negative and satisfy the following conditions: ac # bb; ba # ab; af # gd; bf # ge; jbd 2 aej # gb; ð6þ 0 # A þ B þ C, a þ b; g # D þ E with F. 0 ifg ¼ D þ E: ð7þ Then (1) has a unique fixed point x. 0 that attracts all positive solutions.

4 Journal of Difference Equations and Applications 459 Proof. We first show that if the inequalities (6) hold then the function f ðu 1 ;...; u kþ1 Þ¼ Au2 1 þ Bu 1u kþ1 þ Cu 2 kþ1 þ Du 1 þ Eu kþ1 þ F ð8þ is nondecreasing in each of its coordinates u 1 ;...; u kþ1 (i.e. f is coordinate-wise monotonic). Since f in (8) is independent of u 2 ;...; u k hence nondecreasing in those coordinates, we need only prove monotonicty in coordinates i ¼ 1; k þ 1. For this purpose, we compute the partial derivatives f i ¼ f = u i and determine when each is non-negative. For i ¼ 1, direct calculation shows that f 1 ¼ f = u 1 $ 0 iff aau 2 1 þ 2bAu 1u kþ1 þðbb 2 acþu 2 kþ1 þ 2gAu 1 þðgb þ bd 2 aeþu kþ1 þ gd 2 af $ 0: The above inequality holds for u 1 ; u kþ1. 0if ac # bb; gb þ bd 2 ae $ 0; af # gd: ð9þ Similarly, f kþ1 ¼ f = u kþ1 $ 0 if and only if bcu 2 kþ1 þ 2aCu 1u kþ1 þðab 2 baþu 2 1 þ 2gCu kþ1 þðgb þ ae 2 bdþu 1 þ ge 2 bf $ 0 which is true for all u 1 ; u kþ1. 0if ba # ab; gb þ ae 2 bd $ 0; bf # ge: ð10þ The middle inequalities in (9) and (10) combine into the single inequality jbd 2 aej # gb. Therefore, we have shown that conditions (6) are sufficient for f to be nondecreasing in each of its coordinates. Next, assume that (7) holds and define a ¼ A þ B þ C a þ b ; b ¼ D þ E a þ b ; c ¼ F a þ b ; d ¼ g a þ b : Then the function h in (5) takes the form hðtþ ¼ at 2 þ bt þ c : t þ d Now x is a fixed point of (1) if and only if x is a solution of the equation hðtþ ¼t, i.e. ð1 2 aþx 2 2 ðb 2 dþx 2 c ¼ 0: ð11þ Since by (7) a, 1 and d # b with c. 0ifd ¼ b, a unique positive fixed point is obtained as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ b2dþ ðb2dþ 2 þ4ð12aþc 2ð12aÞ ; or multiplying by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ DþE2gþ ðdþe2gþ 2 þ4ðaþb2a2b2cþf 2ðaþb2A2B2CÞ : aþb aþb ; ð12þ

5 460 C.M. Kent and H. Sedaghat Next, we verify that conditions (5) hold. Note that h may be written as hðtþ ¼fðtÞt; where fðtþ ¼ at þ b þ c=t : t þ d Since f 0 ðtþ ¼ ad 2 b 2 ðc=tþð2 þ d=tþ ðt þ dþ 2 and by (7) ad 2 b, d 2 b # 0, it follows that f is decreasing (strictly) for all t [ I. Therefore, with fðxþ ¼ hðxþ=x ¼ 1 we find that t, x implies hðtþ. fðxþt ¼ t; t. x implies hðtþ, fðxþt ¼ t: Now using Lemma 1 completes the proof. A Remarks. 1. Periodic solutions. If some of the inequalities in Theorem 1 do not hold then its conclusion is easily seen to be false. In particular, if A ¼ D ¼ F ¼ 0 and B ¼ a; C ¼ b; E ¼ g; ð13þ then the inequalities in (6) hold but those in (7) do not. If (13) holds then equation (1) reduces to x nþ1 ¼ x n2k in which case every positive solution (non-negative solution if g. 0) is periodic with period k þ 1: 2. Inequalities (6) are not necessary for global attractivity. In Ref. [1], a special case of (1) is discussed where A ¼ C ¼ D ¼ E ¼ g ¼ 0; a ¼ b ¼ B. 0; F $ 0: ð14þ p ffiffiffi In this case, it is shown that every positive solution converges to the unique fixed point F. Note that conditions (14) contradict some of the inequalities in (6) if F Global asymptotic stability of the origin When F ¼ 0 and g. 0 in (1) then the origin is a fixed point of (1) and the class of solutions can be expanded to include the non-negative solutions. The question arises as to whether the origin can be globally attracting for the non-negative solutions. In this section, we study the global attractivity of the origin for the more general equation (3). We need the following result from Ref. [10] (Theorem 4.3.1) that for convenience we quote as a lemma. Recall that a fixed point x is asymptotically stable relative to a set S if x is locally stable and attracts all orbits with their initial points in S.

