- Email: [email protected]
On ordinary differential equations in locally convex spaces
036?-546X% $3 00 j .W Pergamon foumals Ltd.
ON ORDINARY DIFFERENTIAL EQUATIONS IN LOCALLY CONVEX SPACES ROLAND LEMMERT Mathematisches Institut I, Universit%t Karlsruhe, Kaiserstr. 12, D-7500 Karlsruhe 1, FRG (Receioed
12 August 1985; received for publicurion 21 October
1985)
Key wards and phrases: Ordinary differential equations, infinite systems.
SECTION
LET E BE a locally convex Hausdorff consider the initial value problem
1
space, f : [0, l] x E --, E a function,
u’(r) = f& u(r)) >
and ttOE E. We
tE [O, 11, (1)
u(0) = ug
and establish conditions on f and E such that existence and uniqueness theorems are valid for (l), where linear dependence of fwith respect to the second variable is explicitly admissible. In the case E = R’,f(t, u) = Lu our results are best possible. For the specific behaviour of the solutions of (1) even in this case the reader may be referred to Hille [2] and Deimling [l], resp., as well as to the references given there. SECTION
2
First we consider the case E = C’ with an arbitrary index set J # 0, CJ being a locally convex Hausdorff space when endowed with the product topology. The topological dual of @’ is isomorphic to Cj : = (JJ = (qi)ie,, at most finitely many Qi are different from zero}. The weak* topology a(C,, C’) is the finest locally convex topotogy on CI (cf., for example, Schaefer (4]), so that, in particular, bounded sets lay in finite-dimensional subspaces and are bounded therein. A continuous linear map L : CJ + Cl may be understood as a “row-finite” matrix (lq)i,je, (i.e. for every i E J there is an at most finite number of j’s such that /, # 0); the transpose ‘L is then given by the transposed matrix (Z;~)i.i~l,being “column-finite”. The behaviour of the solutions to ’ ’ u’(r) = Lufr),
rEW,
u(0) = ug
(2)
in C’ and to u’(r) = TLu(t),
(3)
o(0) = ug in CJ, resp., was investigated
tE R,
in [3] for J = N. 1385
R. LEMMERT
1386
THEOREM [3]. For J = N the following assertions are equivalent: (a) the spectrum of L is at most countable; (b) problem (2) is uniquely solvable for each u. E CN; the series 2 C L”uo converges in @” for each u. E Q=%,t E R; n=!Jn!
problem (3) is uniquely solvable for each 00 E d=,; = I” the series x - TLnuO converges in CN for each u. E C”, t E A; n=On! rL is locally algebraic;
assertion (f) means that, for each u. E CN, there exists a polynomial 4 with q(‘L)uo = 0. This implies that the sequence of partial sums of an arbitrary series
lies in a finite-dimensional subspace (being dependent of u,,) of C, which is invariant under TL: for if q is such a polynomial of minimal degree (say, of degree n) with leading coefficient 1, then ‘LnuO E U for the subspace U being spanned by uo, TLuo,. . . , rLn-luo, and the elements uo, rLuo,. . ., TL”-luo are linearly independent. In particular, the restriction of ‘L on U with respect to this basis has the matrix representation: 0
0
0
1 0
0
.. a0
...
a,
TL=
where the a, are the coefficients of q. For arbitrary index sets J we have (cf. [3, theorem 31) the following. THEOREM. The following assertions are equivalent: (a) (3) is solvable; (b) there exists a polynomial *
t”
(c) nzo ;
*LnuO converges
Remarks. (1) Problem
q with q(TL)uo = 0 for uo;
in C, for 1 E W.
(3) has at most one solution.
(2) (c) is equivalent to the fact that there is a to # 0 such that $o:
TLnuo converges in C,.
(3) If the coordinates of u. and the elements I, are real and if (3) is solvable, then the solution has only real coefficients.
