ON A SECOND ORDER RATIONAL DIFFERENCE EQUATION

Transcription

1 Hacttp Journal of Mathmatics and Statistics Volum 41(6) (2012), ON A SECOND ORDER RATIONAL DIFFERENCE EQUATION Nourssadat Touafk Rcivd 06:07:2011 : Accptd 26:12:2011 Abstract In this papr, w invstigat th stability of th following diffrnc quation x n+1 = ax4 n +bx nx 3 n 1 +cx 2 nx 2 n 1 +dx 3 nx n 1 +x 4 n 1, Ax 4 n +Bx nx 3 n 1 +Cx2 nx 2 n 1 +Dx3 nx n 1 +Ex 4 n 1 n = 0,1,..., whr th paramtrs a, b, c, d,, A, B, C, D, E ar positiv ral numbrs and th initial valus x 0,x 1 ar arbitrary positiv numbrs. Kywords: Diffrnc quations, Global stability AMS Classification: 39 A Introduction and prliminaris Considr th following scond-ordr diffrnc quation (1.1) x n+1 = ax4 n +bx nx 3 n 1 +cx 2 nx 2 n 1 +dx 3 nx n 1 +x 4 n 1, n = 0,1,..., Ax 4 n +Bx nx 3 n 1 +Cx2 nx 2 n 1 +Dx3 nx n 1 +Ex 4 n 1 whr th initial conditions x 0, x 1 (0, ) and th paramtrs a, b, c, d,, A, B, C, D, E (0, ). In this papr w study th global stability of th uniqu positiv quilibrium point, and th bounddnss and th convrgnc of th solutions of Equation (1.1). Nonlinar diffrnc quations appar naturally, for xampl, from crtain modls in cology, conomy, automatic control thory, and thy ar of grat importanc in applications whr th (n+1) st stat of th modl dpnds on th prvious k stats. Rcntly, thr has bn a lot of attntion givn to studying th global bhavior of nonlinar diffrnc quations by many authors, S for xampl [1-3,5-19] and th rfrncs citd thrin. Now, w rviw som dfinitions (s for xampl [11-12]), which will b usful in th squl. Dpartmnt of Mathmatics, Jijl Univrsity, Algria. nstouafk@yahoo.fr

2 868 N. Touafk Lt I b an intrval of ral numbrs and lt F : I I I b a continuously diffrntiabl function. Considr th diffrnc quation (1.2) x n+1 = F(x n,x n 1) with initial valus x 1, x 0 I Dfinition. A point x I is calld an quilibrium point of (1.2) if x = F(x,x) Dfinition. Lt x b an quilibrium point of (1.2). Th quilibrium x is calld locally stabl if for vry ǫ > 0, thr xist δ > 0 such that for all x 1, x 0 I with x 1 x + x 0 x < δ, w hav x n x < ǫ, for all n 1. Th quilibrium x is calld locally asymptotically stabl if it is locally stabl, and if thr xists γ > 0 such that if x 1, x 0 I and x 1 x + x 0 x < γ thn lim xn = x. n + Th quilibrium x is calld a global attractor if for all x 1, x 0 I, w hav lim xn = x. n + Th quilibrium x is calld global asymptotically stabl if it is locally stabl and a global attractor. Th quilibrium x is calld unstabl if it is not stabl. Lt p = F x (x,x) and q = F y (x,x). Thn th quation (1.3) y n+1 = py n +qy n 1, n = 0,1,... is calld th linarizd quation of (1.2) about th quilibrium point x. Th nxt rsult, which was givn by Clark [4], provids a sufficint condition for th locally asymptotically stability of (1.2) Thorm. Considr th diffrnc quation (1.3). Thn, p + q < 1 is a sufficint condition for th locally asymptotically stability of (1.2) Dfinition. Th diffrnc quation (1.2) is said to b prmannt if thr xist numbrs α,β with 0 < α β < such that for any initial valus x 1, x 0 I thr xists a positiv intgr N which dpnds on th initial conditions such that α x n β for all n N. 2. Main rsults Lt us dfin th following ral numbrs: r 1 = ab ba, r 2 = ac ca, r 3 = ad da, r 4 = ae A, r 5 = be B, r 6 = cb bc, r 7 = ce C, r 8 = db bd, r 9 = dc cd, r 10 = de D Rmark. Equation (1.1) has a uniqu positiv quilibrium point which is givn by x = a+b+c+d+ A+B +C +D +E. Lt f : (0,+ ) 2 (0,+ ) b th function dfind by f(x,y) = ax4 +bxy 3 +cx 2 y 2 +dx 3 y +y 4 Ax 4 +Bxy 3 +Cx 2 y 2 +Dx 3 y +Ey 4.

