- Email: [email protected]
New criteria of stability and boundedness for discrete systems
Nonlinear Analysis 41 (2000) 779 – 785
www.elsevier.nl/locate/na
New criteria of stability and boundedness for discrete systems Fu Yuli a , Liao Xiaoxin a;∗ , Luo Qi b a
Department of Automatic Control, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China b Department of Basic Science, Wuhan Yejin University of Science and Technology, Wuhan 430072, People’s Republic of China Received 6 July 1998; accepted 8 July 1998
Keywords: Discrete dynamic systems; Uniform stability; Uniform boundedness
1. Introduction In practice, many systems are described in the form of discrete dynamic systems. Stability and boundedness are two important properties in the theoretical study and applications of dynamic systems. So there is vast literature in this area [1– 5]. Here, we present some new criteria of uniform stability and uniform boundedness via the Liapunov function method, however, the restrictions on time di erences of Liapunov functions are weakened. Also, we give some examples to show the e ectiveness of the criteria. Now, we consider the nonlinear discrete dynamical systems: xk+1 − xk = f(k; xk );
(1.1)
where xk ∈ Rn ; k ∈ I + ; f : I + × Rn 7→ Rn . In this paper, we always assume that the solution of Eq. (1.1) with the initial condition xk0 = ’0 ;
∗
Corresponding author. E-mail addresses: [email protected] (F. Yuli), [email protected] (L. Xiaoxin)
0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 3 0 9 - 5
(1.2)
780
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
where ’0 ∈ Rn is a constant vector. We always describe the solution of Eq. (1.1) with Eq. (1.2) by xk in this paper. k · k is the norm in Rn . Deÿnition. (1) Under the condition of f(k; 0) ≡ 0, the zero solution of Eq. (1.1) is said to be uniformly stable, if for any k0 ≥0 and any ¿0, there exists ()¿0; kxk0 k ≤(), the inequality kxk k¡ holds for all k ≥k0 . (2) The solutions of Eq. (1.1) are said to be uniformly bounded, if for any k0 ≥0 and any Á¿0, there exists B(Á)¿0; kxk0 k ≤Á, and the inequality kxk k¡B(Á) holds for all k ≥k0 . A function ’ : R+ 7→ R+ is said to be a wedge function, if ’(0) = 0; ’(r) increases strictly and tends to +∞.
2. Main results First, for system (1:1) a criterion of uniform stability for system (1:1) is obtained. Theorem 2.1. For system (1:1), under the condition of kf(k; xk )k ≤M kxk k, where M ¿0, if there exists a continuous function V deÿned by V : I + × Rn 7→ R+ = [0; +∞), and there exist wedge functions u; w satisfying the following conditions: (1) u(kxk k)≤V (k; xk )≤w(kxk k). (2) There exist positive real number sequences {H i } and {Hi }; H i ¡Hi ; i ∈ I + , such that limt→+∞ Hi = 0; [H i ; Hi ] ∩ [ H j ; Hj ] = ∅ for i 6= j and u−1 (H i )¡
1 w−1 (Hi ); 1+M
i ∈ I +:
(2.1)
(3) For the solution xk of Eq. (1.1) with Eq. (1.2), ∀k0 ≥0, if V (k; xk ) ∈ [ H i ; Hi ]; k; i ∈ I + , V (k; xk ) := V (k + 1; xk+1 ) − V (k; xk ) = V (k + 1; xk + f(k; xk )) − V (k; xk )≤0 holds. Then the zero solution of Eq. (1.1) is uniformly stable. Proof. For any ¿0, there exists a positive integer i0 such that 0¡Hi0 ≤u(). Obviously, we have u(kxk0 k)≤V (k0 ; xk0 )≤w(kxk0 k)≤Hi0 ≤u(); i.e. kxk0 k≤; for kxk0 k≤() := w−1 (Hi0 ). If there exists a positive integer k1 ¿k0 such that V (k1 ; x1 )¿Hi0 , without loss of generality, we assume that k1 also guarantees V (k1 − 1; xk1 −1 )≤Hi0 . Therefore, V (k1 − 1; xk1 )¿0:
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
