7 Statement of the principle of the stationary point
Corollary 4 Corollary 4 (criterion for a point spectrum). An operator A has a point spectrum on the imaginary axis if and only if the homogeneous
2. We give a construction of a scalar equation u' +a(t)u = f(t) (a
and f are almost periodic functions with an integer two-term basis), which has bounded solutions but not almost periodic ones. For sets 12 and A co is introduced in § 3.4) we have f2 o .9e and A = 9r. Other properties will be mentioned during the course of the construction.
For convenience in reading we give separately a slightly modified version of the classical example of Bohr.
Bohr's example. Let k be the two-dimensional torus, realised as a square of side 2 1T , and let A 1 , A2 be linearly independent numbers.
9 See Levitan [73], p. 109.
Solvability in the Besicovitch class 145
We put
h = {x i , x2}, h t = {xi+ al, x2+ tA.2}, ho= {0, 0}.
We choose numbers tom = i mi A i + /L1 2 (l,n1 and / 2m are integers) so that WI 7 e
M 2/3 < 2com < 2m 2"3 . we set
a(h)= E 04,22 sin (/ m1 x i +/x 2),
a(t)= a(hot )= E 07, 4 2 sin com t.
Obviously, a(t) is almost periodic. Let n (t) =Ç a(s) ds. Then (t) = E (on, (1 — cos (om t)
=E con, sin2 (com t/2).
By using the inequality Isin xi Ix 1/2 for 1x1--... 1 we have
-ri(t).--. E con, sin2 (com t12)
(t2/4) m_,Eiti3/2 (i/m2)
42/4) r ds
Jiti3/2-1 5 2 t2
4(1t1 312 -1)'
that is, lim t _.. n(t) = +co. The numbers l mi can be taken to be odd;
then a(h) is antiperiodic (in the first argument) with period 7T, that is, a(x l +z-, x2) = –a(xi, x2).
Now we come to our construction. We use the notation of Bohr's example. In addition, for 0(h) (h € k) we set by definition
n)=A1—+A. 2 .
at axi ax2
Let a(t) be the function in Bohr's example, and z(t)=
exp Hot a(s) ds); z(t) satisfies the equation z'+a(t)z = 0 and decreases with its derivative faster than any power. We take an interval 4 cf of length 2s small enough that the corresponding portion d = ItA i , tA21= {hot} (t E 4) lies strictly inside the square {0 .--_
x 1 .--.2 ,77-, 0.---. x 2 --. 27}. By using a partition of unity for J we can write
z(t) in the form
c° 1 z(t)=
-,,, n
where the un (t) are concentrated on 4 and are uniformly bounded together with their derivatives un i(t). We put zm =
ET—. (1/n 2)un(t + En). It is clear that z,n (t)= z(t) on some interval
[– Tm, Tm] and that Tm
146 Favard theory
Next, it is not difficult to realise the construction of periodic functions on(h) with the following properties:
(1) On (h ) and a4i(h)/at are continuous everywhere apart from d where they have a discontinuity of the first kind with jumps u(t),
un '(t), respectively;
(2) t/i(h ) and a/i(h)/at are uniformly bounded.
Then we set
4).(h)= E --.0(h"), ml , n
c° I _con It is easy to see that
—Ch') for lt, {hot }. (15)
at -00 n at
We define fm (h) and f(h) by
fm (h)=-0m (h)+ a(h)(1)„t (h), a at
f(h)=-0(h)+ a(h)0(h). a at
By construction, fm(h) is continuous at each point ht {hot}. On the compact part of {hot }, where It' ----. T., 0. and ackmlat have jumps equal to z,, and zn.,', respectively. Since z'(t)+ a (t)z (t) = z in i(t)+
a (hot )z„,(t) = 0 for Iti ----. Tm , the function fm (h) does not have discon- tinuities on this interval. Since the convergence fn, -+ f is uniform on k, f(h) is continuous on
Thus, for the equation
ts' - - P a(h t )u = f(h t ) (16)
the function o(h) is an invariant section with discontinuities (which are not removable) only on {hot }. This means that the solution c k (ht),
where ht {hot }, is N-almost periodic but not almost periodic.
We show that (16) does not have almost periodic solutions. Sup- pose this were not so, then for h 1 = {7r, 0} the equation u' + a(hi t )u =
f(hit) would have an almost periodic solution u(t). The difference
q5(h i t)_ u( t ) is a non-trivial bounded solution of the homogeneous equation. But since a(h) is antiperiodic with period Ir, we have a(hi t )= –a(t), and so the homogeneous equation does not have a non-trivial bounded solution.
