Global Behavior of a Higher Order Rational Difference Equation

Transcription

1 International Journal of Difference Euations ISSN , Volume 10, Number 1, (2015) htt://camus.mst.edu/ijde Global Behavior of a Higher Order Rational Difference Euation Raafat Abo-Zeid The Valley Higher Institute of Engineering & Technology Deartment of Basic Science Cairo, Egyt abuzead73@yahoo.com Abstract The aim of this aer is to investigate the global stability the existence of unbounded solutions of the difference euation x n+1 = A + Bx n 2r 1 C + Dx n 2l n = 0, 1,... where the initial values are nonnegative real numbers A, B are nonnegative real numbers, C, D > 0 r, l, are nonnegative integers. AMS Subject Classifications: 39A20, 39A21, 39A30. Keywords: Difference euation, eriodic solution, unbounded solution. 1 Introduction Preliminaries The mathematical modeling of a hysical or economical roblem very often leads to difference euations (for artial review of the theory of difference euations their alications see [3, 6, 7, 9]). The study of nonlinear rational difference euations of higher order is of aramount imortance, since we still now so little about such euations. It is worthwhile to oint out that although several aroaches have been develoed for finding the global character of difference euations, a relatively a large number of difference euations has not been thoroughly understood yet [5, 7, 10, 11]. In [1], we have investigated the global asymtotic stability of the difference euation x n+1 = Bx n 2 1 C + D i=l x, n = 0, 1,... n 2i Received November 7, 2014; Acceted December 22, 2014 Communicated by Eugenia Petrooulou

2 2 R. Abo-Zeid Also in [4], we have discussed the global asymtotic stability of the difference euation x n+1 = Ax n 2r 1 B + Cx n 2l x n 2, n = 0, 1,... In this aer, we study the global asymtotic stability the existence of unbounded solutions of the difference euation x n+1 = A + Bx n 2r 1, n = 0, 1,... (1.1) C + Dx n 2l where A, B are nonnegative real numbers C, D > 0 r, l, are nonnegative integers. When r = l = 0 = 1, euation (1.1) is reduced to the difference euation x n+1 = A + Bx n 1 C + Dx n, n = 0, 1,... which has been investigated in [10]. When r = l = 0, = 2, euation (1.1) is reduced to the euation x n+1 = A + Bx n 1, n = 0, 1,... C + Dx 2 n which we have discussed in [2], where A, B are nonnegative real numbers C, D > 0. We give some reliminaries which will be needed in this aer. Consider the difference euation x n+1 = f(x n, x n 1,..., x n ), n = 0, 1,... (1.2) where f : R +1 R. Definition 1.1 (See [9]). An euilibrium oint for euation (1.2) is a oint x R such that x = f( x, x,..., x). Definition 1.2 (See [9]). 1. An euilibrium oint x for euation (1.2) is called locally stable if for every ɛ > 0, there exists a δ > 0 such that every solution {x n } with initial conditions x, x +1,..., x 0 ] x δ, x + δ[ is such that x n ] x ɛ, x + ɛ[ for all n N. Otherwise x is said to be unstable. 2. The euilibrium oint x of euation (1.2) is called locally asymtotically stable if it is locally stable there exists γ > 0 such that for any initial conditions x, x +1,..., x 0 ] x γ, x + γ[, the corresonding solution {x n } tends to x, a global attractor if every solution {x n } converges to x as n. 3. The euilibrium oint x for euation (1.2) is called globally asymtotically stable if it is locally asymtotically stable global attractor.

3 Global Behavior of a Higher order 3 Suose that f is continuously differentiable in some oen neighborhood of x. Let a i = f ( x,..., x), for i = 0, 1,..., x n i denote the artial derivatives of f(x n, x n 1,..., x n ) with resect to x n i evaluated at the euilibrium oint x of euation (1.2). Then the euation z n+1 = a i z n i, n = 0, 1,... (1.3) i=0 is called the linearized euation associated with euation (1.2) about the euilibrium oint x, the euation λ +1 a i λ i = 0 (1.4) i=0 is called the characteristic euation associated with euation (1.3) about the euilibrium oint x. Theorem 1.3 (See [9]). Assume that f is a C 1 function let x be an euilibrium oint of euation (1.2). Then the following statements are true: 1. If all roots of euation (1.4) lie in the oen dis λ < 1, then x is locally asymtotically stable. 2. If at least one root of euation (1.4) has absolute value greater than one, then x is unstable. Theorem 1.4 (See [8]). Assume that a i < 1. Then every root of euation (1.4) has absolute value less than one. i=0 Definition 1.5 (See [10]). A ositive semicycle of a solution {x n } n= of euation (1.2) consists of a string of terms {x l, x l+1,..., x m }, all greater than or eual to the euilibrium x, with l m such that either l =, either m =, or l > x l 1 < x or m < x m+1 < x. Definition 1.6 (See [10]). A negative semicycle of a solution {x n } n= of euation (1.2) consists of a string of terms {x l, x l+1,..., x m }, all less than or eual to the euilibrium x, with l m such that either l =, either m =, or l > x l 1 x or m < x m+1 x.