3 On a Scond Ordr Rational Diffrnc Equation Lmma. (1) Assum that (a) a max( d, c, b ), (b) min( d, c, b ), (c) 3r 1 +r 9 0, (d) 2r 4 +r 8 0, () r 6 +3r Thn f is incrasing in x for ach y and it is dcrasing in y for ach x. (2) Assum that (a) a min( d, c, b ), (b) max( d, c, b ), (c) 3r 1 +r 9 0, (d) 2r 4 +r 8 0, () r 6 +3r Thn f is dcrasing in x for ach y and it is incrasing in y for ach x. Proof. (1) W hav, 3r 1 +r 9 0, 2r 4 +r 8 0 and r 6 +3r Using th fact that, a max( d, c, b ) and min( d, c, b ), w gt: r2, r3, r5, r7 0. Now, th rsult follows from th formulas f x (x,y) = r 3yx 6 + 2r 2 y 2 x 5 + (r 9 + 3r 1 )y 3 x 4 + (4r 4 + 2r 8 )y 4 x 3 + (3r 10 + r 6 )y 5 x 2 + (2r 7 )y 6 x + r 5 y 7 (Ax 4 + Bxy 3 + Cx 2 y 2 + Dx 3 y + Ey 4 ) 2, f y (x,y) = r 3x 7 2r 2 yx 6 (r 9 + 3r 1 )y 2 x 5 (4r 4 + 2r 8 )y 3 x 4 (3r 10 + r 6 )y 4 x 3 2r 7 y 5 x 2 r 5 y 6 x (Ax 4 + Bxy 3 + Cx 2 y 2 + Dx 3 y + Ey 4 ) 2. (2) Th proof of (2) is similar and will b omittd. Th locally stability of th positiv quilibrium point x = a+b+c+d+ A+B+C+D+E dscribd in th following thorm Thorm. Assum that 3r1 +2r2 +r3 +4r4 +r5 +r6 +2r7 +2r8 +r9 +3r10 2 < 1. (a+b+c+d+)(a+b +C +D +E) Thn th positiv quilibrium point x = a+b+c+d+ A+B+C+D+E stabl. Proof. Th linarizd quation of (1.1) about x = a+b+c+d+ A+B+C+D+E is whr and x n+1 = px n +qx n 1 p = q = 3r1 +2r2 +r3 +4r4 +r5 +r6 +2r7 +2r8 +r9 +3r10 (a+b+c+d+)(a+b +C +D+E) 3r1 +2r2 +r3 +4r4 +r5 +r6 +2r7 +2r8 +r9 +3r10. (a+b+c+d+)(a+b +C +D+E) By using Thorm 1.3, w gt that x is locally asymptotically stabl if of (1.1) is of (1.1) is locally asymptotically 3r1 +2r2 +r3 +4r4 +r5 +r6 +2r7 +2r8 +r9 +3r10 2 < 1. (a+b+c+d+)(a+b +C +D +E)

