Difference Equations Article ID 936302 6 pages http://dx.doi.org/10.1155/2014/936302 Research Article Asymptotic Behavior of the Solutions of System of Difference Equations of Exponential Form Vu Van Khuong 12 and Tran Hong Thai 1 1 Department of Mathematics Hung Yen University of Technology and Education Khoai Chau Hung Yen 393008 Vietnam 2 Department of Mathematical Analysis University of Transport and Communications Dong Da Hanoi 10200 Vietnam Correspondence should be addressed to Vu Van Khuong; vuvankhuong@gmail.com Received 31 May 2014; Revised 10 September 2014; Accepted 24 September 2014; Published 13 October 2014 Academic Editor: Honglei Xu Copyright 2014 V. Van Khuong and T. Hong Thai. 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. The goal of this paper is to study the boundedness the persistence and the asymptotic behavior of the positive solutions of the system of two difference equations of exponential form: x n1 (abe y n /(dhy n y n1 (abe x n /( n where a b c dandh are positive constants and the initial values x 0 y 0 are positive real values. Also we determine the rate of convergence of a solution that converges to the equilibrium E(x y of this system. 1. Introduction In [1] the authors studied the boundedness the asymptotic behavior the periodicity and the stability of the positive solutions of the difference equation: y n1 αβey n (1 γy n1 where α β γ are positive constants and the initial values y 1 y 0 are positive numbers. Motivated by the above paper we will extend the above difference equation to a system of difference equations; our goal will be to investigate the boundedness the persistence and the asymptotic behavior of the positive solutions of the following system of exponential form: x n1 abey n dhy n y n1 abex n n where a b c d h are positive constants and the initial values x 0 y 0 are positive real values. (2 Difference equations and systems of difference equations of exponential form can be found in [2 6]. Moreover as difference equations have many applications in applied sciences there are many papers and books that can be found concerning the theory and applications of difference equations; see [7 9] and the references cited therein. 2. Global Behavior of Solutions of System (2 In the first lemma we study the boundedness and persistence ofthepositivesolutionsof(2. Lemma 1. Every positive solution of (2 is bounded and persists. Proof. Let (x n y n be an arbitrary solution of (2. From (2 we can see that x n abc d y n abc n12... (3 d
2 Difference Equations In addition from (2 and(3 we get x n abe(abc/d ce (abc/d dh ((abc /d y n abe(abc/d ce (abc/d dh ((abc /d n23... Therefore from (3 and (4 the proof of the lemma is complete. In order to prove the main result of this section we recall the next theorem without its proof. See [10 11]. Theorem 2. Let R [a 1 b 1 ] [c 1 d 1 ] and (4 f:r [a 1 b 1 ] g : R [c 1 d 1 ] (5 be a continuous functions such that the following hold: (a f(x y is decreasing in both variables and g(x y is decreasing in both variables for each (x y R; (b if (m 1 M 1 m 2 M 2 R 2 is a solution of M 1 f(m 1 m 2 m 1 f(m 1 M 2 M 2 g(m 1 m 2 m 2 g(m 1 M 2 then m 1 M 1 and m 2 M 2. Then the following system of difference equations (6 x n1 f(x n y n y n1 g(x n y n (7 has a unique equilibrium (x y and every solution (x n y n of the system (7 with (x 0 y 0 R converges