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On the other hand, if V (k1 − 1; xk1 −1 )¡H i0 , we have kxk1 −1 k¡u−1 (H i0 ). Hence, by condition (2:1), V (k1 ; xk1 ) ≤ w(kxk1 k)≤w(kxk1 −1 k + kf(k1 − 1; xk1 −1 )k) ≤ w((M + 1)kxk1 −1 k)≤Hi0 : This contradicts the deÿnition of k1 . So we have V (k1 − 1; xk1 −1 ) ∈ [H i0 ; Hi0 ]: By condition (3), it leads to a contradiction. Thus, V (k; xk )≤Hi0 , for any k ≥k0 , hence, kxk k≤u−1 (Hi0 )¡: The proof is completed. For system (1:1), we can give two criteria of uniform boundedness as follows. Theorem 2.2. For system (1:1), under the condition of kf(k; xk )k ≤M; M ¿0, if there exists a continuous function V deÿned by V : I + × Rn 7→ R+ = [0; +∞), and there exist wedge functions u; w satisfying the following conditions: (1) u(kxk k)≤V (k; xk )≤w(kxk k). (2) There exist positive real number sequences {H i } and {Hi }; H i ¡Hi ; i ∈ I + , such that lim H i = +∞; [H i ; Hi ] ∩ [ H j ; H j ] = ∅
t→+∞
for i 6= j
and
u(Hi )¿w(H i + M ):
(2.2)
(3) If kxk k ∈ [H i ; Hi ]; k; i ∈ I + ; V (k; xk )≤0 holds. Then the solution of Eq. (1.1) are uniformly bounded. Proof. For any Á¿0 there exists an integer i0 ¿0 such that H i0 ≥Á. Then we have u(Hi0 )¿w(Á). Obviously, kxk0 k≤Á≤Hi0 . If, without loss of generality, there exists an integer k1 ¿k0 such that V (k1 ; xk1 )¿u(Hi0 )
and
it is obvious that V (k1 − 1; xk1 −1 )¿0:
V (k1 − 1; xk1 −1 )≤u(Hi0 );
782
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
On the other hand, we have kxk1 −1 k≤Hi0 since V (k1 − 1; xk1 −1 )≤u(Hi0 ). If kxk1 −1 k¡H i0 , from condition (2:2), by the following inference: V (k1 ; xk1 ) ≤ w(kxk1 k)≤w(kxk1 −1 k + kf(k1 − 1; xk1 −1 )k) ≤w(M + kxk1 k) ≤ w(M + H i0 )≤u(Hi0 ); it leads to a contradiction of the deÿnition of k1 . Thus kxk1 −1 k ∈ [ H i0 ; Hi0 ] must hold, hence, by condition (3), V (k1 − 1; xk1 −1 )≤0. This contradiction implies that V (k; xk ) ≤u(Hi0 ) for any k ≥k0 . It is obvious to conclude that kxk k≤B(Á) := Hi0
for all k ≥k0 :
The proof is completed. Theorem 2.3. For system (1:1), kf(k; xk )k ≤M kxk k; M ¿0, if there exists a continuous function V deÿned by V : I + × Rn 7→ Rn = [0; +∞), and there exist wedge functions u; w satisfying the following conditions: (1) u(kxk k)≤V (k; xk )≤w(kxk k). (2) There exist positive real number sequences {H i } and {Hi }; H i ¡Hi ; i ∈ I + , such that lim H i =+∞;
t→+∞
[H i ; Hi ]∩[H j ; Hj ] = ∅
for i 6= j
and
u(Hi )¿w(H i (1 + M )):
(2.3)
(3) If kxk k ∈ [H i ; Hi ]; k; i ∈ I + , V (k; xk )≤0: holds. Then the solutions of Eq. (1.1) are uniformly bounded. Proof. For any Á¿0 there exists an integer i0 ¿0 such that (1 + M )H i0 ≥Á. Then we have u(Hi0 )¿w(Á). Obviously, kxk0 k≤Á ≤Hi0 . If, without loss of generality, there exists an integer k1 ¿k0 such that V (k1 ; xk1 )¿u(Hi0 )
and
V (k1 − 1; xk1 −1 )≤u(Hi0 );
it is obvious that V (k1 − 1; xk1 −1 )¿0: On the other hand, we have kxk1 −1 k≤Hi0 since V (k1 − 1; xk1 −1 )≤u(Hi0 ). If kxk1 −1 k¡H i0 , from Eq. (2.3), we have V (k1 ; xk1 ) ≤ w(kxk1 k)≤w(kxk1 −1 k + kf(k1 − 1; xk1 −1 k) ≤w((M + 1)kxk1 k) ≤ w((M + 1)H i0 )≤u(Hi0 );
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
783
which leads to a contradiction of the deÿnition of k1 . Thus kxk1 −1 k ∈ [ H i0 ; Hi0 ] must hold, hence, by condition (3), V (k1 − 1; xk1 −1 )≤0. This contradiction implies that V (k; xk ) ≤u(Hi0 ) for any k ≥k0 . It is obvious to conclude that kxk k≤B(Á) := Hi0
for all k ≥k0 :
The proof is completed.