On ordinary
differential
equations
in locally
convex
spaces
1387
If C’ is ordered by the cone !%8’,= {X= (ei)ls/: g, E I&?,s”i2 0}, then L is quasimonotone increasing in the sense of Volkmann [6], if f, E W, 1, L 0 for i fj. The expected “monotonicity theorem” if u’(t) z Lu(t), L quasimonotone
increasing, u,-, 2 0, then u(t) 3 0 fort 2 0
does not hold, in general, as shown by the shift matrix 1, = 6,.j_, (i E N) in the case J = N: In this case every solution of (2) is of the form u(t) = (n(r), r?‘(t), rl”(0, . . J with any function 77: R-+ @, being arbitrarily often differentiable and u. = (q(O), q’(O), r1”(0),. . .). From the validity of the above mentioned “monotonicity theorem” the uniqueness of the solution of (2) would follow, which is not correct. In the following we assume C, to be ordered by the dual cone (R+), = {Y = (vi): Y E @I, 77iE w, 7i 2 01. LEMMA. If L is quasimonotone
increasing and if (3) is solvable with u. 2 0, then u(t) 2 0
fort 2 0.
Proof. For fixed T > 0, {u(t) : 0 G t s 7J c C, is compact, hence bounded and contained in a finite-dimensional subspace of C,. Therefore, there are finitely many ii, . . . , ik E J with U,(t) = 0 for i f ii, . . . , ik, t E [0, T], and d : [0, T] + Ck solves the system
r7’= Md,
a(0) = 00
with ~4 = (m,), m, = l;,i,, (Co), = (uo)i,, 1 s r, s s k. Since A4 is quasimonotone increasing, d 2 0 on [0, T] according to the monotonicity theorem of Mtiller, hence follows (since T is arbitrary) the proposition of the lemma. THEOREM. If L E Y(@‘)
is quasimonotone
increasing and if rL is locally algebraic, then from
u;(t) 2 Lu(t),
t E [O, T),
u(r) 3 0
follows the inequality u(r) 5 0, t E [0, T) (u;
is
the right-hand side derivative).
Proof. Choose t E (0, 7) and u. E (iw,), arbitrary, (3) we consider
h(s) = (u(s), u(t - s)),
but fixed. With u being the solution of OSsSt;
h is differentiable from the right. (Remark that h is equal to a finite sum of functions being differentiable from the right.) We have h;(s)
= (U;(S), u(t - S)) - (u(s), u’(t - S)) 5 (Lu(s), u(t - s)) - (u(s), ZLu(t - s)) = 0,
1388
R. LEMMERT - .
hence 0 s h(O) = &I, u(r)) 6 h(t) = (u(t), ug), and, since I, u. are arbitrary,
the proposition
of the theorem follows.
SECTION
3
Now, assume E to be a real locally convex space and (pi)ic, to be a system of seminorms which produces the topology of E. To every x E E there is (e.g. according to SchrGder [5]) the element (p,(x)) E Oa’,which we denote by &xl. We have (a) (b) (c) (d)
function from E to rW’, lhll 3 0, llxll= 0 iff x = 0, Ik + ~11c Ikll + It_yll> x, Y E E, ll~ll = IAl+ll, x, Y E E, A E R.
/]*]I is a continuous
Furthermore, it is easily shown that a linear endomorphism A of E is continuous if there exists a row-finite matrix L = (fij)i,j~, with 1, > 0 and Ikil c Llkll> If the set of linear continuous
if and only
x E E.
maps from E into iw’ is denoted by E”, then there follows:
THEOREM. To every x0 E E there is a Q E E” with @(x0) =
lkoll
Q(X) < IIxII,x E E.
and
Proof. According to the Hahn-Banach 4+(x) ~PP~(x), and set a(x) = (&(x)>~,J.