Solvability in the Besicovitch class 147 Now we prove the stronger assertion: Ch ) is the unique (essen- tially) bounded invariant section. By assuming non-uniqueness we have a non-trivial bounded section for the homogeneous equations;
we denote it by a (h).
Returning to Bohr's example we set g(h)= —E Wm cos (lm' xi+ / 2.x2).
The series E 0),„2 converges and so g(h) is square integrable over k.
The relation g(h t )— g(h)= f(t, a (h ) ds (for almost all h € Ye) is readily verified by considering partial sums for a(h) and g(h).
The functions a(h) and exp (—g(h)) are non-trivial sections for one and the same scalar equation u'+ a(h t )u =0; consequently, their ratio a (h) exp (g(h)) is invariant under translation on 9e. Since trans- lations are strictly ergodic, this ratio is constant almost everywhere on Ye. We find that exp (—g(h)) is essentially bounded.
Let g = g+ —g be the decomposition into positive and negative parts. From exp (—g) 1—g it follows that g - is bounded, while the antiperiodicity of g implies that g + is bounded.
Thus, we have found that exp (—g) and exp (g) are bounded.
Then there is a point h2 e 9e for which u(t)= exp (—g(h 2t )) and 1/u(t) are bounded. This means that the equation u' + a(h2 t )u =0 has a bounded solution that is separated from zero. But then u; + a (hot )u = 0 also must have such a solution, which contradicts the original properties of Bohr's example. This completes the construction of the required example. We can also give for the homogeneous equation an example of an invariant section that is not equivalent to a continuous one. For this we must consider the equation u; + ia(Ou =0, where a(h) is the function in Bohr's example. The required section is defined by 0(h) = exp (—ig(h)).
Comments and references to the literature
§ 2. The minimax method was given by Favard [106] in the finite- dimensional case and in the case of a uniformly convex space by Amerio (see Amerio & Prouse [2]). Both Favard and Amerio assumed the condition of (two-sided) separation and not of semiseparation, and they studied almost periodic solutions and not N-almost peri- odic ones. The general formulation of the `minimax' Theorem 3 and also the example of a non-compact solution is due to the authors.
For finite-dimensional spaces the condition of semiseparation implies that of separation (see Chapter 7, § 4). This is not so in the infinite-dimensional case. As an example we can take the semigroup of right translations in Y 2(0, co).
148 Favard theory
§§ 3 and 4. Lemma 2 (in a more general form) is in Gel'fand's article [36]. The other results are taken from the works of Zhikov. Note that the problem of completely non-distal extensions and related questions of the existence of N-almost periodic solutions (see § 3, Corollary 1) are open.
§ 5. Questions of compactness occupy a central place in the investiga- tions of Amerio, who started from the well-known works of Sobolev
[102] on the homogeneous wave equation. He succeeded in studying the non-homogeneous wave equation, and in proving the theorem on the integral (see Chapter 6, § 1) and certain abstract results about compactness within the framework of the minimax method (all these results are contained, for example, in Theorem 5). Our approach to questions of compactness differs from Amerio's. The important argu- ment about a point of strong continuity is given in an article by Kadets [63], where there is a proof of Corollary 2. (This argument was used independently by Zhikov [50], p. 184, in connection with the criterion for a point spectrum.) For a unitary group this criterion can be proved on the basis of a spectral resolution (see Lax & Phillips [69], p. 139).
§ 6. Example 3 is in the fundamental article of Bochner & von Neumann [29], who discuss the case of compact solutions. The Bochner—von Neumann method is based on the rather complicated techniques of generalised harmonic analysis. The original formula- tion of Theorem 6 (with a derivation of the results of Bochner &
von Neumann) is given in an article of Zhikov [47]. K. V. Valikov (private communication) and Perov and Ta Kuang Khai [97] have made some simplifications to the original proof.
§ 7. The problem of constructing a linear equation solvable in the class of bounded functions but not in the class of almost periodic functions was discussed in the original article of Favard [106]. But Favard only succeeded in showing (on the basis of Bohr's example) that the separation property does not carry over automatically to the limiting equations.
9 The method of monotonic operators
1 General properties of monotonic operators
1. Let V be a separable reflexive Banach space, V* be its dual, and (y, x) denote the value of y E V*at x E V.
Definition. An operator A: V -> V* is called monotonic if
Re (Ax i -Ax2, x i - x2) ?.-- 0 (x i , x2 E V).