3. Examples In this section, we give some examples to show the e ectiveness of the criteria above. Example 3.1. For the discrete system xk+1 − xk = v1 (xk ; yk );
(3.1)
yk+1 − yk = v2 (xk ; yk ); where 2
2
2
2
v1 (xk ; yk ) = 14 xk [(−1)(xk +yk ) + sin2 (xk2 + yk2 )]; v2 (xk ; yk ) = 14 yk [(−1)(xk +yk ) + sin2 (xk2 + yk2 )]; (r) = 1;
r ∈ [2i ; 32 2i ];
i ∈ I +;
(r) = 0;
in the other cases;
i ∈ I +:
By the Liapunov function V (xk ; yk ) = (xk2 + yk2 ), and taking u = w = V in Theorem 2.1, we have V (xk ; yk ) = [(xk+1 + xk )(xk+1 − xk ) + (yk+1 + yk )(yk+1 − yk )] 2
2
= (xk2 + yk2 )(2 + 14 sin2 (xk2 + yk2 ) + 14 (−1)(xk +yk ) ) 2
2
× 14 (sin2 (xk2 + yk2 ) + (−1)(xk +yk ) ); V (xk ; yk )≤0
for xk2 + yk2 ∈ [2i ; 32 2i ]; i ∈ I + ;
where H i = 2i ;
Hi = 32 2i
and
M = 12 :
By Theorem 2.3, we conclude that the solutions of system (3:1) is uniformly bounded.
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F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
Example 3.2. Consider the control system xk+1 − xk = u(xk ; yk );
(3.2)
yk+1 − yk = v(xk ; yk ); where u(xk ; yk ) =
xk sin(xk2 + yk2 ); 1 + xk2
v(xk ; yk ) =
yk sin(xk2 + yk2 ): 1 + yk2
(3.3)
Using Theorem 2.2, we take the function V and the wedge functions as V (xk ; yk ) = 12 (xk2 + yk2 ) We have V (xk ; yk ) =
and
u = w = V:
2 xk 1 1 2 2 2 + sin(x + y ) k k 2 1 + xk2 1 + xk2 yk2 + 1 + yk2
1 2+ sin(xk2 + yk2 ) 1 + yk2
sin(xk2 + yk2 )
≤0 for xk2 + yk2 ∈ [(2i + 1); 2(i + 1)]: For system (3) the constant M in the Theorem 2.2 is 12 ; Hi − H i = ≥M + 1 = 32 , and the other conditions of Theorem 2.2 are obviously satisÿed. The solutions of system (3:3) are uniformly bounded. Example 3.3. Deÿne a function (r) = 1;
r ∈ [ 21i ; 32 ·
1 2i ];
(r) = 0;
in the other cases:
We consider the following discrete system: 2
2
xk+1 − xk = 14 xk [cos2 (xk2 + yk2 ) + (−1)(xk +yk ) ]; 2
(3.4)
2
yk+1 − yk = 14 yk [cos2 (xk2 + yk2 ) + (−1)(xk +yk ) ]:
Also, by using the Liapunov function V (xk ; yk ) = (xk2 + yk2 ), and u = w = V , we have V (xk ; yk ) = [(xk+1 + xk )(xk+1 − xk ) + (yk+1 + yk )(yk+1 − yk )] 2
2
= (xk2 + yk2 )(2 + 14 cos2 (xk2 + yk2 ) + 14 (−1)(xk +yk ) ) 2
2
× 14 (cos2 (xk2 + yk2 ) + (−1)(xk +yk ) );
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
V (xk ; yk )≤0
for xk2 + yk2 ∈
785
1 3 1 · ; ; i ∈ I +; 2i 2 2i
where Hi=
1 ; 2i
Hi =
3 1 · 2 2i
and
1 M= : 2
By Theorem 2.1, we conclude that the zero solution of system (3:4) is uniformly stable. 