theorem
choose
&, i E J, with @,(x0) = p,(x,),
THEOREM (Mazur). If u : [0, 1) + E is differentiable, then Ilu(t is differentiable right-hand side, and there is a @+ E E” with 0(x) G Ikll, x E E, and
from the
MM- = @+(u’(Q), @+W>> = IbWll. Proof. The proof follows again from the validity of the corresponding seminorms: to each i E J there exists pi,+ E E’ with [Pi(u(t))]k
= @i.+(u’(t)>
f
Pi(K(t>)
=
@i.+
(u(t)>
7
4i,+
Cx>
c
assertions
for
PiCx>,
and it is sufficient to choose @+ = (~i,+)iE,. F: E+ E is called L-dissipative, if from x, y E E, qbE E”, @(x - y) = IIx- yll, Q(z) s llz//, z E E it follows that @(F(x) - F(Y)) s Lllx - ~11. THEOREM. Let f: [0, 1) x E ---,E be L-dissipative with respect to the variable x, let L be quasimonotone increasing and rL be locally algebraic, then (1) has at most one solution. Proof. If ul, u2 are two solutions and if d(r) = Ilul(r) - u2(r)ll, then we have according to the
On ordinary differential equations in locally convex spaces
theorem of Mazur (t 2 0): There-exists
a+ E E”, Q+(x) s /bll. 0+(ui(t)
d;(r) = @+(4(r)
1389
- u:(r)) = d(r), and
- 4(r)>
= Q+((f(r, Ui(f)) -907
4)))
G U(t), d(O)= 0,
hence d(t) s 0, according to the theorem in Section 2. THEOREM. Let E be sequentially
complete and let f: [0, l] x E-,
I\f(r, 4 - f(r, Y)\\s ~~~~- ~11,
x, YE Et
E be continuous.
tE[O,l],
If
LSO,
where ‘L is locally algebraic, then (1) has exactly one solution. Proof. Assume u,(t) = uo. We consider the sequence of successive approximations by u:,+,(r)
=f(c 4$>),
n 2 1,
f E [O,11,
4l+,Ko = uo* Since functions being continuous in E are Riemann integrable, u,+,(t)
= uo +
it is equivalent
I‘f(h a>>b, 0
so that the iteration is meaningful.
As usual, the estimation
IIkl+i(0 - 4z(~)ll s L”_’
&l,,1,~,
follows, and we only have to prove the uniform convergence
- ul(t>ll of
for fixed E E R’. For 17E C, we have
According to the assumption 7L is locally algebraic. Hence
converges uniformly on [0, 11, i.e. $ Therefore,
(u,,);,,
Lk ; E.
converges uniformly on [0, 11, and the limit solves (1).
that
defined
1390
R. LEMMERT
Remark. The assertion of the abovFtheorem urges the following “fixed point theorem” which can be proven with the same method. We did not succeed, however, in deriving the existence of the solution of (1) from this fixed point theorem, except in the case of o(~L) being bounded. FIXED POINTTHEOREM. Assume (F,(qJis,) to be a locally convex space, M C F a sequentially M a map with
complete subset, and T: &I-,
l/TX - Qll s L/lx - Yll,
x, y E A4
Let L = (fii)i.j~, be nonnegative, ‘L locally algebraic, and let the spectrum o(~L) lay on the open unit circle of @. Then T has exactly one fixed point in M, and the successive approximations converge. REFERENCES 1. DEIMLINGK., Ordinary Differential Equations in Banach Spaces, Springer, Berlin (1977).
2.HILLE E., Pathology of infinite systems of linear first order differential equations with constant coefficients, Anna/i Mat. pura appl. 5.5, 133-148 (1961). 3. LEMMERTR. & WECKBACH A., Charakterisierungen zeilenendlicher Matrizen mit abzlhlbarem Spektrum, Math. Z. 188, 119-124 (1984). 4. SCHAEFERH. H., Topological Vector Spaces, Springer, Berlin (1971). 5. SCHR~DERJ., Das Iterationsverfahren bei allgemeinerem Abstandsbegriff, Math. Z. 66, 111-116 (1956). 6. VOLKMANNP., GewBhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorrlumen, Math. Z. 127, 157-164 (1972).