6 Journal of Difference Equations and Applications 461 Lemma 2. Let x be a fixed point of the difference equation (4) in a closed, invariant set T, R m and define the set M as {ðu 1 ;...; u m Þ : j f ðu 1 ;...; u m Þ 2 xj, max{ju 1 2 xj;...; ju m 2 xj}} < {ðx;...; xþ}: Then ðx;...; xþ is asymptotically stable relative to the largest invariant subset S of M > T such that S is closed in T. Theorem 2. Assume that the following conditions are satisfied for equation (3) for some d [ ½0; 1Š: A þ db # a; C þð1 2 dþb # b; D 1 þ þ D m, g: ð15þ Then the origin is the unique non-negative fixed point of (3) and it is asymptotically stable relative to ½0; 1Þ m. Proof. Let D ¼ D 1 þ þ D m. The origin is clearly a fixed point of (3) since g. D $ 0 by (15). Also by (15), A þ B þ C # a þ b. If this inequality strict then there exists a negative fixed point g 2 D x ¼ 2 2ða þ b 2 A 2 B 2 CÞ ; while if A þ B þ C ¼ a þ b then there are no nonzero fixed points. It follows that the origin is the only non-negative fixed point of (3). Next, if gðu 1 ;...; u m Þ¼ Au2 1 þ Bu 1u kþ1 þ Cu 2 kþ1 þ D 1u 1 þ þ D m u m with m. k then defining m ¼ max{u 1 ;...; u m } for all u 1 ;...; u m $ 0 gðu 1 ;...; u m Þ # B min{u 1; u kþ1 }max{u 1 ; u kþ1 } þ mðau 1 þ Cu kþ1 þ DÞ # m ½Bd þ Bð1 2 dþšmin{u 1; u kþ1 } þ Au 1 þ Cu kþ1 þ D # m ða þ dbþu 1 þ½c þð1 2 dþbšu kþ1 þ D, m; if ðu 1 ;...; u m Þ ð0;...; 0Þ; where the strict inequality holds because of conditions (15). Now we apply Lemma 2 with S ¼ M ¼ T ¼½0; 1Þ m to conclude the proof. A Next, we show that even when the origin is the only non-negative fixed point in equation (3) or its special case (1), it may not be globally attracting under certain conditions. Note that one of the inequalities in (15) is contradicted in Theorem 3 below, which we present after a few preliminaries. For further details in this direction,

