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AD-Am" 155 BtO~i UNkIV PROVIDENCE RI LEFSCI4ETZ CENTER FOR DYNAMI-flO F/f 12/1AN APPROXIMATION SCIEME FOR DELAY EQUATIONS (U)JUN 80 F KAPPEL DAA629-79-C-0161
UNCLASSIFIED AFOSR-TR-80-0924 It
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ICROOP REOUIO IET HR
UNCLASSiI IDSECURITY CLASSIPF.CATION OF THIS PAGE (*%Oen Vat:,Entered) /
REPORT DOCUMENTATION PAGE HEF6R _GeFORM1
A4EPORT N I(fM 9 OT CESO O R PE E
1N APPROXIMATION SC&iEDE FOR DELAY EQJATIONS. Interim ,'PERFORMING 070 RE0ONUMBER
Franz/Kappel[7-39
Providence, RI 02912
00 Boiling AFB, Washington, DC 20332 -f-UBRO AE
14. MONITOIiNG AGENCY NAME & AOORESS(it different from Controilin-1Olh ce) IS. SECURITY CLASS. (of this8 report)
SCHEDULE
16. DITIUINSTATEMENTt (o this Report)ATO/OWNRDIO
Aproe foIubi elae distribution unlimited.17. DISTR19UTION STATEMENT (of the abstract entered In Block 20, It different from Report)
III. SUPPLEMENTARY NOTES D TFCICr19. KEY WORDS (Continue on reverse side It necessary and identify by block number)A
20. ASST RACT (Continue on reverse aide Ui necessary and identify by block number)
WE PRESENT A GENERAL APPROXIMATION SCHEME FOR AUTONOMOUSFDE's WITH LIPSCHITZEAN RIGHT-HAND SIDE. OUR APPROACH IS BASED ONAPPROXIMATION RESULTS IN SEMIGROUP THEORY. THE RESULTS AREAPPLIED To NONLINEAR AUTONOMOUS FDE's IN THE STATE SPACE ]Rnl
SAD O LER JUTONOMOIS FDE's OF NEUTRAL TYPE IN THESTATE
LjI. SPACES I x L' and W1#2. KD I JAN 73 1473 EDITION OW I NOV 68 IS OBSOLETE CSIIE
AFSR.-Th. 8 0 -0 92 4
AN APPROXIMATION SCHEME FOR DELAY EQUATIONS +
by
F. Kappel*
Mathematisches InstitutUniversit~t Graz
Elisabethstrasse 11A 8010 Graz (Austria)
and
Lefschetz Center for Dynamical Systems- Division of Applied Mathematics
/Brown UniversityPr idence, R. I. 02912
June 16, 1980 DIS7 oBU1ON STA IMWApp'ovd for publo nlnMu
Distulbutio Uniniltd
This research was supported in part by the Air Force Office of ScientificResearch under contract 4-AFOSR 76-30920, and in part by the UnitedStates Army Research Office under contract #ARO-DAAC29-79-C-0161.
+ Invited lecture, International Conference on Nonlinear Phenomena inMathematical Sciences, Univ. Texas, Arlington, Texas, June 16-20, 1980.
Appw@Y@ 809 22 106IC
AN APPROXIMATION SCHEME FOR DELAY EQUATIONS
by
ABSTRACT
We present a general approximation scheme for autonomous FDE's
with Lipschitzean right-hand side. Our approach is based on approximation
results in semigroup theory. The results are applied to nonlinear
autonomous FDE's in the state space en x L 2 and to linear autonomous
FDE's of neutral type in the state spaces 10'x L and W ' 2
4R FURCE UFJ'C OF SCZENTIFIC RESE kCd (ASc)WJOTICE OF TRANSMITTAL TO DDC'TLS technical report has been revieweamW'fved for publ.ic release IAM 19.. and 1s
