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A note on global asymptotic stability of non-linear difference equations
Economics Letters North-Holland
26 (1988) 349-352
349
A NOTE ON GLOBAL ASYMPTOTIC DIFFERENCE EQUATIONS Ferenc Karl Marx
STABILITY
OF NON-LINEAR
SZIDAROVSZKY University, Budapest IX, Hungary
Koji OKUGUCHI Tokyo Metropolitan Received Accepted
University,
Tokyo, Japan
16 December 1987 4 January 1988
This paper introduces a generalization also verify that in the case of linear coincide with the well known necessary unit circle.
of the global asymptotic stability theorem of Brock and Scheinkman (1975). We shall first order difference equations with constant coefficients our sufficient conditions and sufficient condition that all eigenvalues of the matrix of coefficients are inside the
1. Introduction Let
v(t) =h( Y(t - 1))
(1)
be a system of difference equations, where h : R” + R” is a function independent of t. The most frequently applied criteria for global asymptotic stability of difference eq. (1) can be formulated as follows: Theorem I. Assume that function z(y) of h(y) satisfies the relation (Vye
IIJ( Y) II < 1,
h(y) is continuously
differentiable
in R”; furthermore,
the Jacobian
R”).
(2)
Then equation y = h(y) has a unique solution in R”, which is globally asymptotically
stable.
This important result is a consequence of the mean value theorem and the contraction mapping theory [see Ortega and Rheinboldt (1970)]. A nice and useful generalization of this result has been presented by Brock and Scheinkman (1975),’ which can be given as: ’ See Fujimoto
0165-1765/88/$3.50
(1987a.
1987b) and Okuguchi
(1976, 1977) for some extentions
0 1988, Elsevier Science Publishers
B.V. (North-Holland)
of Brock and Scheinkman’s
(1975) result.
F. Szidurouszky,
350
that h(0)
Theorem 2. Assume furthermore that (a)
II_J(0)~ll~
(b)
If
K. Okuguchi
/ Stability
= 0, and function
of non-lrneardifference equations
h(y)
is continuously
differentiable
in R”. Assume
for allyER”i
O+ll~ll2=Ilh(~ll2~
then
II_J(y)~ll2
where _J( y) is the Jacobian of h( y), and II y 112 = $3. Then the zero solution is globally asymptotically stable. The applications of this result in analyzing practical problems have two difficulties. First of all, conditions (a) and (b) may be satisfied if a different vector norm other than II . II 2 is used. The second difficulty is based on the fact that, even in the special case of h(y) = _Hu (with H being a constant matrix) the conditions of Theorem 2 are weaker than the well-known necessary and sufficient conditions for global asymptotic stability, that all eigenvalues of matrix _H are inside the unit circle of the complex plane. In the next section of this paper conditions (a) and (b) will be weakened further in a way such that the new conditions could be applied even in the case of a more general norm family. In addition, our conditions will coincide with the well-known sufficient and necessary condition in the special case of a linear difference equation with constant coefficients.
2. Generalized
stability
Let G be a constant
condition
non-singular
matrix.
Then
eq. (1) is equivalent
to the relation
cy(t)=Gh(_G-‘Gy(r-l)),
and by introducing
the new variables
x(t) = G y( t) (t 2 0) we get the equation
x(t)=g(x(t-I)), where
g(x) = Gh(G-‘x),
(VXE
R”).