to the unique equilibrium (x y. In addition the equilibrium (x y is globally asymptotically stable. Now we state the main theorem of this section. Theorem 3. Consider system (2. Suppose that the following relation holds true: bc<d. (8 Then system (2 has a unique positive equilibrium (x y and every positive solution of (2 tends to the unique positive equilibrium (x y as n. In addition the equilibrium (x y is globally asymptotically stable. Proof. We consider the functions f (u V abev ce u dhv (9 g (u V abeu ce V dhu where a(bc e(abc/d u V I[ ch ((abc /d abc ]. (10 d It is easy to see that f(u V g(u V are decreasing in both variables for each (u V I I. In addition from (9 and (10 wehavef(u V I g(u V Ias (u V I Iand so f:i I I g:i I I. Now let m 1 M 1 m 2 M 2 be positive real numbers such that M 1 abem 2 ce m 1 dhm 2 M 2 abem 1 ce m 2 dhm 1 m 1 abem 2 ce M 1 dhm 2 m 2 abem 1 ce M 2 dhm 1. (11 Moreover arguing as in the proof of Theorem 2itsufficesto assume that From (11we get which imply that m 1 M 1 m 2 M 2. (12 be m 2 ce m 1 (dhm 2 M 1 a be M 2 ce M 1 (dhm 2 m 1 a be m 1 ce m 2 (dhm 1 M 2 a be M 1 ce M 2 (dhm 1 m 2 a d(m 1 m 1 h(m 1 m 2 M 2 m 1 b(e m 2 e M 2 c(e m 1 e M 1 be m 2M 2 (e M 2 e m 2 ce m 1M 1 (e M 1 e m 1 d(m 2 m 2 h(m 2 m 1 M 2 m 2 b(e m 1 e M 1 c(e m 2 e M 2 (13 be m 1M 1 (e M 1 e m 1 ce m 2M 2 (e M 2 e m 2. (14 Moreover we get e M 1 e m 1 e α (M 1 m 1 m 1 α M 1 e M 2 e m 2 e β (M 2 m 2 m 2 β M 2. Then by adding the two relations (14weobtain d(m 1 m 1 d(m 2 m 2 (bc (15 [e m 1M 1 α (M 1 m 1 e m 2M 2 β (M 2 m 2 ]. (16
Difference Equations 3 Therefore from (16we have (M 1 m 1 [d(bc e m 1M 1 α ] (M 2 m 2 [d(bc e m 2M 2 β ]0. (17 Then using (8 (12 and (17 givesusm 1 M 1 and m 2 M 2.HencefromTheorem 2 system (2hasauniquepositive equilibrium (x y and every positive solution of (2 tends totheuniquepositiveequilibrium(x y as n.in addition the equilibrium (xy is globally asymptotically stable. This completes the proof of the theorem. 3. Rate of Convergence In this section we give the rate of convergence of a solution that converges to the equilibrium E (x y of the system (2 for all values of parameters. The rate of convergence of solutions that converge to an equilibrium has been obtained for some two-dimensional systems in [12 13]. The following results give the rate of convergence of solutions of a system of difference equations: x n1 [AB(n] x n (18 where x n is a k-dimensional vector A C k k is a constant matrix and B:Z C k k is a matrix function satisfying B (n 0 when n (19 where denotes any matrix norm which is associated with the vector norm; also denotes the Euclidean norm in R 2 given by x (x y x 2 y 2. (20 Theorem 4 (see [14]. Assume that condition (19 holds. If x n isasolutionofsystem(18 then either x n 0for all large n or ρ lim n n x n (21 exists and is equal to the modulus of one of the eigenvalues of matrix A. Theorem 5 (see [14]. Assume that condition (19 holds. If x n isasolutionofsystem(18 then either x n 0for all large n or ρ lim x n1 n x (22 n exists and is equal to the modulus of one of the eigenvalues of matrix A. The equilibrium point of the system (2 satisfies the following system of equations: x abey ce x dh y (23 y abex ce y. dh x If bc < dwecaneasilyseethatthesystem(23 hasan unique equilibrium E(x x. The map T associated with the system (2is abe y ce x T(xy( f(xy g(xy ( dhy. (24 abe x ce y The Jacobian matrix of T is dhy J T ( be x ( h (abe x ce y ( 2 be y (d hy h (a be y ce x (d hy 2 ce y. (25 By using the system (23 value of the Jacobian matrix of T at the equilibrium point E(x y (x x is J T ( be x ( h [a(bc e x ] ( 2 be x ( h [a(bc e x ] ( 2. (26