aftnaotaton in unhi tea.0 (
f) 1 DDC UtB
Distwifeaton
A_._il .rd/or
1. Introduction.
In recent years one can see considerable interest in approximation of
delay systems by ordinary differential equations. Results of this type
proved to be very useful for the numerical solution of optimal control
problems and identification problems (see [21,[5], [63). A very successful
approach uses abstract approximation results in semigroup theory and applies
these to the semigroups associated with autonomous delay systems. This
approach goes back to [4] (see also [5]) where control problems involving
linear autonomous functional-differential equations and approximation of the
state by step functions are considered. A considerable improvement over the
scheme developed in [53 was obtained in [7] where for the same class of delay
systems a scheme is developed which allows approximation of the state by spline
functions. In this paper we give an abstract formulation of the scheme
developed in [71 which can be applied to autonomous delay systems with
globally Lipschitzean right-hand side. We also indicate the application of
the linear version of the scheme to autonomous neutral delay systems. Since
by lack of space we cannot give a detailed discussion of the relevant literature,
we include a rather complete list of papers in the references ([1,3,5,7,13,14,15,
16,17,18,19,20,22,23,24]) and refer to the discussions given there (see es-
pecially Section 5 of [5]).
2. An Approximation Scheme for Semigroups of Nonlinear Transformations.
Throughout this section X will be a Hilbert space with inner product
(.') and norm 11. A-family T(t), t > 0, of globally Lipschitzean
operators X - X is called a semigroup of type w, w EIR, if the following
properties are satisfied:
2.
(i) T.(O) I.
(ii) T(t+s) - T(t)T(s) for all t,s > 0.
(iii) For any x E X the map defined by t * T(t)x is
continuous on [0,-).
(iv) IT(t)x - T(t)yI < e'tlx-yl for all x,y E X and t > 0.
If for a not necessarily single-valued operator A on X, the operator
_ is single-valued and defined on X for X sufficiently small-, we
say that T(.) is generated by A if for any x E X
t -
T(t)x - lim (I- t A) -nx, t > O,n
uniformly with respect to t in bounded intervals. Of fundamental importance
is the generation theorem by Crandall-Liggett ([121):
Let A be a densely defined operator on X. If A - wl is dissipative
for some w EIR and if range (I-XA) - X for sufficiently small X,
then A generates a semigroup of type w on X.
If A is single-valued then A - wl is dissipative if and only if
(Ax-Ay,x-y) <Wlx-y1 2 for all x,y C dom A. But note, that the Crandall-
Liggett Theorem is valid in general Banach spaces. Fundamental for our approach
is the following approximation theorem for semigroups of type W:
Theorem 1 ([10]). Let AN, N - 1,2,..., and A be single-valued operators
on X such that dom AN dom A for all N and dom A X. Assume that the
.........
3.
following conditions are satisfied:
(i) There exists a XO > 0 such that
range (I-XA) - range (I-XAA) - X
for N - 1,2,... and all A E (O,X0).
(ii) There exist real constants WN, N - 1,2,..., and w such
that the sequence (WN) is bounded above and AN - WN'
and A - wi are dissipative.
. (Aiii) There exists_a subset G of dom A such that (I-TA)G X K
for X sufficiently small and
x - Ax as N+W
for all x E G.
Then AN and A generate semigroups TN(-) and T(.) of type wN
and W, respectively, and for all x E X
lim T,(t)x - T(t)xN-w
uniformly on bounded t-intervals.
This theorem follows immediately from Theorems 4.1 and 3.1 in [101 and
is also true in general Banach spaces.
Definition 1. Assume that A is an operator on X. A sequence {XN,PN,AN},
N - 1,2,..., is called an approximation scheme related to A if
(i) are finite dimensional linear subspaces of dom A, N = 1,2,...
(ii) N: X 9X N is the orthogonal proj%%etion onto X N - 1,2,...
(iii) A; - PN N, N " 1,2,...
4.
Condition (i) in this definition is the most restrictive one. It could
mean that only the trivial approximation scheme (0,0,0) for all N is
related to A. If A generates a semigroup of some type W it is of interest
to know when the operators AN generate semigroups TN(.) converging to T(.).
Theorem 2. let A be a densely defined single-valued operator on X such
that A - wl is dissipative for some w EX and range (I-)A) = X for X
suff iciently small. Furthermore, let {XNPN,AN} be an approximation scheme
related to A. Assume that the following conditions are satisfied:
(i) him PNx - x for all x E X.
(ii) There exists a set G C doam A such that (I-XA)G = X
for X sufficiently small and
lm APNx = Ax for all x E G.. N-' m
(iii) For each N there exists a AN > 0 such that
range (I-AX) W X
for all X E (o,AN).
Then the following is true:
a) AN is continuous on X and generates a semigroup of type w
on X, N m 1,2,...
b) TN(t)XN c XN for t > 0, N - 1,2,...
c) lim TN(t)PNx - T(t)x for all x C X uniformly on bounded
t-intervals, where T(.) is the semigroup generated by A.
i 5.