7 462 C.M. Kent and H. Sedaghat see Ref. [7]. Recall from (8) that f ðu 1 ;...; u kþ1 Þ¼ Au2 1 þ Bu 1u kþ1 þ Cu 2 kþ1 þ Du 1 þ Eu kþ1 þ F : Also the partial derivatives f u1 and f ukþ1 of f used in the next result are f u1 ðu 1 ; u 2 ;...; u kþ1 Þ¼ Aau 2 1 þ 2Abu 1u kþ1 þ 2Agu 1 þðbb 2 CaÞu 2 kþ1 þðbg þ Db 2 EaÞu kþ1 þ Dg ðþ 2 ð16þ and f ukþ1 ðu 1 ; u 2 ;...; u kþ1 Þ¼ Cbu 2 kþ1 þ 2Cau 1u kþ1 þ 2Cgu kþ1 þðba 2 AbÞu 2 1 þðbg þ Ea 2 DbÞu 1 þ Eg ðþ 2 : ð17þ Definition Let N [ ð0; 1Þ kþ1 denotes the region in which the partial derivatives of f satisfy f u1 $ 0 and f ukþ1 $ Let V [ ð0; 1Þ kþ1 denotes the region in which f u1 # Let Y denotes the set {ðu 1 ; u 2 ;...; u k ; u kþ1 Þ [ ð0; 1Þ kþ1 : u 1 ¼ 0}. Lemma 3. Assume that the following inequalities hold for the parameters of equation (1): F ¼ 0; D þ E, g; A þ B þ C # a þ b: Then, in Y, we have f ð0; u 2 ;...; u k ; aþ # f ð0; u 2 ;...; u k ; bþ, for any 0 # a, b. Proof. Let ðu 1 ; u 2 ;...; u kþ1 Þ [ Y. Then u 1 ¼ 0. If we let u kþ1 ¼ a and we assume that 0, a, b, then f ð0; u 2 ;...; u k ; aþ # f ð0; u 2 ;...; u k ; bþ, where f ð0; u 2 ;...; u k ; aþ # f ð0; u 2 ;...; u k ; bþ, Ca 2 þ Ea ba þ g # Cb 2 þ Eb bb þ g, Cba 2 b þ Cga 2 þ Ebab þ Ega # Cbab 2 þ Cgb 2 þ Ebab þ Egb, Cbabðb 2 aþþcgðb þ aþðb 2 aþþegðb 2 aþ $ 0; which is true by assumption (with equality holding if C ¼ E ¼ 0). A

8 Journal of Difference Equations and Applications 463 Theorem 3. Assume that the following inequalities hold for the parameters of equation (1): F ¼ 0; D þ E, g; A þ B þ C # a þ b; b, C; ac, bða þ BÞ; Bb, Ca or Bg þ Db 2 Ea, 0; Ba $ Ab and Bg þ Ea 2 Db $ 0: Then lim x n ¼ 1 n!1 for every solution {x n } of (1) with initial values x 0 ; x 21 ;...; x 2k [ ðm; 1Þ where m ¼ max g 2 E C 2 b ; p; q ; and p and q are defined as follows: 8 p < ffiffiffiffi 2ð2AgþBgþDb2EaÞþ D p 2ðAaþ2AbþBb2CaÞ ; D p $ 0; p ¼ : 0; D p, 0; and 8 p < ffiffiffiffi 2ðAgþBgþDb2EaÞþ D q 2ðAbþBb2CaÞ ; D q $ 0; q ¼ : 0; D q, 0; with D p ¼ð2Ag þ Bg þ Db 2 EaÞ 2 2 4ðAa þ 2Ab þ Bb 2 CaÞDg; D q ¼ðAg þ Bg þ Db 2 EaÞ 2 2 4ðAb þ Bb 2 CaÞDg: Proof. Under our assumptions f ukþ1 $ 0onð0; 1Þ kþ1. We observe that the condition Ca, bða þ BÞ and equation (16) imply the following: (O1): f u1 ðu 1 ; u 2 ;...; u k ; u kþ1 Þ. 0 for all u 1 $ u kþ1. p, and thus N. {ðu 1 ; u 2 ;...; u k ; u kþ1 Þ [ ð0; 1Þ kþ1 : u 1 $ u kþ1. p}; where N is as defined in Definition 1. (O2): Let u 1 ¼ u kþ1 ¼ v. q and let u 2 ;...; u k. 0 be arbitrary. Then f ðv; u 2 ;...; u k ; u k ; vþ. f ð0; u 2 ;...; u k ; vþ; where f ðv; u 2 ;...; u k ; vþ. f ð0; u 2 ;...; u k ; vþ, ða þ B þ CÞv 2 þðd þ EÞv ða þ bþv þ g. Cv 2 þ Ev bv þ g,ðaþbþcþvðbvþgþþðdþeþðbv þ gþ. ðcv þ EÞ½ða þ bþv þ gš,ðab þ Bb 2 CaÞv 2 þðag þ Bg þ Db 2 EaÞv þ Dg. 0;

9 464 C.M. Kent and H. Sedaghat which is true since the condition that Ca, bða þ BÞ holds (so that Ab þ Bb 2 Ca. 0) and by the definition of q We make further observations. (O3): Let V; N be as defined in Definition 1. Note that V < N ¼ð0; 1Þ kþ1 1. Suppose that ðu 1 ; u 2 ;...; u k ; u kþ1 Þ [ V. Then f u1 # 0, by definition, but also f ukþ1 $ 0, where the conditions that b, C; Bb, Ca or Bg þ Db 2 Ea, 0; and Ba $ Ab and Bg þ Ea 2 Db $ 0 all hold. Hence, if u 1, u kþ1, then f ðu 1 ;...; u k ; u kþ1 Þ. f ðu 1 ; u 2 ;...; u k ; u 1 Þ) f ðu 1 ;...; u k ; u kþ1 Þ. min{f ðu 1 ;...; u k ; u 1 Þ; f ðu kþ1 ; u 2 ;...; u kþ1 Þ}: 2. Suppose that ðu 1 ; u 2 ;...; u k ; u kþ1 Þ [ N (so that f u1 $ 0; f ukþ1 $ 0). The following is true: (a) If u 1, u kþ1, then f ðu 1 ; u 2 ;...; u k ; u kþ1 Þ. f ðu 1 ; u 2 ;...; u k ; u 1 Þ $ min{f ðu 1 ;...; u k ; u 1 Þ; f ðu kþ1 ; u 2 ;...; u kþ1 Þ}: (b) If u 1 $ u kþ1, then f ðu 1 ; u 2 ;...; u k ; u kþ1 Þ $ f ðu kþ1 ; u 2 ;...; u k ; u kþ1 Þ $ min{f ðu 1 ;...; u k ; u 1 Þ; f ðu kþ1 ; u 2 ;...; u kþ1 Þ}: 3. Thus, in general, f ðu 1 ; u 2 ;...; u k ; u kþ1 Þ $ min{f ðu 1 ; u 2 ;...; u k ; u 1 Þ; f ðu kþ1 ; u 2 ;...; u k ; u kþ1 Þ} for ðu 1 ; u 2 ;...; u k ; u kþ1 Þ [ ð0; 1Þ kþ1 (O4): Let u 1 ¼ 0 and u kþ1 ¼ v. ðg 2 EÞ=ðC 2 bþ, and let u 2 ;...; u k [ ð0; 1Þ be arbitrary. Then f ð0; u 2 ;...; u k ; vþ. v, since given the condition that b, C, we have f ð0; u 2 ;...; u k ; vþ. v, Cv 2 þ Ev bv þ g. v, g 2 E C 2 b, v: (O5): m. 0 since ðg 2 E=C 2 bþ. 0 by the conditions that D þ E, g and b, C.