(3)
If h(0) = 0, then g(0) = 0, and if h( y) is continuously differentiable in R”, then the same holds for function g(x). Thus the zero solution of (1) is globally asymptotically stable if conditions (a) and (b) hold for function g(x). Since the Jacobian of g(x) is C ._J((G-‘x) . _C-‘, where _J( y) is the Jacobian of h, conditions (a) and (b) for function g(x) can be rewritten as IIGmK-‘~lI,~
IIXIIZ~
(VXE
R”)
and 0 # 11x II 2 = 11C. h(C_-‘x) y=CP1x,andthegeneralvectornorm
11 2 implies
II G-JWG-‘x
II G,
IlxlI2=
IIGYll2=
lIGh(G-‘+I,=
II 2 = II J(O)Y II YIIG.3 llGh(~)ll,=
IP(Y)IIG
that II_C._J(((;-‘x)_C-‘x IlylI,=IIQII,.Then
II 2 -C /I x 112. Introduce
the notation
351
F. Sridarouszky, K. Okuguchi / Stability of non-linear drfference equations
Hence we have the following
result:
Theorem 3. Assume that h(0) = 0, and function constant non-singular matrix. Assume that (a’>
h(y) is continuously
differentiable
in R”. Let G be a
llJ(O)~ll~< II .YII~ for ally&R”;
@‘I If
O# II YII(;= Ilh(y)ll,,
then II_J(Y)YII,<
Then the zero solution is globally asymptotically Remark following h,(y) = satisfies sequence
II ~11,.
stable.
The conditions of Theorem 2 can be obtained by selecting _C= I. Introduce next the functions: h( y), and for all k 2 1, hk( y) = (hohk_l)( y). Assume that for some k 2 1, function hk( y) the conditions of Theorem 3. Then we know that starting from arbitrary vector z0 E R”, the generated by the iteration scheme I.
z(t)=h,&(t-1))
(4)
converges to the zero solution for t + co. We can easily verify that in this case the entire sequence (1) is also convergent for arbitrary y(0) E R”. For 1 = O,l,, . . . , k - 1. define z(O)y’ then from (4) we have that z(t) = y(tk + 1). Hence for all 1 = O,l,. . . , k - 1, ycrk+l) -+ 0 as t + co. That is, the entire sequence y(t) converges to zero. Thus we have verified the following: Theorem 4. Assume that h(0) = 0 and function h(y) is continuously differentiable in R”. If for some k 2 1, function h, satisfies conditions (a’) and (6’) of Theorem 3. Then the zero solution is globally asymptotically stable. Remark
2.
Theorem
4 obviously
generalizes
Theorem
3 if one selects k = 1.
Remark 3. Consider now the special case of h(y) = H y, where H is a constant matrix. We shall verify that in this case the conditions of Theorem 4 holdif and onlyif all eigenvalues of _H are inside the unit disk of the complex plane. That is, in this special case, Theorem 4 gives sufficient and necessary condition. Assume first that for some k 2 1, function hk( y) = H”y satisfies conditions (a’) and (b’). Then 11_Hk IIG < 1, which implies that all eigenvalues of _Hk are inside the unit disk. If A, (i 2 1) are the eigenvalues of _H, then (A: 1-c 1, that is, 1A, I -c 1. Assume next that all eigenvalues of H are inside the unit disk. Then Hk -+ 0 for k + co. Hence for sufficiently large k, I(_Hk 11_G< 1. Consequently conditions (a’) and (b’) are satisfied by function hk( y).
References Brock, W.A. Economic
and J.A. Scheinkman, Theory
10, 265-268.
1975, Some results
on global
asymptotic
stability
of difference
equations,
Journal
of
352
F. Szidarouszky,
K. Okuguchi
/ Stability of non-linear
difference equations
Fujimoto, T., 1987a, Global asymptotic stability of non-linear difference equations, Economics Letters 22, 247-250. Fujimoto, T., 1987b. Global asymptotic stability of non-linear differential equations II, Economics Letters 23, 275-277. Okuguchi, K., 1976, Expectations and stability in oligopoly models (Springer-Verlag, Berlin/Heidelberg/New York). Okuguchi. K., 1977, Mathematical foundations of economic analysis (in Japanese) (McGraw-Hill Kogakusha, Tokyo). Ortega, J.M. and W.C. Rheinboldt, 1970, Iterative solution of nonlinear equations in several variables (Academic Press, New York/London). Szidarovszky, F. and S. Yakowitz, 1978, Principles and procedures of numerical analysis (Plenum Press, New York).