Proof. Condition (ii) of Theorem 1 is satisfied with wN - w. This follows
from
(ANx-1y,x-y) - (PN [APNx-APNY] ,x-y) =
(APNX-APNY,PNX-PNy) < WIPNX-PNy 2 2wx-yl
for all x,y in X. Here we have used PN P For x E G we have
JA ANx-A_ _ !. I NPN x-PNAx1 + IPN - i
< I X I + IP Ax -Ax.
By conditions (ii) and (i) both terms on the right-hand side vanish as N .
Thus, also condition (iii) of Theorem 1 is satisfied. It only remains to
prove that range (I-XA) = X for N - 1,2,... and all X E (O,X0 ), I0 a
positive number not dependent on N, and that A is continuous on X.
We have range (I-X(AN-wI)) - range (I- X AM) - X for all X with
0 < X(l+xw)- < X N Since A - wI is dissipative, we immediately get
(cf. [8; p.73] or [9; p.23]) range (I-)(AN-wI)) = X for all X > 0, i.e.,
range (I-A,) - X for X E 1 This proves condition (i) of Theorem 1
with Xo = " Since A. - WI 'is m-dissipative and defined on all of X, it
is also demicontinuous, i.e., continuous from X with the norm topology into
X with the weak topology. But AX c N and do. N < - imply that AN
is continuous. Finally, conclusion b) follows from AN C XN.
Corollary 1. If A is linear, then the conclusion of Theorem 2 holds without
assuming condition (iii) explicitly. Moreover, TN(t) is given by
TN(t) - exp ANt.
N .,.. . .. . .. .. . w iii i I Ili
6.
Proof. The conditions on A imply (also in the nonlinear case) that A is
closed (cf. [8; p.75]). Then also APN is closed and defined on all of X.
Therefore, by the Closed-Graph Theorem APN is a bounded linear operator.
The rest of the proof is clear.
In the linear case we also can replace the condition "range (I-XA) - X
for A sufficiently small" by "A is the infinitesimal generator of a. C-semi-
group,.- This is a-consequence-of the Tmer-Phillips Theorem (see for ins-tance-
[21; p.161).
The following corollary covers a situation taylored for delay systems:
Corollary 2. let AO be a closed linear operator with dom A0 X and A,
be an operator with do% A, dom Ao such that
lx-.lyl _ L 11lx-yl
for all x,y 6 dor%, where L > 0 and 11.11 is a norm on dom %. Then
condition (iii) of Theorem 2 holds for any approximation scheme {UNPN,-A}
related to A.- A0 + A.
Proof. The conditions on', AO '=ply that A.,0 - PNAoPN is a bounded linear
operator, A - l-,ol. We have to prove that (I-A.Vy - x has a solution
*y for any x E X provided A is sufficiently small. This equation is
equivalent to
y "X-(I-A N 0 'P;Pjy + (I-XMO) "1 : h(y)
if X C (O, 0 1 ). Then for all y,y 2 C X
7.
Ih(y1)-h(y2) I AX(l-Y)-IAlPNyl-AiPNY2I
< AL(1V)-c II PNY-PNY2 < 0NL(l X)-1 yl-y2I.
Here we have used the fact that two norms on a finite dimensional space are
equivalent. Note, that PNy1 and PNy2 are in XN. The estimate shows
that h is a contraction for X sufficiently small.
Using, for instance, Theorem II of [12] and the fact that the Peano-
Existence Theorem is valid in finite dimensional spaces, we see that
u(t) TN(t)PNx for any x E X is the solution of
u(t) = ANu(t), t > 0,
u(0) - PNx,
which is an initial value problem for an ordinary differential equation on
X. For calculations one has to choose a basis for XN and to represent A
and PNx with respect to this basis (see [7] and [15] for more details).
3. Applications to Functional Differential Equations.
a) Nonlinear Equations in Rn x L
We consider autonomous equations
x(t) - f(xt), t > 0, (3.1)
with initial data
x(O) - A Cln, x(s) * *(s) a.e. on [-r,0], (3.2)
2,4
8.
r > 0, where E L (-r,OR). For a function x: [-rc) ,JRn, a > 0, x t
is defined by xt(s) - x(t+s), s E [-r,0]. A solution of (3.1), (3.2) is a
function x on an interval [-r,a), a > 0, such that (3.2) holds and x(t) =
n + ds for t E [O,a). We assume that f is a map .Y2(-r,O 1R) +IR
(_V2 is the linear space of square integrable functions) satisfying
(hl) For any function. x in L2(-r,a$R n), a > 0, which is continuous on
-... . . [0 ) t -If(x )- uniquelydefines a map in Ll(OOa'n).