4 4 R. Abo-Zeid When = 0, Euation (1.1) reduces to the linear nonhomogeneous difference euation x n+1 = B C + D x n 2r 1 + A, n = 0, 1,... C + D A The euilibrium oint x = of this euation is globally asymtotically C + D B stable when B < C + D unstable when B C + D. When 1, the change of variables x n = C D y n reduces euation (1.1) to the difference euation y n+1 = + y n 2r 1, n = 0, 1,... (1.5) 1 + yn 2l where = A D C C, = B C. When = 1, euation (1.5) reduces to the difference euation y n+1 = + y n 2r y n 2l, n = 0, 1,... This euation was discussed in [7]. In the following, we assume that 2. 2 Linearized Stability Analysis Now we determine the euilibrium oints of euation (1.5) discuss their local asymtotic behavior. It is clear that the values of the euilibrium oints deends on. The euilibrium oints of euation (1.5) are the zeros of the function f(x) = x +1 + (1 )x. When > 1, euation (1.5) has a uniue ositive euilibrium oint ȳ > 1. When < 1, euation (1.5) has a uniue ositive euilibrium oint ȳ such that ȳ > ( ) +1 ( ) if > < ȳ < 1 1 if <. 1 Now assume that K = max{2l, 2r + 1}. Also let t be the largest nonnegative integer such that 0 < 2t + 1 K s be the largest nonnegative integer such that 0 2s K. The linearized euation associated with euation (1.5) about the ositive euilibrium ȳ is z n ȳ z n 2r 1 + ȳ 1 + ȳ z n 2l = 0, n = 0, 1,... (2.1) The characteristic euation associated with this euation is λ K ȳ λk 2r 1 + ȳ 1 + ȳ λk 2l = 0. (2.2)

5 Global Behavior of a Higher order 5 We summarize the results of this section in the following two theorems. Theorem 2.1. Assume that K = 2l let ȳ be the uniue ositive euilibrium oint of euation (1.5). Then the following statements are true. 1. If > 1, then ȳ is an unstable euilibrium oint. 2. If < 1, then we have the following: ( 1 (a) If < 1 ( 1 (b) If > 1 ) +1, then ȳ is locally asymtotically stable. ) +1, then ȳ is an unstable euilibrium oint. Proof. When K = 2l, the associated characteristic euation (2.2) becomes λ 2l ȳ λ2l 2r 1 + ȳ = 0. (2.3) 1 + ȳ 1. If > 1, then ȳ > 1. It follows that the characteristic euation (2.3) has a root in the interval (, 1) so ȳ is an unstable euilibrium oint. 2. Assume that < 1 let ȳ be the ositive euilibrium oint of euation (1.5). Then we have the following: (a) If < ( ) +1 1, then ȳ < 1. By Theorem (1.4), we have ȳ + ȳ 1 + ȳ = + ȳ 1 + ȳ < 1 the result follows. ( ) +1 1 (b) If >, then ȳ > 1. Therefore, euation (2.3) has a 1 1 root λ < 1 with λ > 1 so ȳ is an unstable euilibrium oint. Theorem 2.2. Assume that K = 2r + 1 let ȳ be the uniue ositive euilibrium oint of euation (1.5). Then the following statements are true. 1. If > 1, then ȳ is a saddle oint. 2. If < 1, then we have the following:

6 6 R. Abo-Zeid ( 1 (a) If < 1 ( 1 (b) If > 1 ) +1, then ȳ is locally asymtotically stable. ) +1, then ȳ is a saddle oint. Proof. When K = 2r + 1, the associated characteristic euation (2.2) becomes λ 2r+2 + ȳ 1 + ȳ λ2r+1 2l = 0. (2.4) 1 + ȳ 1. If > 1, then ȳ > 1. It is sufficient to see that the characteristic euation (2.4) has a root in the interval (0, 1) a other root in the interval (, 1). 2. Assume that < 1 let ȳ be the ositive euilibrium oint of euation (1.5). Then we have the following: ( ) +1 1 (a) If <, then ȳ < 1. By Theorem (1.4), we have ȳ + ȳ 1 + ȳ = + ȳ 1 + ȳ < 1 the result follows. ( ) +1 1 (b) If >, then ȳ > 1. Therefore, the characteristic 1 1 euation (2.4) has a root in the interval (0, 1) a other root in the interval (, 1). 3 Global Behavior of (1.5) In this section, we show that under a certain conditions, a solution of euation (1.5) oscillates with semicycles of length one. We study the global stability of the ositive euilibrium oint ȳ. Also we show the existence of unbounded solutions. Theorem 3.1. Assume that ȳ denote the uniue ositive euilibrium of euation (1.5) let {y n } n= K be a nontrivial solution of euation (1.5). If either one of the conditions (C 1 )y 2t 1, y 2t+1,..., y 1 < ȳ y 2s, y 2s+2,..., y 0 or (C 2 ) y 2s, y 2s+2,..., y 0 < ȳ y 2t 1, y 2t+1,..., y 1 is satisfied, then {y n } n= K oscillates about ȳ with semicycles of length one.

7 Global Behavior of a Higher order 7 ( ) +1 1 Theorem 3.2. Assume that < 1 <. Then the ositive euilibrium oint 0 < ȳ < is globally asymtotically stable Proof. Let {y n } n= K be a solution of euation (1.5). Then y n+1 = + y n 2r y n 2l < + y n 2r 1, n = 0, 1,.... Then there exists a real number β > 0 such that y n < β, n = 1, 2,.... This imlies that y n+1 = + y n y n 2l > 1 + β. Let λ = lim inf y n Λ = lim su y n. Hence we have + λ 1 + Λ λ Λ + Λ 1 + λ. This imlies that Then Then we get that That is + λ λ + λλ Λ + Λλ + Λ. λ 1 + λ λ + λ Λ Λ + Λ λ Λ 1 + Λ. λ 1 + λ ( 1) Λ 1 + Λ ( 1). (1 )λ λ 1 (1 )Λ Λ 1. (3.1) Consider the function h(x) = (1 )x x 1. We claim that ( 1) (1 ) < ȳ.

8 8 R. Abo-Zeid Proof of the claim: We have that, the euilibrium oint ȳ is the ositive zero of the function f(x) = x +1 + (1 )x. Now, ( 1) 1) ( 1) f( ) = (( (1 ) (1 ) )+1 + (1 ) (1 ) ( 1) ( 1) = ( (1 ) )+1 + ( 1) = ( (1 ) )+1 = (( ) ( 1 1 )+1 1) < (( 1 1 )+1 ( 1 1 )+1 1) = 0, ( < As the function f(x) is increasing everywhere when < 1, we get ( 1) (1 ) < ȳ. ( ) +1 1 ). 1 The claim is roved. ( 1) 1 Now, we have (1 ) < ȳ < 1) h(x) is increasing on (( 1 (1 ), ). In view of euation (3.1), we have a contradiction. Therefore λ = Λ = ȳ ȳ is a global attractor. In view of Theorem (2.1) Theorem (2.2), ȳ is globally asymtotically stable. This comletes the roof. Lemma 3.3. Assume that > 2 > 0. Then the following statements are true: 1. If x > If x > 1 y > 1, then 1 > x y, then y > x x. Theorem 3.4. Assume that > 2. Then euation (1.5) has solutions which are neither bounded nor ersist. Proof. Let {y n } n= K be a solution of euation (1.5) with initial conditions y 2t < y 2s < y 2s+2 <... < y 0 < < y 2t 1 < y 2t+1 <... < y 1.