4. Conclusion The results mentioned above weaken the previous criteria of uniform stability and boundedness in the restrictions of time di erences of Liapunov functions. We do not need the time di erences of the Liapunov functions to be nonpositive on the whole area. The examples shown in Section 3 explain that it is easy to determine the stability or boundedness behavior of discrete systems by using these new criteria and very simple Liapunov functions, while it is dicult to determine the behaviors by using the previous criteria and the simple Liapunov functions. References [1] I. Gyori, G. Ladas, P.N. Vlahos, Global attractivity in a delay di erence equation, Nonlinear Anal. 17 (5) (1991) 473– 479. [2] W.G. Kelley, A.C. Peterson, Di erence Equations: An Introduction with Applications, Academic Press, New York, 1991. [3] V. Lakshmikantham, S. Leela, A.A. Martynyuk, Stability Analysis of Nonlinear System, Marcel Dekker, New York, 1989. [4] X. Liao, Absolute Stability of Nonlinear Control Systems, Kluwer Academic Publishers, Netherlands, 1993. [5] J.R. Yan, C.X. Qian, Oscillation and comparison results for delay di erence equations, J. Math. Anal. Appl. 165 (1992) 346 –360.
www.elsevier.nl/locate/na
New criteria of stability and boundedness for discrete systems Fu Yuli a , Liao Xiaoxin a;∗ , Luo Qi b a
Department of Automatic Control, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China b Department of Basic Science, Wuhan Yejin University of Science and Technology, Wuhan 430072, People’s Republic of China Received 6 July 1998; accepted 8 July 1998
Keywords: Discrete dynamic systems; Uniform stability; Uniform boundedness
1. Introduction In practice, many systems are described in the form of discrete dynamic systems. Stability and boundedness are two important properties in the theoretical study and applications of dynamic systems. So there is vast literature in this area [1– 5]. Here, we present some new criteria of uniform stability and uniform boundedness via the Liapunov function method, however, the restrictions on time di erences of Liapunov functions are weakened. Also, we give some examples to show the e ectiveness of the criteria. Now, we consider the nonlinear discrete dynamical systems: xk+1 − xk = f(k; xk );
(1.1)
where xk ∈ Rn ; k ∈ I + ; f : I + × Rn 7→ Rn . In this paper, we always assume that the solution of Eq. (1.1) with the initial condition xk0 = ’0 ;
∗
Corresponding author. E-mail addresses: [email protected] (F. Yuli), [email protected] (L. Xiaoxin)
0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 3 0 9 - 5
(1.2)
780
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
where ’0 ∈ Rn is a constant vector. We always describe the solution of Eq. (1.1) with Eq. (1.2) by xk in this paper. k · k is the norm in Rn . Deÿnition. (1) Under the condition of f(k; 0) ≡ 0, the zero solution of Eq. (1.1) is said to be uniformly stable, if for any k0 ≥0 and any ¿0, there exists ()¿0; kxk0 k ≤(), the inequality kxk k¡ holds for all k ≥k0 . (2) The solutions of Eq. (1.1) are said to be uniformly bounded, if for any k0 ≥0 and any Á¿0, there exists B(Á)¿0; kxk0 k ≤Á, and the inequality kxk k¡B(Á) holds for all k ≥k0 . A function ’ : R+ 7→ R+ is said to be a wedge function, if ’(0) = 0; ’(r) increases strictly and tends to +∞.