(h2) There exist constants L > 0 and rj, J - 0,...,m, 0= r0 < r1 < ... < rm r,
such that for any 0,* in .V2(-r,oJRn)
f f(*)j < L( I 10(-r )-*(-r )I + 10-0 2).j0J L
Under conditions (hl), (h2) problem (3.1), (3.2) has a unique solution
n 2 nx(t;n,O) on [-r,o) for any (n,O) E X =IR x L (-r,0 In). The family of
operators T(.) defined by T(t)(n,o) = (x(t;n, ),xt(0,)), t > 0, (n,O) E X,
is a strongly continuous semigroup of globally Lipschitzean operators on X
(cf. (17]). X is a Hilbert space with norm I~l,*)Ig = (l 2 + Jl(s)i2g(s)ds)1/2-r
and corresponding inner product, where I- denotes the Euclidean norm in ,n
and the weighting function g is defined by
g(s) - m-j+l for s E (-r, J - ,...,m
Define the operator A by
doam A - {(,(0),,)I, E Wl 2 (1r,0Rn)},
A- (f(O)A), * E
9.
Theorem 3. Let {XNPN'AN} be an approximation scheme related to A such
that
(i) lim PNx = x for all x E X.N-m
(ii) There exists an integer k > I such that for 4E ck(-r,0iRn)
O(0) and - 4) in L2 (-r,O"Rn)
as N - o. Here 0N E W1,2 ( -r,ORn) is defined by
PN(O(o).,O) - (N( 0),'N) E dom A.
Then, for each N, AN generates a semigroup TN(-) on X such that for all
xEX
ha TN(t)Psx = T(t)xN4)-
uniformly on bounded t-intervals.
Indication of proof -(a complete representation of the results
concerning equation (3.1) will appear elsewhere (15]).
Step 1: We have dom A - X. For the set G in condition (ii) of Theorem 2
we take G = {(k(O),)I, C Ck(-r,OeItn)). Using (h2) one shows that there exists
an W E R such that A - WI is dissipative in X. The idea to introduce a
weighting function is due to G.F. Webb [25] and is essential here. In order
to verify the range condition on A one observes that (I-XA)(O(O),O)
(n,) E X is equivalent to O(s) - eS/0(0) - e(S-)/(T)dT, s E [-r,01,Er
and *(0) - h(*(O)), where h(a) - n + Xf(e* a - . f (")/.*(T)dT) for
a E M n . Using (h2) one shows that h is a contraction on en for X
sufficiently small.
10.
Step 2: By definition of A we see that APNx A x in D is equivalent to
f-N* f($) and *N 4 in L2 (-r,0"Rn). But f<> N f(O) follows from
(ii) and (h2). Therefore, condition (ii) of Theorem 2 is satisfied.
: In order to apply Corollary 2 we write A - A + Al, where
o( (0),O) = (0,€) and A1 (0(0),O) - (f(O),O) for 0 E Wl92(-r,ORn). Then
A0 is a densely defined and closed linear operator (see [17]). (h2) implies
- that- A- satisfies the -conditton in Corollary 2 with I1((o),1) = sup 10(s)t.-r<s<0
Thus, condition (iii) of Theorem 2 is satisfied.
Step 4: The semigroup generated by A is indeed T(.) (cf. [17]).
The conclusion of Theorem 3 in the case of spline approximation was also
obtained by H.T. Banks in [3] by a different method under the additional
assumption that f is differentiable.
b) Equations of Neutral Type in IRn X L2 (-r,O$Rn)
ni nLet D and L be bounded linear functionals C(-r,0. R) IR
defined by
m 10D( ) 0 (O) - I B (-rj- B(s) b(s)ds,
I(* A A,(-r ) + A(s) (s)ds,
where 0 =r 0 < rI < ... < rm r. 'AJ,B are n x n matrices and A(.),B(')0 1 m
L2 0 nxnare in L(-r,01 ).
~-~-v
- *.. -* '
~11.