9 Global Behavior of a Higher order 9 Then y 1 = + y 2r y 2l > + y 2r 1 = + y 2r 1, where y 2l < 1, y 2 = + y 2r < y 1 + y 2l+1 2r where y 2l+1 > 1 y 2r > y 2t > y 2l Now consider the subseuences {y (2r+2)n 2r+2j 1 } n=0 {y (2r+2)n 2r+2j } n=0, 0 j r. We claim that for each integer j such that 0 j r, {y (2r+2)n 2r+2j 1 } n=0 is monotonically increasing subseuence of {y n } n= K {y (2r+2)n 2r+2j } n=0 is monotonically decreasing subseuence of {y n } n= K. Proof of the claim: Consider the case when 0 r < l. For n = 1, we have for 0 j r where y 2j 2l < 1, y 2j+1 = + y 2j 2r y 2j 2l > + y 2j 2r 1 = + y 2j 2r 1, y 2j+2 = + y 2j 2r 1 + y 2j 2l+1 where y 2j 2l+1 > 1, y 2j 2r > Also, we have the following: < y 2j 2r, y 2t > y 2j 2l where y 0 < 1, y 2l+1 = + y 2l 2r y 0 > + y 2l 2r 1 = + y 2l 2r 1, y 2l+2 = + y 2l 2r 1 + y 1 < + y (2l 2r) 2r y 2r 1 = y 2l 2r, where y 1 > y 2r 1 y 2l 2r < y (2l 2r) 2r 2. Now suose that for a certain n we have < y (2r+2)(n 1) 2r+2j 1 < y (2r+2)n 2r+2j 1 y (2r+2)n 2r+2j < y (2r+2)(n 1) 2r+2j < 1.

10 10 R. Abo-Zeid Then y (2r+2)(n+1) 2r+2j 1 = + y (2r+2)n 2r+2j y (2r+2)n+2j 2l > + y (2r+2)(n 1) 2r+2j y (2r+2)(n 1)+2j 2l = y (2r+2)n 2r+2j 1, y (2r+2)(n+1) 2r+2j = + y (2r+2)n 2r+2j 1 + y (2r+2)n+2j 2l+1 < + y (2r+2)(n 1) 2r+2j 1 + y (2r+2)(n 1)+2j 2l+1 = y (2r+2)n 2r+2j. In the case when 0 l < r, there exist integers i m with m 1 0 i < l such that r = ml + i. The roof of the claim in this case is similar will be omitted. This comletes the roof of the claim. It follows by induction from the claim that for each nonnegative integer j with 0 j r y (2r+2)n 2r+2j 1 > + y (2r+2)(n 1) 2r+2j 1 > > n + y 2r+2j 1. This imlies that lim y (2r+2)n 2r+2j 1 =, 0 j r, n so Therefore lim y (2r+2)n 2r+2j = 0, 0 j r. n lim y 2n 1 = lim y 2n = 0. n n This comletes the roof. Acnowledgement The author is grateful to the anonymous referee for her/his hel constructive suggestions.

11 Global Behavior of a Higher order 11 References [1] R. Abo-Zeid, Global asymtotic stability of a higher order difference euation, Bull. Allahabad Math. Soc. 25 (2) (2010), [2] R. Abo-Zeid, Global asymtotic stability of a second order rational difference euation, J. Al. Math. & Inform. 28 (3) (2010), [3] R.P. Agarwal, Dfference euations ineualities, First edition, Marcel Deer, [4] M.A. Al-Shabi, R. Abo-Zeid, Global asymtotic stability of a higher order difference euation, Al. Math. Sci., 4 (17) (2010), [5] E, Camouzis G. Ladas, Dynamics of third-order rational Difference Euations; With Oen Problems Conjectures, Chaman Hall/HRC Boca Raton, [6] S. N. Elaydi, An Introduction to Difference Euations, Undergraduate Texts in Mathematics, Sringer, New Yor, NY, USA, [7] E.A. Grove G. Ladas, Periodicities in Nonlinear Difference Euations, Chaman Hall/CRC, [8] E.A. Grove, G. Ladas, M. Predescu, M. Radin, On the global character of the difference euation x n+1 = α + γx n δx n 2l A + x n 2l J. Diff. E. A., 9 (2) (2003), [9] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Euations of Higher Order with Alications, Kluwer Academic, Dordrecht, [10] M.R.S. Kulenović G. Ladas, Dynamics of second order rational Difference Euations; With Oen Problems Conjectures, Chaman Hall/HRC Boca Raton, [11] H. Sedaghat, Nonlinear Difference Euations, Theory Alications to Social Science Models, Kluwer Academic Publishers, Dordrecht, 2003.