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Asymptotic behaviour of the solutions of differential equations with a small parameter in Banach space
0041-5553/79/0601-0231$07.50/O
U.S.S.R. Compur. Maths. Math. Phys. Vol. 19, pp. 231-233 0 Pergamon Press Ltd. 1980. Printed in Great Britain.
ASYMPTOTICBEHAVIOUROFTHESOLUTIONSOF DIFFERENTIALEQUATIONSWITHASMALLPARAMETER INBANACHSPACE" N.A. MAGNITSKII Moscow (Receirjed 26 Januav 1978)
THE ASYMPTOTIC behaviour of the solutions of differential equations with a small parameter coefficient of the derivative in an arbitrary Banach space is described when there is no uniform asymptotic stability of the corresponding operator equation. We consider in the Banach space B a linear differential equation with a small parameter before the derivative
(1)
We make the following assumptions: a) the vector-function f(t) with values in B and the operator-function A(r) with values in [B] are continuous on [0, T] ; b) the function
.iIA (t) I = lim h4O
Ill+hA (t) I(- 1 h
is strictly less-than zero in the interval 0 < r < T and has zeros of the m-th and n-th orders, respectively, at the points t = 0 and t = T (a proof of the existence of the function A [A(r)] can be found in [ 1] ); c) the degenerate equation 0 = A(r)u t f(r) has a unique continuous differentiable solution x(t), and t~z’(t)
*Zh. vjkhisl. Mu. mat. FL!.,
=
max /z’(t) iI< M. 0<1CT
19, 3, 774-776, 1979. 231
232
N. A. Magnitskii
It is natural to pose the following question: will the solutions x(t, /.A)of Eq. (I) with the initial condition x(0, cl) = xo converge as cc+ 0 to the solution x(t) of the degenerate equation, and if they will, then at what rate? This formulation of the question is close to formulations of the asymptotic behaviour of the solutions of systems of differential equations with a small parameter before the derivative considered in [2,3 ] and elsewhere. In the theorems given in [2,3] uniform convergence (with an accuracy up to the corresponding boundary layer) as I-(+ 0 of the solutions of Eq. (1) to the solution of the degenerate equation with order O@) was ensured by the condition of uniform asymptotic stability of the quiescent point of the corresponding associated system of equations. In our case this condition is not satisfied even for the scalar equation, since the function A(r) = A [A(r)] h as zeros in the interval [0, T] ; therefore it is natural to expect that the order of convergence of the functions x(t, cc)to x(r) as p + 0 and the properties of the corresponding boundary function will depend essentially on the orders of the zeros of the function A [A(r)] at the ends of the segment [0, 7’j . Theorem
If conditions a) - c) are satisfied the representation
holds, where k=max(m, n),
‘r:(l,~)i.=O(~:,‘(k+‘l),
(3)
Proof: Let the operator U(r, /A)be the Cauchy operator of Eq. (l), that is, a solution of the equation au -=dt
A (r)
u(o,~)=l.
u,
P
We represent the function x(t, 1) in the form (2). Then the function ~(t, P) will satisfy the equation dx(t)
du P -=A((t)c-pdt
~KJ,cL)=o,
df
from which it follows that : v(t,p)=
-
JCT(4 0
T‘t p)
-
dx (~1 dr
dT.
From the latter equation, taking into account condition c), we obtain the estimate
liU(t,T,p);:dT.
Ilv(f.p)/:df
J 0
We now use the following two lemmas.
233
Short communications
Lemma 1
For the norm of the evolutionary (in the terminology of [43) operator of Eq. (1) the following estimate holds:
A proof of Lemma I can be found in [l] . Lemma 2
Let the continuous function g(t) be strictly greater than zero in the interval 0 < f < T and have zeros of the m-th and n-th orders, respectively, at the points t = 0 and t = T. Then as p + 0 the following estimate holds:
A proof of Lemma 2 can be found in [5]. Now applying to the inequality for the norm of the function v(r, ,u) Lemmas 1 and 2 and taking into account condition b), we obtain the estimate (3). Lemma 1 also implies the estimate (4) for the norm of the Cauchy operator U(r, cc), the function U(t, II) [Q--I(O) ] being the boundary function in this problem. The theorem is proved. Remark. From what has been proved above it is easy to see that if the function A[A(r)] is less than zero everywhere on [O, 7’j , then :‘I,(I, p) =0(p) Tramlated bv J. Berry. REFERENCES 1.
LOZINSKII, S. M. Estimation of the errors of the numerical integration of ordinary differential equations. 12~. Vuror, 5, 52-90, 1958.
2.
TIKHONOV, A. N. Systems of differential equations containing small parameters before the derivatives. Matem. sb., 31 (?3), 3,575-586, 1952.
3.
VASIL’EVA, A. B. and BUTUZOV, V. F. Asymptotic expansions of the solutions of singuZar2y perrurbed equations. (Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykh uravnenii), “Nauka”, Moscow. 1973.
4.
DALETSKII, Yu. 1. and KREIN, M. G. Stability of the solutions of differential equations in Banach space (Ustoichivost reshenii differentsial’nykh uravnenii v banakhovom prostranstve), “Nauka”, Moscow, 1970.
5.
MAGNITSKII, N. A. On the approximate solution of some Volterra internal equations of the first kind, VestiMGU. Ser. vjQzhisl. matem. kibernetika, NO. 1,91-96, 1978.