2. Main results First, for system (1:1) a criterion of uniform stability for system (1:1) is obtained. Theorem 2.1. For system (1:1), under the condition of kf(k; xk )k ≤M kxk k, where M ¿0, if there exists a continuous function V deÿned by V : I + × Rn 7→ R+ = [0; +∞), and there exist wedge functions u; w satisfying the following conditions: (1) u(kxk k)≤V (k; xk )≤w(kxk k). (2) There exist positive real number sequences {H i } and {Hi }; H i ¡Hi ; i ∈ I + , such that limt→+∞ Hi = 0; [H i ; Hi ] ∩ [ H j ; Hj ] = ∅ for i 6= j and u−1 (H i )¡
1 w−1 (Hi ); 1+M
i ∈ I +:
(2.1)
(3) For the solution xk of Eq. (1.1) with Eq. (1.2), ∀k0 ≥0, if V (k; xk ) ∈ [ H i ; Hi ]; k; i ∈ I + , V (k; xk ) := V (k + 1; xk+1 ) − V (k; xk ) = V (k + 1; xk + f(k; xk )) − V (k; xk )≤0 holds. Then the zero solution of Eq. (1.1) is uniformly stable. Proof. For any ¿0, there exists a positive integer i0 such that 0¡Hi0 ≤u(). Obviously, we have u(kxk0 k)≤V (k0 ; xk0 )≤w(kxk0 k)≤Hi0 ≤u(); i.e. kxk0 k≤; for kxk0 k≤() := w−1 (Hi0 ). If there exists a positive integer k1 ¿k0 such that V (k1 ; x1 )¿Hi0 , without loss of generality, we assume that k1 also guarantees V (k1 − 1; xk1 −1 )≤Hi0 . Therefore, V (k1 − 1; xk1 )¿0:
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
781
On the other hand, if V (k1 − 1; xk1 −1 )¡H i0 , we have kxk1 −1 k¡u−1 (H i0 ). Hence, by condition (2:1), V (k1 ; xk1 ) ≤ w(kxk1 k)≤w(kxk1 −1 k + kf(k1 − 1; xk1 −1 )k) ≤ w((M + 1)kxk1 −1 k)≤Hi0 : This contradicts the deÿnition of k1 . So we have V (k1 − 1; xk1 −1 ) ∈ [H i0 ; Hi0 ]: By condition (3), it leads to a contradiction. Thus, V (k; xk )≤Hi0 , for any k ≥k0 , hence, kxk k≤u−1 (Hi0 )¡: The proof is completed. For system (1:1), we can give two criteria of uniform boundedness as follows. Theorem 2.2. For system (1:1), under the condition of kf(k; xk )k ≤M; M ¿0, if there exists a continuous function V deÿned by V : I + × Rn 7→ R+ = [0; +∞), and there exist wedge functions u; w satisfying the following conditions: (1) u(kxk k)≤V (k; xk )≤w(kxk k). (2) There exist positive real number sequences {H i } and {Hi }; H i ¡Hi ; i ∈ I + , such that lim H i = +∞; [H i ; Hi ] ∩ [ H j ; H j ] = ∅
t→+∞
for i 6= j
and
u(Hi )¿w(H i + M ):
(2.2)
(3) If kxk k ∈ [H i ; Hi ]; k; i ∈ I + ; V (k; xk )≤0 holds. Then the solution of Eq. (1.1) are uniformly bounded. Proof. For any Á¿0 there exists an integer i0 ¿0 such that H i0 ≥Á. Then we have u(Hi0 )¿w(Á). Obviously, kxk0 k≤Á≤Hi0 . If, without loss of generality, there exists an integer k1 ¿k0 such that V (k1 ; xk1 )¿u(Hi0 )
and
it is obvious that V (k1 − 1; xk1 −1 )¿0:
V (k1 − 1; xk1 −1 )≤u(Hi0 );
782
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