26 (1988) 349-352
349
A NOTE ON GLOBAL ASYMPTOTIC DIFFERENCE EQUATIONS Ferenc Karl Marx
STABILITY
OF NON-LINEAR
SZIDAROVSZKY University, Budapest IX, Hungary
Koji OKUGUCHI Tokyo Metropolitan Received Accepted
University,
Tokyo, Japan
16 December 1987 4 January 1988
This paper introduces a generalization also verify that in the case of linear coincide with the well known necessary unit circle.
of the global asymptotic stability theorem of Brock and Scheinkman (1975). We shall first order difference equations with constant coefficients our sufficient conditions and sufficient condition that all eigenvalues of the matrix of coefficients are inside the
1. Introduction Let
v(t) =h( Y(t - 1))
(1)
be a system of difference equations, where h : R” + R” is a function independent of t. The most frequently applied criteria for global asymptotic stability of difference eq. (1) can be formulated as follows: Theorem I. Assume that function z(y) of h(y) satisfies the relation (Vye
IIJ( Y) II < 1,
h(y) is continuously
differentiable
in R”; furthermore,
the Jacobian
R”).
(2)
Then equation y = h(y) has a unique solution in R”, which is globally asymptotically
stable.
This important result is a consequence of the mean value theorem and the contraction mapping theory [see Ortega and Rheinboldt (1970)]. A nice and useful generalization of this result has been presented by Brock and Scheinkman (1975),’ which can be given as: ’ See Fujimoto
0165-1765/88/$3.50
(1987a.
1987b) and Okuguchi
(1976, 1977) for some extentions
0 1988, Elsevier Science Publishers
B.V. (North-Holland)
of Brock and Scheinkman’s
(1975) result.
F. Szidurouszky,
350
that h(0)
Theorem 2. Assume furthermore that (a)
II_J(0)~ll~
(b)
If
K. Okuguchi
/ Stability
= 0, and function
of non-lrneardifference equations
h(y)
is continuously
differentiable
in R”. Assume
for allyER”i
O+ll~ll2=Ilh(~ll2~
then
II_J(y)~ll2
where _J( y) is the Jacobian of h( y), and II y 112 = $3. Then the zero solution is globally asymptotically stable. The applications of this result in analyzing practical problems have two difficulties. First of all, conditions (a) and (b) may be satisfied if a different vector norm other than II . II 2 is used. The second difficulty is based on the fact that, even in the special case of h(y) = _Hu (with H being a constant matrix) the conditions of Theorem 2 are weaker than the well-known necessary and sufficient conditions for global asymptotic stability, that all eigenvalues of matrix _H are inside the unit circle of the complex plane. In the next section of this paper conditions (a) and (b) will be weakened further in a way such that the new conditions could be applied even in the case of a more general norm family. In addition, our conditions will coincide with the well-known sufficient and necessary condition in the special case of a linear difference equation with constant coefficients.
2. Generalized
stability
Let G be a constant
condition
non-singular
matrix.
Then
eq. (1) is equivalent
to the relation
cy(t)=Gh(_G-‘Gy(r-l)),
and by introducing
the new variables
x(t) = G y( t) (t 2 0) we get the equation
x(t)=g(x(t-I)), where
g(x) = Gh(G-‘x),
(VXE
R”).