ON ORDINARY DIFFERENTIAL EQUATIONS IN LOCALLY CONVEX SPACES ROLAND LEMMERT Mathematisches Institut I, Universit%t Karlsruhe, Kaiserstr. 12, D-7500 Karlsruhe 1, FRG (Receioed
12 August 1985; received for publicurion 21 October
1985)
Key wards and phrases: Ordinary differential equations, infinite systems.
SECTION
LET E BE a locally convex Hausdorff consider the initial value problem
1
space, f : [0, l] x E --, E a function,
u’(r) = f& u(r)) >
and ttOE E. We
tE [O, 11, (1)
u(0) = ug
and establish conditions on f and E such that existence and uniqueness theorems are valid for (l), where linear dependence of fwith respect to the second variable is explicitly admissible. In the case E = R’,f(t, u) = Lu our results are best possible. For the specific behaviour of the solutions of (1) even in this case the reader may be referred to Hille [2] and Deimling [l], resp., as well as to the references given there. SECTION
2
First we consider the case E = C’ with an arbitrary index set J # 0, CJ being a locally convex Hausdorff space when endowed with the product topology. The topological dual of @’ is isomorphic to Cj : = (JJ = (qi)ie,, at most finitely many Qi are different from zero}. The weak* topology a(C,, C’) is the finest locally convex topotogy on CI (cf., for example, Schaefer (4]), so that, in particular, bounded sets lay in finite-dimensional subspaces and are bounded therein. A continuous linear map L : CJ + Cl may be understood as a “row-finite” matrix (lq)i,je, (i.e. for every i E J there is an at most finite number of j’s such that /, # 0); the transpose ‘L is then given by the transposed matrix (Z;~)i.i~l,being “column-finite”. The behaviour of the solutions to ’ ’ u’(r) = Lufr),
rEW,
u(0) = ug
(2)
in C’ and to u’(r) = TLu(t),
(3)
o(0) = ug in CJ, resp., was investigated
tE R,
in [3] for J = N. 1385
R. LEMMERT
1386
THEOREM [3]. For J = N the following assertions are equivalent: (a) the spectrum of L is at most countable; (b) problem (2) is uniquely solvable for each u. E CN; the series 2 C L”uo converges in @” for each u. E Q=%,t E R; n=!Jn!
problem (3) is uniquely solvable for each 00 E d=,; = I” the series x - TLnuO converges in CN for each u. E C”, t E A; n=On! rL is locally algebraic;
assertion (f) means that, for each u. E CN, there exists a polynomial 4 with q(‘L)uo = 0. This implies that the sequence of partial sums of an arbitrary series
lies in a finite-dimensional subspace (being dependent of u,,) of C, which is invariant under TL: for if q is such a polynomial of minimal degree (say, of degree n) with leading coefficient 1, then ‘LnuO E U for the subspace U being spanned by uo, TLuo,. . . , rLn-luo, and the elements uo, rLuo,. . ., TL”-luo are linearly independent. In particular, the restriction of ‘L on U with respect to this basis has the matrix representation: 0
0
0
1 0
0
.. a0
...
a,
TL=
where the a, are the coefficients of q. For arbitrary index sets J we have (cf. [3, theorem 31) the following. THEOREM. The following assertions are equivalent: (a) (3) is solvable; (b) there exists a polynomial *
t”
(c) nzo ;
*LnuO converges
Remarks. (1) Problem
q with q(TL)uo = 0 for uo;
in C, for 1 E W.
(3) has at most one solution.
(2) (c) is equivalent to the fact that there is a to # 0 such that $o:
TLnuo converges in C,.
(3) If the coordinates of u. and the elements I, are real and if (3) is solvable, then the solution has only real coefficients.