On the other hand, we have kxk1 −1 k≤Hi0 since V (k1 − 1; xk1 −1 )≤u(Hi0 ). If kxk1 −1 k¡H i0 , from condition (2:2), by the following inference: V (k1 ; xk1 ) ≤ w(kxk1 k)≤w(kxk1 −1 k + kf(k1 − 1; xk1 −1 )k) ≤w(M + kxk1 k) ≤ w(M + H i0 )≤u(Hi0 ); it leads to a contradiction of the deÿnition of k1 . Thus kxk1 −1 k ∈ [ H i0 ; Hi0 ] must hold, hence, by condition (3), V (k1 − 1; xk1 −1 )≤0. This contradiction implies that V (k; xk ) ≤u(Hi0 ) for any k ≥k0 . It is obvious to conclude that kxk k≤B(Á) := Hi0
for all k ≥k0 :
The proof is completed. Theorem 2.3. For system (1:1), kf(k; xk )k ≤M kxk k; M ¿0, if there exists a continuous function V deÿned by V : I + × Rn 7→ Rn = [0; +∞), and there exist wedge functions u; w satisfying the following conditions: (1) u(kxk k)≤V (k; xk )≤w(kxk k). (2) There exist positive real number sequences {H i } and {Hi }; H i ¡Hi ; i ∈ I + , such that lim H i =+∞;
t→+∞
[H i ; Hi ]∩[H j ; Hj ] = ∅
for i 6= j
and
u(Hi )¿w(H i (1 + M )):
(2.3)
(3) If kxk k ∈ [H i ; Hi ]; k; i ∈ I + , V (k; xk )≤0: holds. Then the solutions of Eq. (1.1) are uniformly bounded. Proof. For any Á¿0 there exists an integer i0 ¿0 such that (1 + M )H i0 ≥Á. Then we have u(Hi0 )¿w(Á). Obviously, kxk0 k≤Á ≤Hi0 . If, without loss of generality, there exists an integer k1 ¿k0 such that V (k1 ; xk1 )¿u(Hi0 )
and
V (k1 − 1; xk1 −1 )≤u(Hi0 );
it is obvious that V (k1 − 1; xk1 −1 )¿0: On the other hand, we have kxk1 −1 k≤Hi0 since V (k1 − 1; xk1 −1 )≤u(Hi0 ). If kxk1 −1 k¡H i0 , from Eq. (2.3), we have V (k1 ; xk1 ) ≤ w(kxk1 k)≤w(kxk1 −1 k + kf(k1 − 1; xk1 −1 k) ≤w((M + 1)kxk1 k) ≤ w((M + 1)H i0 )≤u(Hi0 );
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
783
which leads to a contradiction of the deÿnition of k1 . Thus kxk1 −1 k ∈ [ H i0 ; Hi0 ] must hold, hence, by condition (3), V (k1 − 1; xk1 −1 )≤0. This contradiction implies that V (k; xk ) ≤u(Hi0 ) for any k ≥k0 . It is obvious to conclude that kxk k≤B(Á) := Hi0
for all k ≥k0 :
The proof is completed.
3. Examples In this section, we give some examples to show the e ectiveness of the criteria above. Example 3.1. For the discrete system xk+1 − xk = v1 (xk ; yk );
(3.1)
yk+1 − yk = v2 (xk ; yk ); where 2
2
2
2
v1 (xk ; yk ) = 14 xk [(−1)(xk +yk ) + sin2 (xk2 + yk2 )]; v2 (xk ; yk ) = 14 yk [(−1)(xk +yk ) + sin2 (xk2 + yk2 )]; (r) = 1;
r ∈ [2i ; 32 2i ];
i ∈ I +;
(r) = 0;
in the other cases;
i ∈ I +:
By the Liapunov function V (xk ; yk ) = (xk2 + yk2 ), and taking u = w = V in Theorem 2.1, we have V (xk ; yk ) = [(xk+1 + xk )(xk+1 − xk ) + (yk+1 + yk )(yk+1 − yk )] 2
2
= (xk2 + yk2 )(2 + 14 sin2 (xk2 + yk2 ) + 14 (−1)(xk +yk ) ) 2
2
× 14 (sin2 (xk2 + yk2 ) + (−1)(xk +yk ) ); V (xk ; yk )≤0
for xk2 + yk2 ∈ [2i ; 32 2i ]; i ∈ I + ;
where H i = 2i ;
Hi = 32 2i
and
M = 12 :
By Theorem 2.3, we conclude that the solutions of system (3:1) is uniformly bounded.