(3)
If h(0) = 0, then g(0) = 0, and if h( y) is continuously differentiable in R”, then the same holds for function g(x). Thus the zero solution of (1) is globally asymptotically stable if conditions (a) and (b) hold for function g(x). Since the Jacobian of g(x) is C ._J((G-‘x) . _C-‘, where _J( y) is the Jacobian of h, conditions (a) and (b) for function g(x) can be rewritten as IIGmK-‘~lI,~
IIXIIZ~
(VXE
R”)
and 0 # 11x II 2 = 11C. h(C_-‘x) y=CP1x,andthegeneralvectornorm
11 2 implies
II G-JWG-‘x
II G,
IlxlI2=
IIGYll2=
lIGh(G-‘+I,=
II 2 = II J(O)Y II YIIG.3 llGh(~)ll,=
IP(Y)IIG
that II_C._J(((;-‘x)_C-‘x IlylI,=IIQII,.Then
II 2 -C /I x 112. Introduce
the notation
351
F. Sridarouszky, K. Okuguchi / Stability of non-linear drfference equations
Hence we have the following
result:
Theorem 3. Assume that h(0) = 0, and function constant non-singular matrix. Assume that (a’>
h(y) is continuously
differentiable
in R”. Let G be a
llJ(O)~ll~< II .YII~ for ally&R”;
@‘I If
O# II YII(;= Ilh(y)ll,,
then II_J(Y)YII,<
Then the zero solution is globally asymptotically Remark following h,(y) = satisfies sequence
II ~11,.
stable.
The conditions of Theorem 2 can be obtained by selecting _C= I. Introduce next the functions: h( y), and for all k 2 1, hk( y) = (hohk_l)( y). Assume that for some k 2 1, function hk( y) the conditions of Theorem 3. Then we know that starting from arbitrary vector z0 E R”, the generated by the iteration scheme I.
z(t)=h,&(t-1))
(4)
converges to the zero solution for t + co. We can easily verify that in this case the entire sequence (1) is also convergent for arbitrary y(0) E R”. For 1 = O,l,, . . . , k - 1. define z(O)y’ then from (4) we have that z(t) = y(tk + 1). Hence for all 1 = O,l,. . . , k - 1, ycrk+l) -+ 0 as t + co. That is, the entire sequence y(t) converges to zero. Thus we have verified the following: Theorem 4. Assume that h(0) = 0 and function h(y) is continuously differentiable in R”. If for some k 2 1, function h, satisfies conditions (a’) and (6’) of Theorem 3. Then the zero solution is globally asymptotically stable. Remark
2.
Theorem
4 obviously
generalizes
Theorem
3 if one selects k = 1.
Remark 3. Consider now the special case of h(y) = H y, where H is a constant matrix. We shall verify that in this case the conditions of Theorem 4 holdif and onlyif all eigenvalues of _H are inside the unit disk of the complex plane. That is, in this special case, Theorem 4 gives sufficient and necessary condition. Assume first that for some k 2 1, function hk( y) = H”y satisfies conditions (a’) and (b’). Then 11_Hk IIG < 1, which implies that all eigenvalues of _Hk are inside the unit disk. If A, (i 2 1) are the eigenvalues of _H, then (A: 1-c 1, that is, 1A, I -c 1. Assume next that all eigenvalues of H are inside the unit disk. Then Hk -+ 0 for k + co. Hence for sufficiently large k, I(_Hk 11_G< 1. Consequently conditions (a’) and (b’) are satisfied by function hk( y).
References Brock, W.A. Economic
and J.A. Scheinkman, Theory
10, 265-268.
1975, Some results
on global
asymptotic
stability
of difference
equations,
Journal
of
352
F. Szidarouszky,
K. Okuguchi
/ Stability of non-linear
difference equations
Fujimoto, T., 1987a, Global asymptotic stability of non-linear difference equations, Economics Letters 22, 247-250. Fujimoto, T., 1987b. Global asymptotic stability of non-linear differential equations II, Economics Letters 23, 275-277. Okuguchi, K., 1976, Expectations and stability in oligopoly models (Springer-Verlag, Berlin/Heidelberg/New York). Okuguchi. K., 1977, Mathematical foundations of economic analysis (in Japanese) (McGraw-Hill Kogakusha, Tokyo). Ortega, J.M. and W.C. Rheinboldt, 1970, Iterative solution of nonlinear equations in several variables (Academic Press, New York/London). Szidarovszky, F. and S. Yakowitz, 1978, Principles and procedures of numerical analysis (Plenum Press, New York).