4 870 N. Touafk Th nxt thorm is dvotd to th prmannc of th diffrnc quation (1.1) Thorm. Lt {x n} + n= 1 b a positiv solution of quation (1.1). (1) Assum that (a) a max( d, c, b ), (b) min( d, c, b ). Thn, E xn a A for all n 1. (2) Assum that (a) a min( d, c, b ), (b) max( d, c, b ). Thn, a A xn E for all n 1. Proof. (1) W hav x n+1 a A = x n+1 E = r 3x n 1x 3 n r 2x 2 n 1x 2 n r 1x 3 n 1x n r 4x 4 n 1 A(Ax 4 n +Bx nx 3 n 1 +Cx2 nx 2 n 1 +Dx3 nx n 1 +Ex 4 n 1 ), r 4x 4 n +r 10x n 1x 3 n +r 7x 2 n 1x 2 n +r 5x 3 n 1x n E(Ax 4 n +Bx nx 3 n 1 +Cx2 nx 2 n 1 +Dx3 nx n 1 +Ex 4 n 1 ). Now, it suffics to gt r 1, r 2, r 3, r 4, r 5, r 7, r 10 0, which rsult from a A max( d D, c C, b B ) and E min( d D, c C, b B ). (2) Similarly w can asily prov (2). Hr w study th global asymptotic stability of quation (1.1) Thorm. Lt Assum that p 1 = (A + D)+ Ab Ea, p 2 = (A + C + D) Ed + Ac + (A + D)b (B + E)a, p 3 = (A + B + C + D) + (A B E)d+(A + D E)c+(A + C + D)b (B + C + E)a, p 4 = (A + B + C + D + E) + (A B C + D E)d + (A B + C + D E)c + (A + B + C + D E)b + (A B C D E)a. (1) a max( d, c, b ), (2) min( d, c, b ), (3) 3r 1 +r 9 0, (4) 2r 4 +r 8 0, (5) r 6 +3r 10 0, (6) 2 3r 1+2r 2 +r 3 +4r 4 +r 5 +r 6 +2r 7 +2r 8 +r 9 +3r 10 < 1, (a+b+c+d+)(a+b+c+d+e) (7) p 1, p 2, p 3, p 4 0. Thn th quilibrium point x = a+b+c+d+ A+B+C+D+E of (1.1) is globally asymptotically stabl. Proof. Lt {x n} + n= 1 b a solution of quation (1.1). In viw of thorm 2.3 w nd only to prov that x is a global attractor. Lt m = lim n + infxn

5 On a Scond Ordr Rational Diffrnc Equation 871 and To prov that M = lim n + supxn. lim xn = x, n + it suffics to show that m = M. Lt ǫ ]0,m[ thn thr xist n 0 N such that for all n n 0 w gt m ǫ x n M +ǫ. Thus by using Lmma 2.2, Part (1); w gt for all n n 0 +1 x n+1 a(m ǫ)4 + b(m ǫ)(m + ǫ) 3 + c(m ǫ) 2 (M + ǫ) 2 + d(m ǫ) 3 (M + ǫ) + (M + ǫ) 4 A(m ǫ) 4 + B(m ǫ)(m + ǫ) 3 + C(m ǫ) 2 (M + ǫ) 2 + D(m ǫ) 3 (M + ǫ) + E(M + ε) 4, x n+1 a(m + ǫ)4 + b(m + ǫ)(m ǫ) 3 + c(m + ǫ) 2 (m ǫ) 2 + d(m + ǫ) 3 (m ǫ) + (m ǫ) 4 A(M + ǫ) 4 + B(M + ǫ)(m ǫ) 3 + C(M + ǫ) 2 (m ǫ) 2 + D(M + ǫ) 3 (m ǫ) + E(m ǫ) 4. Thn w gt th following inqualitis m a(m ǫ)4 + b(m ǫ)(m + ǫ) 3 + c(m ǫ) 2 (M + ǫ) 2 + d(m ǫ) 3 (M + ǫ) + (M + ǫ) 4 A(m ǫ) 4 + B(m ǫ)(m + ǫ) 3 + C(m ǫ) 2 (M + ǫ) 2 + D(m ǫ) 3 (M + ǫ) + E(M + ε) 4, M a(m + ǫ)4 + b(m + ǫ)(m ǫ) 3 + c(m + ǫ) 2 (m ǫ) 2 + d(m + ǫ) 3 (m ǫ) + (m ǫ) 4 A(M + ǫ) 4 + B(M + ǫ)(m ǫ) 3 + C(M + ǫ) 2 (m ǫ) 2 + D(M + ǫ) 3 (m ǫ) + E(m ǫ) 4. Ths inqualitis yild So, Hnc m am4 +bmm 3 +cm 2 M 2 +dm 3 M +M 4 Am 4 +BmM 3 +Cm 2 M 2 +Dm 3 M +EM 4, M am4 +bmm 3 +cm 2 m 2 +dm 3 m+m 4 AM 4 +BMm 3 +CM 2 m 2 +DM 3 m+em 4. mm M am4 +bmm 3 +cm 2 M 2 +dm 3 M +M 4 Am 4 +BmM 3 +Cm 2 M 2 +Dm 3 M +EM 4, mm m am4 +bmm 3 +cm 2 m 2 +dm 3 m+m 4 AM 4 +BMm 3 +CM 2 m 2 +DM 3 m+em 4. M am4 + bmm 3 + cm 2 M 2 + dm 3 M + M 4 Am 4 + BmM 3 + Cm 2 M 2 + Dm 3 M + EM 4 m am 4 + bmm 3 + cm 2 m 2 + dm 3 m + m 4 AM 4 + BMm 3 + CM 2 m 2 + DM 3 m + Em 4 0 which can b writtn Sinc w gt (M m) A(m8 + M 8 ) + p 1 mm(m 6 + M 6 ) + p 2 m 2 M 2 (M 4 + m 4 ) + p 3 m 3 M 3 (M 2 + m 2 ) + p 4 m 4 M 4 (Am 4 + BmM 3 + Cm 2 M 2 + Dm 3 M + EM 4 )(AM 4 + BMm 3 + CM 2 m 2 + DM 3 m + Em 4 ) 0. A(m 8 + M 8 ) + p 1 mm(m 6 + M 6 ) + p 2 m 2 M 2 (M 4 + m 4 ) + p 3 m 3 M 3 (M 2 + m 2 ) + p 4 m 4 M 4 (Am 4 + BmM 3 + Cm 2 M 2 + Dm 3 M + EM 4 )(AM 4 + BMm 3 + CM 2 m 2 + DM 3 m + Em 4 ) > 0 M m. So, m = M = x. By th sam argumnts, w can prov th following thorm.