784
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
Example 3.2. Consider the control system xk+1 − xk = u(xk ; yk );
(3.2)
yk+1 − yk = v(xk ; yk ); where u(xk ; yk ) =
xk sin(xk2 + yk2 ); 1 + xk2
v(xk ; yk ) =
yk sin(xk2 + yk2 ): 1 + yk2
(3.3)
Using Theorem 2.2, we take the function V and the wedge functions as V (xk ; yk ) = 12 (xk2 + yk2 ) We have V (xk ; yk ) =
and
u = w = V:
2 xk 1 1 2 2 2 + sin(x + y ) k k 2 1 + xk2 1 + xk2 yk2 + 1 + yk2
1 2+ sin(xk2 + yk2 ) 1 + yk2
sin(xk2 + yk2 )
≤0 for xk2 + yk2 ∈ [(2i + 1); 2(i + 1)]: For system (3) the constant M in the Theorem 2.2 is 12 ; Hi − H i = ≥M + 1 = 32 , and the other conditions of Theorem 2.2 are obviously satisÿed. The solutions of system (3:3) are uniformly bounded. Example 3.3. Deÿne a function (r) = 1;
r ∈ [ 21i ; 32 ·
1 2i ];
(r) = 0;
in the other cases:
We consider the following discrete system: 2
2
xk+1 − xk = 14 xk [cos2 (xk2 + yk2 ) + (−1)(xk +yk ) ]; 2
(3.4)
2
yk+1 − yk = 14 yk [cos2 (xk2 + yk2 ) + (−1)(xk +yk ) ]:
Also, by using the Liapunov function V (xk ; yk ) = (xk2 + yk2 ), and u = w = V , we have V (xk ; yk ) = [(xk+1 + xk )(xk+1 − xk ) + (yk+1 + yk )(yk+1 − yk )] 2
2
= (xk2 + yk2 )(2 + 14 cos2 (xk2 + yk2 ) + 14 (−1)(xk +yk ) ) 2
2
× 14 (cos2 (xk2 + yk2 ) + (−1)(xk +yk ) );
F. Yuli et al. / Nonlinear Analysis 41 (2000) 779 – 785
V (xk ; yk )≤0
for xk2 + yk2 ∈
785
1 3 1 · ; ; i ∈ I +; 2i 2 2i
where Hi=
1 ; 2i
Hi =
3 1 · 2 2i
and
1 M= : 2
By Theorem 2.1, we conclude that the zero solution of system (3:4) is uniformly stable. 4. Conclusion The results mentioned above weaken the previous criteria of uniform stability and boundedness in the restrictions of time di erences of Liapunov functions. We do not need the time di erences of the Liapunov functions to be nonpositive on the whole area. The examples shown in Section 3 explain that it is easy to determine the stability or boundedness behavior of discrete systems by using these new criteria and very simple Liapunov functions, while it is dicult to determine the behaviors by using the previous criteria and the simple Liapunov functions. References [1] I. Gyori, G. Ladas, P.N. Vlahos, Global attractivity in a delay di erence equation, Nonlinear Anal. 17 (5) (1991) 473– 479. [2] W.G. Kelley, A.C. Peterson, Di erence Equations: An Introduction with Applications, Academic Press, New York, 1991. [3] V. Lakshmikantham, S. Leela, A.A. Martynyuk, Stability Analysis of Nonlinear System, Marcel Dekker, New York, 1989. [4] X. Liao, Absolute Stability of Nonlinear Control Systems, Kluwer Academic Publishers, Netherlands, 1993. [5] J.R. Yan, C.X. Qian, Oscillation and comparison results for delay di erence equations, J. Math. Anal. Appl. 165 (1992) 346 –360.