4 Difference Equations Our goal in this section is to determine the rate of convergence of every solution of the system (2in the regions where the parameters a b c d h (0 (b c < d and initial conditions x 0 and y 0 are arbitrary nonnegative numbers. Theorem 6. The error vector e n ( e1 n ( x nx e 2 y n n y of every solution x n 0 of (2 satisfies both of the following asymptotic relations: lim n n e n λ i (J T (E for some i 1 2 lim e (27 n1 n e n λ i (J T (E for some i 1 2 where λ i (J T (E is equal to the modulus of one of the eigenvalues of the Jacobian matrix evaluated at the equilibrium J T (E. Proof. First we will find a system satisfied by the error terms. The error terms are given as x n1 x abey n dhy n abey ce x dhy ((abe y n (dhy (abe y ce x (dhy n ((d hy n (dhy 1 (bd(e y n e y cd(e x n e x ah(yy n bh(e y n ye y y n ch(e x n ye x y n ((d hy n (dhy 1 bd (e y n e y e yny (d hy n (dhy cd (e x n e x e xnx (d hy n (dhy bh (d hy n (dhy (e y n ye y n y n e y n y n e y y n ch (d hy n (dhy (e x n ye x n y n e x n y n e y y n ah (d hy n (dhy (y n y b e y ny (d hy (ey n e y c e x nx (d hy (ex n e x h abe y n (d hy n (dhy (y n y b e y n (d hy (e y ny 1 c e x n (d hy (e x nx 1 h abe y n (d hy n (dhy (y n y b e y n (d hy [(y n y O 2 ((y n y 2 ] c e x n (d hy [(x n x O 1 ((x n x 2 ] h abe y n (d hy n (dhy (y n y c e x n (d hy (x n x bey n (d hy n h(abe y n (d hy n (dhy (y n y O 1 ((x n x 2 O 2 ((y n y 2. By calculating similarly we get y n1 y bex n (d hx n h(abe x n (d hx n ( c e y n ( (y n y O 3 ((x n x 2 O 4 ((y n y 2. From (28and(29wehave x n1 x c e x n (d hy (x n x (x n x bey n (d hy n h(abe y n (d hy n (dhy (y n y (28 (29
Difference Equations 5 Set y n1 y bex n (d hx n h(abe x n (d hx n ( (x n x c e y n ( (y n y. (30 e 1 n x n x e 2 n y n y. (31 Then system (30 can be represented as where e 1 n1 a ne 1 n b ne 2 n (32 e 2 n1 c ne 1 n d ne 2 n a n c e x n (d hy b n bey n (d hy n h(abe y n (d hy n (dhy c n bex n (d hx n h(abe x n (d hx n ( d n c e y n (. (33 Taking the limits of a n b n c n andd n as n weobtain lim a c n n e x ( that is lim b n n bex ( h [a (bc e x ] ( 2 lim c n n bex ( h [a (bc e x ] ( 2 lim d c n n e x ( ; a n c e x ( α n b n bex ( h [a (bc e x ] ( 2 β n c n bex ( h [a (bc e x ] ( 2 γ n d n c e x ( δ n (34 (35 where α n 0 β n 0 γ n 0andδ n 0as n. Now we have system of the form (18: where e n1 (AB(n e n (36 A( be x ( h [a(bc e x ] ( 2 be x ( h [a(bc e x ] ( 2 (37 β n α n B (n ( δ n γ n B (n 0 as n. Thus the limiting system of error terms can be written as ( e 1 n1 e 2 n1 A( e 1 n e 2 n. (38 The system is exactly linearized system of (2 evaluated at the equilibrium E(x y (x x. Then Theorems 4 and 5 imply the result. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [1] I. Ozturk F. Bozkurt and S. Ozen On the difference equation y n1 (αbe yn /(γy n1 Applied Mathematics and Computationvol.181no.2pp.1387 13932006. [2] D. C. Zhang and B. Shi Oscillation and global asymptotic stability in a discrete epidemic model Mathematical Analysis and Applicationsvol.278no.1pp.194 2022003.
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