As before we take X En x L2(-r,0eRn). Following an idea given in
[11] we define the operator A by
dom A - {(D(,),)I, E Wl' 2(-r,on),
A(D(O),O) = (L(,),O), 0 E W192
X will be supplied with the norm I(i,) , where the weighting function g
now is also dependent on D:
g(s) - m-J+lcti(s+ri-l) for s E (-rj ,-rj) j =,
a m(m+2) IB 12 j =
rj I -r J - 1 , .
Theorem 4.
a) A generates a C0-semigroup T() of bounded linear operators on X.
b) Let {NPNAN) be an approximation scheme related to A satisfying:
(i) lim P x - x for all x E X.N4
(ii) There exists an integer k > 1 such that for
E ck(-ron) 0N(O) -+ 0(0) and N 0 in L (-rO'In)
as N -p , where 0N E W1,2 (-r,O"n) is defined by
PN(D(O),O) - (D(ON),ON).
Then, for each N, AN is a bounded linear operator which generates the Co-semi-
group TN(t) - exp ANt, t > 0, and for all xC X
lira TN(t)PNx T(t)x
uniformly on bounded t-intervals.
12.
The proof of this theorem proceeds in similar steps as that for Theorem
3 and can be found in [141. If z - (D(f),f) with f E C(-r,ORn) then
T(t)z = (D(xt) ,xt), t > 0, where x(t) is the solution of the neutral equation
d-at D(xt) - L(xt), t > 0,
with x(s) - *(s), s E [-r,0]. Theorem 4 provides an approximation of
t + (D(xt),xt) t > 0, in X. If we could prove that TN(t)PNz + T(t)z also
with respect to the sup-norm, then we would get an approximation of x(t)
itself (see (14], Section 6 for details and preliminary results).
c) Equations of Neutral Type in W 12(-rO n )
Let the operators D and L be as under b) and put D(O) (0) -D().
The Cauchy problem
m 0O
x(t) - L(xt) + I B x(t)-r + B(s)x(t+s)ds - L(xt) + D(xt), t> ,j-l J -r
x(s) - O(s), s E [-r,0,
has for any 0 E Wl 2 (-r,O'*n) a unique solution x(t; ) on [-r,co). We take
X = W1'2(-r,o.1n) and define the family T(.) of operators by
T(t) O xt(0), t > 0, f E Wl 2 (_r,O1Rn).
T(.) is a Co-semigroup of bounded linear operators on X with infinitesimal
generator A given by
' r , °'" ... I' ll ' ° ' - -NONE...
13.
do. A - E1 W (W'2 (-r,O"eI) and 0(0) () + L(),
AO * dom A.
Note, that dom A depends on D and L. We supply X with the norm
.101g -(10()12+ ,()12g(s)ds) 1 /2 , where g is as under b) with
2m2 12
C-J r -rj- IB I j = 1,...,.
Then there exists an w EIR such that A- wlI is dissipative in X (cf.
[16]).
Theorem 5. Let {XNPNAN} be an approximation scheme related to A such
that:
(t) Iim PNO for all 0 E X.N-
(ii) There exists an integer k > 2 such that for 0 E ck(-r,0"gRn )
.. . L2(-r,0 1n)iii(o) (0) and ;N in L 2 -, as N
where N E W2 2 (-r,0"en) is defined by PN- ON'
Then each AN is a bounded linear operator which generates the C0-semigroup
TN(t) - exp ANt on X and for all E X
lim TN(t)PNO - T(t)o
uniformly on bounded t-intervals.
For a proof of this theorem see [16). Theorem 5 provides, for initial
14.
data in W1'2(-r,0JR n), an approximation of xt with respect to the sup-norm
and of xt with respect 'to the L2-norm. On the other hand, the subspaces
and projection PN depend on the right-hand side of the equation even
if D(O) - (0).
4. Spline Approximation
In this section we indicate that subspaces of spline functions can be
used in order to get concrete algorithms using the schemes presented in Section
3.
In the situation of Section 3, a) and b) we can take any sequence of
subdivisions of [-r,O] with mesh points tN, j - 1,...,k, such that
maxIt -tJ1Ii tNtil < 0 < -. In case of Section 3, a) we define XN tojJ j
be the subspace of all elements (0(0),0) such that 0 is a first order,
cubic or cubic Hermite spline with knots at the tN, respectively. We have
dim XN - (kN+l)n, (kN+3 )n or 2 (kN+l)n, respectively. Similarly in the case
of Section 3, b) XN is the space of all elements (D(O),O) with 0 as
above.