U.S.S.R. Compur. Maths. Math. Phys. Vol. 19, pp. 231-233 0 Pergamon Press Ltd. 1980. Printed in Great Britain.
ASYMPTOTICBEHAVIOUROFTHESOLUTIONSOF DIFFERENTIALEQUATIONSWITHASMALLPARAMETER INBANACHSPACE" N.A. MAGNITSKII Moscow (Receirjed 26 Januav 1978)
THE ASYMPTOTIC behaviour of the solutions of differential equations with a small parameter coefficient of the derivative in an arbitrary Banach space is described when there is no uniform asymptotic stability of the corresponding operator equation. We consider in the Banach space B a linear differential equation with a small parameter before the derivative
(1)
We make the following assumptions: a) the vector-function f(t) with values in B and the operator-function A(r) with values in [B] are continuous on [0, T] ; b) the function
.iIA (t) I = lim h4O
Ill+hA (t) I(- 1 h
is strictly less-than zero in the interval 0 < r < T and has zeros of the m-th and n-th orders, respectively, at the points t = 0 and t = T (a proof of the existence of the function A [A(r)] can be found in [ 1] ); c) the degenerate equation 0 = A(r)u t f(r) has a unique continuous differentiable solution x(t), and t~z’(t)
*Zh. vjkhisl. Mu. mat. FL!.,
=
max /z’(t) iI< M. 0<1CT
19, 3, 774-776, 1979. 231
232
N. A. Magnitskii
It is natural to pose the following question: will the solutions x(t, /.A)of Eq. (I) with the initial condition x(0, cl) = xo converge as cc+ 0 to the solution x(t) of the degenerate equation, and if they will, then at what rate? This formulation of the question is close to formulations of the asymptotic behaviour of the solutions of systems of differential equations with a small parameter before the derivative considered in [2,3 ] and elsewhere. In the theorems given in [2,3] uniform convergence (with an accuracy up to the corresponding boundary layer) as I-(+ 0 of the solutions of Eq. (1) to the solution of the degenerate equation with order O@) was ensured by the condition of uniform asymptotic stability of the quiescent point of the corresponding associated system of equations. In our case this condition is not satisfied even for the scalar equation, since the function A(r) = A [A(r)] h as zeros in the interval [0, T] ; therefore it is natural to expect that the order of convergence of the functions x(t, cc)to x(r) as p + 0 and the properties of the corresponding boundary function will depend essentially on the orders of the zeros of the function A [A(r)] at the ends of the segment [0, 7’j . Theorem
If conditions a) - c) are satisfied the representation
holds, where k=max(m, n),
‘r:(l,~)i.=O(~:,‘(k+‘l),
(3)
Proof: Let the operator U(r, /A)be the Cauchy operator of Eq. (l), that is, a solution of the equation au -=dt
A (r)
u(o,~)=l.
u,
P
We represent the function x(t, 1) in the form (2). Then the function ~(t, P) will satisfy the equation dx(t)
du P -=A((t)c-pdt
~KJ,cL)=o,
df
from which it follows that : v(t,p)=
-
JCT(4 0
T‘t p)
-
dx (~1 dr
dT.
From the latter equation, taking into account condition c), we obtain the estimate
liU(t,T,p);:dT.
Ilv(f.p)/:df
J 0
We now use the following two lemmas.
233
Short communications
Lemma 1
For the norm of the evolutionary (in the terminology of [43) operator of Eq. (1) the following estimate holds:
A proof of Lemma I can be found in [l] . Lemma 2
Let the continuous function g(t) be strictly greater than zero in the interval 0 < f < T and have zeros of the m-th and n-th orders, respectively, at the points t = 0 and t = T. Then as p + 0 the following estimate holds:
A proof of Lemma 2 can be found in [5]. Now applying to the inequality for the norm of the function v(r, ,u) Lemmas 1 and 2 and taking into account condition b), we obtain the estimate (3). Lemma 1 also implies the estimate (4) for the norm of the Cauchy operator U(r, cc), the function U(t, II) [Q--I(O) ] being the boundary function in this problem. The theorem is proved. Remark. From what has been proved above it is easy to see that if the function A[A(r)] is less than zero everywhere on [O, 7’j , then :‘I,(I, p) =0(p) Tramlated bv J. Berry. REFERENCES 1.
LOZINSKII, S. M. Estimation of the errors of the numerical integration of ordinary differential equations. 12~. Vuror, 5, 52-90, 1958.
2.
TIKHONOV, A. N. Systems of differential equations containing small parameters before the derivatives. Matem. sb., 31 (?3), 3,575-586, 1952.
3.
VASIL’EVA, A. B. and BUTUZOV, V. F. Asymptotic expansions of the solutions of singuZar2y perrurbed equations. (Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykh uravnenii), “Nauka”, Moscow. 1973.
4.
DALETSKII, Yu. 1. and KREIN, M. G. Stability of the solutions of differential equations in Banach space (Ustoichivost reshenii differentsial’nykh uravnenii v banakhovom prostranstve), “Nauka”, Moscow, 1970.
5.
MAGNITSKII, N. A. On the approximate solution of some Volterra internal equations of the first kind, VestiMGU. Ser. vjQzhisl. matem. kibernetika, NO. 1,91-96, 1978.