6 872 N. Touafk 2.6. Thorm. Lt Assum that q 1 = A+Ed + (B + E)a, q 2 = (A + D) + (B + E)d+Ec Ab + (C + B + E)a, q 3 = (A + C + D)+ (B + C + E)d+( A + B + E)c + ( A D + E)b + (B + C + D + E)a, q 4 = ( A B C D + E) + ( A+ B + C + D + E)d+ ( A+ B + C D + E)c + ( A + B C D + E)b + (A + B + C + D + E)a. (1) a min( d, c, b ), (2) max( d, c, b ), (3) 3r 1 +r 9 0, (4) 2r 4 +r 8 0, (5) r 6 +3r 10 0, (6) 2 3r 1+2r 2 +r 3 +4r 4 +r 5 +r 6 +2r 7 +2r 8 +r 9 +3r 10 < 1, (a+b+c+d+)(a+b+c+d+e) (7) q 1, q 2, q 3, q 4 0. Thn th quilibrium point x = a+b+c+d+ A+B+C+D+E of (1.1) is globally asymptotically stabl Thorm. Assum that q 1, q 2, q 3, q 4 0. Thn, Equation (1.1) has no positiv solution of priod two. Proof. For th sak of contradiction, assum that thr xist distinct positiv ral numbrs α, β, such that..., α, β, α, β,... is a priod two solution of (1.1). Thn, Thus, w hav which implis Sinc α = f(β,α), β = f(α,β). βf(β,α) = αf(α,β), (β α) ae(α8 + β 8 ) + q 1αβ(α 6 + β 6 ) + q 2α 2 β 2 (α 4 + β 4 ) + q 3α 3 β 3 (α 2 + β 2 ) + q 4α 4 β 4 (Aβ 4 + Bβα 3 + Cβ 2 α 2 + Dβ 3 α + Eα 4 )(Aα 4 + Bαβ 3 + Cα 2 β 2 + Dα 3 β + Eβ 4 ) = 0. ae(α 8 + β 8 ) + q 1αβ(α 6 + β 6 ) + q 2α 2 β 2 (α 4 + β 4 ) + q 3α 3 β 3 (α 2 + β 2 ) + q 4α 4 β 4 (Aβ 4 + Bβα 3 + Cβ 2 α 2 + Dβ 3 α + Eα 4 )(Aα 4 + Bαβ 3 + Cα 2 β 2 + Dα 3 β + Eβ 4 ) > 0, w gt α = β, which is a contradiction Rmark. It follows from (1.1), whn r 1 = = r 10 = 0 that x n+1 = α for all n 1 for som constant α. 3. Numrical xampls In ordr to illustrat our rsults and to support our thortical discussions, w considr numrical xampls in this sction Exampl. Lt (a,b,c,d,,a,b,c,d,e) = (3.8,2.2,3.3,3,5,1.5,2,3.5,3.2,5.5) and (x 1,x 0) = (5,2.9). Thn, all th conditions of Thorm 2.5 ar satisfid and w hav th following rsults:

7 On a Scond Ordr Rational Diffrnc Equation 873 n x n x n x n x n x n x Exampl. Lt (a,b,c,d,,a,b,c,d,e) = (1.5,2,3.5,3.2,5.5,3.8,2.2,3.3,3,5) and (x 1,x 0) = (0.15,21). Thn, all th conditions of Thorm 2.6 ar satisfid and w hav th following rsults: n x n x n x n x n x n x Acknowldgmnt Th hlpful suggstions of th anonymous rfr ar gratfully acknowldgd. Rfrncs [1] Bozkurt, F., Ozturk, I. and Ozn, S. Th global bhavior of th diffrnc quation, Stud. Univ. Babs-Bolyai Math. 54(2), 3 12, x [2] Cinar, C. On th positiv solutions of th diffrnc quation x n+1 = n 1, Appl. 1+x nx n 1 Math. Comp. 150(1), 21 24, [3] Cinar, C., Karatas, R. and Yalçinkaya, I. On solutions of th diffrnc quation x n+1 = x n 3, Math. Bohm. 132(3), , x nx n 1 x n 2 x n 3 [4] Clark, C. W., A dlayd rcruitmnt of a population dynamics with an application to baln whal populations, J. Math. Biol. 3, , [5] Chn, D., Li, X. and Wang, Y. Dynamics for nonlinar diffrnc quation x n+1 = αx n k, Adv. Diffr. Equ., Articl ID , 13 pags, β+γx p n l [6] Elabbasy, E.M., El-Mtwally, H. and Elsayd, E.M. On th diffrnc quation x n+1 = bx ax n n, Adv. Diffr. Equ. Articl ID 82579, 10 pags, cx n dx n 1 [7] Elabbasy, E. M. and Elsayd, E. M. On th global attractivity of diffrnc quation of highr ordr, Carpathian J. Math. 24(2), 45 53, [8] Elsayd, E. M. Exprssions of solutions for a class of diffrntial quations, An. Ştiint. Univ. Ovidius Constanta. Sr. Mat. 18(1), , [9] Fur, J. Priodic solutions of th Lynss max quation, J. Math. Anal. Appl. 288(1), , [10] Fur, J. Two classs of picwis-linar diffrnc quations with vntual priodicity thr, J. Math. Anal. Appl. 332(1), , [11] Kocic, V. L. and Ladas, G. Global bhavior of nonlinar diffrnc quations of highr ordr with applications (Kluwr Acadmic Publishrs, Dordrcht, 1993). [12] Kulnovic, M. R. S. and Ladas, G. Dynamics of scond ordr rational diffrnc quations with opn problms and conjcturs (Chapman and Hall, CRC Prss, 2001). [13] Ozturk, I., Bozkurt, F. and Ozn, S., On th diffrnc quation y n+1 = α+β yn γ+y n 1, Appl. Math. Comput. 181(2), , [14] Li, X. and Zhu, D. Global asymptotic stability in a rational quation, J. Diffrnc Equ. Appl. 9, , 2003.

8 874 N. Touafk [15] Li, X. Existnc of solutions with a singl smicycl for a gnral scond-ordr rational diffrnc quation, J. Math. Anal. Appl. 334(1), , [16] Stvić, S. A not on priodic charactr of a highr ordr diffrnc quation, Rostockr Math. Kolloq. 61, 21 30, [17] Stvić, S. On a class of highr-ordr diffrnc quations, Chaos Solitons and Fractals 42, , [18] Yalçinkaya, I., Iricanin, B. D. and Cinar, C. On a max-typ diffrnc quation, Discrt Dyn. Nat. Soc. Articl ID 47264, 10 pags, [19] Yalçinkaya, I. On th diffrnc quation x n+1 = α+ x n 2, Fasc. Math. 42, , x k n