In the situation of Section 3, c) we have to satisfy the boundary
condition 0(0) - D($) + L(O) and to guarantee more smoothness. One has to
take subdivisions of [-r,O such that the points -rj, J - 0,...,m, are
meshpoints and restrict D and L. In the case of cubic splines we have to
assume B1 - ... - 1Bml - 0, B(s) A(s) 0. Then XN is the subspace of
all cubic splines with knots at the meshpoints tN satisfyingm
0(0) - 0Aj(-r) + B $(-r). In the case of cubic Hermite splines we assumeJ=0 3
15.
B(s) A(s) E 0 and take XN to be the subspace of all cubic HermiteNm m
splines with knots at the t such that j(0) = A.O(-r) + I B0(-r)j J.i0 i -i
For details, explicit calculation of bases for ., matrix representa-
tions of AN and numerical examples, see [7], [14] and [16].
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(2] H.T. Banks, Spline-based approximation methods for control and identifica-tion of hereditary systems, Proc. Int. Symp. System Optimization & Analysis,Rocquencourt, France, Dec. 11-13, 1978, Lecture Notes in Control andInformation Sciences 14 (A. Bensoussan and J.L. Lions, Eds.), pp. 314-320,Springer-Verlag, 19797
[3] H.T. Banks, Identification of nonlinear delay systems using splinemethods, Proc. Int. Conference on Nonlinear Phenomena in MathematicalSciences, Univ. Texas, Arlington, Texas, June 16-20, 1980, to appear.
[4] H.T. Banks and J.A. Burns, An abstract framework for approximate solutionsto optimal control problems governed by hereditary systems, in "InternationalConference on Differential Equations" (H. Antosiewicz, Ed.), Academic Press,New York (1975), pp. 10-25.
[51 H.T.Banks and J.A. Burns, Hereditary control problems: Numerical methodsbased on averaging approximations, SIAM J. Control Optimization 16 (1978),169-208.
[61 H.T. Banks, J.A. Burns and E.M. Cliff, A comparison of numerical methodsfor identification and optimization problems involving control systems withdelay, TR 79-7, Lefschetz Center for Dynamical Systems, Brown University,Providence, R. I.
[71 H.T. Banks and F. Kappel, Spline approximations for functional differentialequations, J. Differential Eqs. 34 (1979), 496-522.
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[9] H. Brezis, Operateurs Maximaux Monotones, North-Holland 1973.
[10] H. Brezis and A. Pazy, Convergence and approximation of semigroups ofnonlinear operators in Banach spaces, J. Funct. Analysis 9 (1972), 63-74.
(11] J.A. Burns, T.L. Herdman and H.W. Stech, The Cauchy problem for linearfunctional differential equations, preliminary draft presented at theonf. Integral and Functional Differential Eqs., Morgantown, West
Virginia, June 1979.
[12] M.G. Crandall and T'M.Liggett, Generation of semigroups of nonlineartransformations on general Banach spaces, Am. J. Math. 93 (1971), 265-298.
(13] A. Halanay and V1. Rasvan, Approximations of delays by ordinary differen-tial equations, INCREST - Institutul de Matematica, Preprint series inMathematics No. 22/1978.
[14] F. Kappel, Approximation of neutral functional differential equationsin the state space Rn x L2, to appear in "Colloquia MathematicaSocietatis Jinos Bolyai, Vol. 30: Qualitative Theory of DifferentialEquations", Janos Bolyai Math. Soc. and North Holland Publ. Comp.
[15] F. Kappel, Spline approximation for autonomous nonlinear functionaldifferential equations, Manuscript, June 1980.
[16] F. Kappel and K. Kunisch, Spline approximations for neutral functionaldifferential equations, Inst. fUr Math., University of Graz, PreprintNo. 5, July 1979.
[17] F. Kappel and W. Schappacher, Autonomous nonlinear functional differen-tial equations and averaging approximations,J.Nonlinear Analysis: TheoryMethods and Appl. 2 (1978), 391-422.
[18] F. Kappel and W. Schappacher, Nonlinear functional differential equationsand abstract integral equations, Proc. Royal Soc. Edinburgh 84A (1979),71-91.
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