10 Journal of Difference Equations and Applications 465 Combining Observations (O1) (O4), we have that, for u 1 ; u kþ1. m and u 2 ;...; u k [ ð0; 1Þ, f ðu 1 ; u 2 ;...; u k ; u kþ1 Þ $ min{f ðu 1 ; u 2 ;...; u k ; u 1 Þ; f ðu kþ1 ; u 2 ;...; u k ; u kþ1 Þ}. min{f ð0; u 2 ;...; u k ; u 1 Þ; f ð0; u 2 ;...; u k ; u kþ1 Þ}. min{u 1 ; u kþ1 } $ min{u 1 ; u 2 ;...; u k ; u kþ1 } Therefore, in summary, we can say that f ðu 1 ; u 2 ;...; u k ; u kþ1 Þ. min{u 1 ; u 2 ;...; u k ; u kþ1 }; ðu 1 ; u 2 ;...; u kþ1 Þ [ ðm; 1Þ kþ1 : for ð18þ Now let {x n } 1 n¼2k be a positive solution of equation (1), with k [ {1; 2;...}, and suppose that x 2k ;...; x 21 ; x 0 [ ðm; 1Þ. Next define x * n ¼def min{x n ; x n21 ;...; x n2k }; for all n $ 0: Clearly, min{x n ; x n2k } $ min{x n ; x n21 ;...; x n2k }. Thus, given equation (18) and the assumption that ðx 0 ; x 21 ;...; x 2k Þ [ ðm; 1Þ kþ1, we have By induction, x 1 ¼ f ðx 0 ; x 21 ;...; x 2k Þ. min{x 0 ; x 21 ;...; x 2k } ¼ x * 0. m: x nþ1 ¼ f ðx n ; x n21 ;...; x n2k Þ. min{x n ; x n21 ;...; x n2k } ¼ x * n. m; for all n $ 0; ð19þ and so equation (18) holds for ðu 1 ; u 2 ;...; u kþ1 Þ¼ðx n ; x n21 ;...; x n2k Þ for all n $ 0. Also, from equation (19), we have x * nþ1 ¼ min{x nþ1; x n ;...; x n2kþ1 } $ x * n, for all n $ 0. This means that {x * n }1 n¼0 is a nondecreasing sequence, and so there exists m, L # 1 such that lim n!1 x * n ¼ L and x* n # Lðx* n, L if L ¼ 1Þ for all n $ 0. We show that L ¼ 1 by contradiction. Assume that L, 1. Let us define the function f : ½0; 1Þ! R as follows: fðxþ¼ def ½bðL þ xþ 2 CðL 2 xþšðl 2 xþþgðl þ xþ 2 EðL 2 xþ: Then, from the conditions in the hypotheses and the fact that L is positive (by Observation (O5)), we have that fð0þ ¼½b 2 CŠL 2 þ½g 2 EŠL, 0: By the continuity of f on R, there exists 1. 0, 1, L such that fð1þ, 0. Thus, we have the following: CðL 2 1Þ 2 þ EðL 2 1Þ bðl 2 1Þþg Now, there exists N $ 0 such that for all n $ N,. L þ 1: ð20þ x n2k ;...; x n21 ; x n $ x * n. L 2 1: ð21þ

11 466 C.M. Kent and H. Sedaghat It follows from equations (20) and (21) and Lemma 3 that Cx 2 n2k þ Ex n2k bx n2k þ g $ CðL 2 1Þ2 þ EðL 2 1Þ bðl 2 1Þþg. L þ 1: Hence, from observation (O4) by induction, x n2k ;...; x n21 ; x n. L þ 1 for all n $ N; and so L ¼ lim n!1 x * n ¼ lim n!1 min{x n; x n21 ;...; x n2k } $ lim n!1 min{l þ 1; L þ 1;...; L þ 1} ¼ L þ 1; which is impossible and the proof is complete. A References [1] R. Abu-Saris, C. Cinar, and I. Yalcinkaya, On the asymptotic stability of x n þ 1 ¼ (a þ x n x n2k )/(x n þ x n2k ), Comp. Math. Appl. 56 (2008), pp [2] E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC Press, Boca Raton, FL, [3] M. Dehghan, C.M. Kent, R. Mazrooei-Sebdani, N.L. Ortiz, and H. Sedaghat, Dynamics of rational difference equations containing quadratic terms, J. Differ. Equ. Appl. 14 (2008), pp [4] M. Dehghan, C.M. Kent, R. Mazrooei-Sebdani, N.L. Ortiz, and H. Sedaghat, Monotone and oscillatory solutions of a rational difference equation containing quadratic terms, J. Differ. Equ. Appl. 14 (2008), pp [5] E.A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC Press, Boca Raton, FL, [6] M.L.J. Hautus and T.S. Bolis, Solution to problem E 2721 [1978, 496], Am. Math. Monthly 86 (1979), pp [7] C.M. Kent and H. Sedaghat, Global attractivity in a quadratic-linear rational difference equation with delay, J. Differ. Equ. Appl. 15 (2009), pp [8] V.L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, [9] M.R.S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC Press, Boca Raton,FL, [10] H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, 2003.