On ordinary
differential
equations
in locally
convex
spaces
1387
If C’ is ordered by the cone !%8’,= {X= (ei)ls/: g, E I&?,s”i2 0}, then L is quasimonotone increasing in the sense of Volkmann [6], if f, E W, 1, L 0 for i fj. The expected “monotonicity theorem” if u’(t) z Lu(t), L quasimonotone
increasing, u,-, 2 0, then u(t) 3 0 fort 2 0
does not hold, in general, as shown by the shift matrix 1, = 6,.j_, (i E N) in the case J = N: In this case every solution of (2) is of the form u(t) = (n(r), r?‘(t), rl”(0, . . J with any function 77: R-+ @, being arbitrarily often differentiable and u. = (q(O), q’(O), r1”(0),. . .). From the validity of the above mentioned “monotonicity theorem” the uniqueness of the solution of (2) would follow, which is not correct. In the following we assume C, to be ordered by the dual cone (R+), = {Y = (vi): Y E @I, 77iE w, 7i 2 01. LEMMA. If L is quasimonotone
increasing and if (3) is solvable with u. 2 0, then u(t) 2 0
fort 2 0.
Proof. For fixed T > 0, {u(t) : 0 G t s 7J c C, is compact, hence bounded and contained in a finite-dimensional subspace of C,. Therefore, there are finitely many ii, . . . , ik E J with U,(t) = 0 for i f ii, . . . , ik, t E [0, T], and d : [0, T] + Ck solves the system
r7’= Md,
a(0) = 00
with ~4 = (m,), m, = l;,i,, (Co), = (uo)i,, 1 s r, s s k. Since A4 is quasimonotone increasing, d 2 0 on [0, T] according to the monotonicity theorem of Mtiller, hence follows (since T is arbitrary) the proposition of the lemma. THEOREM. If L E Y(@‘)
is quasimonotone
increasing and if rL is locally algebraic, then from
u;(t) 2 Lu(t),
t E [O, T),
u(r) 3 0
follows the inequality u(r) 5 0, t E [0, T) (u;
is
the right-hand side derivative).
Proof. Choose t E (0, 7) and u. E (iw,), arbitrary, (3) we consider
h(s) = (u(s), u(t - s)),
but fixed. With u being the solution of OSsSt;
h is differentiable from the right. (Remark that h is equal to a finite sum of functions being differentiable from the right.) We have h;(s)
= (U;(S), u(t - S)) - (u(s), u’(t - S)) 5 (Lu(s), u(t - s)) - (u(s), ZLu(t - s)) = 0,
1388
R. LEMMERT - .
hence 0 s h(O) = &I, u(r)) 6 h(t) = (u(t), ug), and, since I, u. are arbitrary,
the proposition
of the theorem follows.
SECTION
3
Now, assume E to be a real locally convex space and (pi)ic, to be a system of seminorms which produces the topology of E. To every x E E there is (e.g. according to SchrGder [5]) the element (p,(x)) E Oa’,which we denote by &xl. We have (a) (b) (c) (d)
function from E to rW’, lhll 3 0, llxll= 0 iff x = 0, Ik + ~11c Ikll + It_yll> x, Y E E, ll~ll = IAl+ll, x, Y E E, A E R.
/]*]I is a continuous
Furthermore, it is easily shown that a linear endomorphism A of E is continuous if there exists a row-finite matrix L = (fij)i,j~, with 1, > 0 and Ikil c Llkll> If the set of linear continuous
if and only
x E E.
maps from E into iw’ is denoted by E”, then there follows:
THEOREM. To every x0 E E there is a Q E E” with @(x0) =
lkoll
Q(X) < IIxII,x E E.
and
Proof. According to the Hahn-Banach 4+(x) ~PP~(x), and set a(x) = (&(x)>~,J.
theorem
choose
&, i E J, with @,(x0) = p,(x,),
THEOREM (Mazur). If u : [0, 1) + E is differentiable, then Ilu(t is differentiable right-hand side, and there is a @+ E E” with 0(x) G Ikll, x E E, and
from the
MM- = @+(u’(Q), @+W>> = IbWll. Proof. The proof follows again from the validity of the corresponding seminorms: to each i E J there exists pi,+ E E’ with [Pi(u(t))]k
= @i.+(u’(t)>
f
Pi(K(t>)
=
@i.+
(u(t)>
7
4i,+
Cx>
c
assertions
for
PiCx>,
and it is sufficient to choose @+ = (~i,+)iE,. F: E+ E is called L-dissipative, if from x, y E E, qbE E”, @(x - y) = IIx- yll, Q(z) s llz//, z E E it follows that @(F(x) - F(Y)) s Lllx - ~11. THEOREM. Let f: [0, 1) x E ---,E be L-dissipative with respect to the variable x, let L be quasimonotone increasing and rL be locally algebraic, then (1) has at most one solution. Proof. If ul, u2 are two solutions and if d(r) = Ilul(r) - u2(r)ll, then we have according to the
On ordinary differential equations in locally convex spaces
theorem of Mazur (t 2 0): There-exists
a+ E E”, Q+(x) s /bll. 0+(ui(t)
d;(r) = @+(4(r)
1389
- u:(r)) = d(r), and
- 4(r)>
= Q+((f(r, Ui(f)) -907
4)))
G U(t), d(O)= 0,
hence d(t) s 0, according to the theorem in Section 2. THEOREM. Let E be sequentially
complete and let f: [0, l] x E-,
I\f(r, 4 - f(r, Y)\\s ~~~~- ~11,
x, YE Et
E be continuous.
tE[O,l],
If
LSO,
where ‘L is locally algebraic, then (1) has exactly one solution. Proof. Assume u,(t) = uo. We consider the sequence of successive approximations by u:,+,(r)
=f(c 4$>),
n 2 1,
f E [O,11,
4l+,Ko = uo* Since functions being continuous in E are Riemann integrable, u,+,(t)
= uo +
it is equivalent
I‘f(h a>>b, 0
so that the iteration is meaningful.
As usual, the estimation
IIkl+i(0 - 4z(~)ll s L”_’
&l,,1,~,
follows, and we only have to prove the uniform convergence
- ul(t>ll of
for fixed E E R’. For 17E C, we have
According to the assumption 7L is locally algebraic. Hence
converges uniformly on [0, 11, i.e. $ Therefore,
(u,,);,,
Lk ; E.
converges uniformly on [0, 11, and the limit solves (1).
that
defined
1390
R. LEMMERT
Remark. The assertion of the abovFtheorem urges the following “fixed point theorem” which can be proven with the same method. We did not succeed, however, in deriving the existence of the solution of (1) from this fixed point theorem, except in the case of o(~L) being bounded. FIXED POINTTHEOREM. Assume (F,(qJis,) to be a locally convex space, M C F a sequentially M a map with
complete subset, and T: &I-,
l/TX - Qll s L/lx - Yll,
x, y E A4
Let L = (fii)i.j~, be nonnegative, ‘L locally algebraic, and let the spectrum o(~L) lay on the open unit circle of @. Then T has exactly one fixed point in M, and the successive approximations converge. REFERENCES 1. DEIMLINGK., Ordinary Differential Equations in Banach Spaces, Springer, Berlin (1977).
2.HILLE E., Pathology of infinite systems of linear first order differential equations with constant coefficients, Anna/i Mat. pura appl. 5.5, 133-148 (1961). 3. LEMMERTR. & WECKBACH A., Charakterisierungen zeilenendlicher Matrizen mit abzlhlbarem Spektrum, Math. Z. 188, 119-124 (1984). 4. SCHAEFERH. H., Topological Vector Spaces, Springer, Berlin (1971). 5. SCHR~DERJ., Das Iterationsverfahren bei allgemeinerem Abstandsbegriff, Math. Z. 66, 111-116 (1956). 6. VOLKMANNP., GewBhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorrlumen, Math. Z. 127, 157-164 (1972).