In what follows, to simplify the notation we assume that V is real, so that the symbol Re in the definition can be dropped. An operator
A is called bounded if it carries bounded sets in V into bounded
sets in V*.
Definition. An operator A: V -> V* is called semicontinuous if the
scalar function t -> (A(u + tv), w) is continuous for any u, v, w E V.
Lemma 1. Let A: V -> V* be a monotonic, semicontinuous operator.
An element u E V satisfies the equation Au = f if and only if
(Av - f, v - u)--.0 (1)
for any v e V.
Proof. Let Au =f. Then we have
(Av -f, v - u)= (Av - Au, v - u) ---0.
Conversely, if in (1) we take v = u + tw (t > 0, tV E V), then we obtain (A(u +tw)-f, tw)?---0.
After cancelling by t and letting t tend to 0 we obtain (Au -f, w)_-.-. 0
for any w e V. Hence it is clear that Au =f
150 The method of monotonic operators
Remark 1. We have assumed that the operator A is defined on the whole of V, but it is clear that Lemma 1 is still valid when the domain D(A) of A is a linear manifold dense in V; it is merely necessary to assume, in addition, that u belongs to D(A).
The next proposition is a consequence of Lemma 1.
Proposition 1. Fora monotonic semicontinuous operator A: V-+ V*
the set of solutions of the equation Au =f is closed and convex.
For a proof it is enough to note that if two elements u li u2 satisfy (1), then (u i + u2)/2 also satisfies (1).
Proposition 2. A monotonic semicontinuous bounded operator A: V -> V* gives a continuous mapping if Vis endowed with the strong
topology and V* with the weak.'
Proof. Let un -> u and fn = A (un ). The sequence ffn l is bounded, and consequently it has a weak limit point, which we denote by y. We can take the limit in the inequality (A v -fn, y - un ) 0, and as a result obtain Au = y, as we required.
Theorem 1. Suppose that a monotonic semicontinuous bounded operator A: V -> V* is such that
(Au, u) 0 (Iluil = ro) (2)
Then there is auE V with lull ro, such that Au = 0.
Proof. Let Vn be an expanding sequence of finite-dimensional subspaces of V whose union is dense in V, and let 3n denote the restriction operator of v E V * to Vn . We consider the continuous finite-dimensional operators
= : Vn -> Vn * .
Since (Au, u) 0 (11u11= ro, u e Va ), we obtain from the finite- dimensional result (see Proposition 3 below) fin E Vn with Ilunil ro such that Au n =0. We have the inequality
(Anv, v - un ) = (Av, v - u n ) 0, (3)
where v e Vm, m n. The bounded sequence {u n } contains a weak limit point which we denote by u. We take the limit as n -> co in (3) for a fixed v e Vff, ; as a result we obtain (A v, v - u) 0 (v E V,n ). Then
This property is called demicontinuity.
General properties 151
by continuity (using Proposition 2) this inequality turns out to hold for any y E V. It remains to apply Lemma 1.
Theorem 2. Suppose that a monotonic semicontinuous bounded operator A: V -> V* has the coercive property
(Au, u)
lim = +00.
Ilull->00 'lull
Then for any fE V* the equation Au =f has at least one solution.
For a proof it is sufficient to recognise that Au -f satisfies (2) for a suitable choice of ro.
Proposition 3. Let P:R m -> Rm. be a continuous mapping of the Euclidean space Irn into itself such that (p(e), 6) 0 for any e in the
sphere llI = ro. Then there is a 6, ro, such that p(e) = O.
Proof. If I() o 0 on the ball K = ileil ro}, then we can consider the operator
P(f)
( )il . C >
which in this case is continuous. Then from the classical fixed point theorem of Bohl-Brouwer it follows that that there is a e such that
- roP(6)
6 = (4)
ii-P(e)ii
But then lei = ro, and by multiplying both sides of (4) scalarly by P() we obtain
(P(6), = - roiiP (e)ll< 0, which is impossible. 2
2. So far we have assumed that the monotonic operator is defined on the whole of the space V. For an application to evolution problems we require to study the problem of the solvability of an equation of the form
Lu Au + Au = f, (5)
where A is, as before, a monotonic semicontinuous bounded operator V-+ V*, and A is an unbounded linear operator D(A).-> V* (here
2 To use Proposition 3 in the proof of Theorem 1 we need to equip the
space Vn with a scalar product. This enables us to identify Vn * with Vi,.
152 The method of monotonic operators
D (A) c V and D (A) is dense in V) subject to the following con- ditions:
(1) A is closed and monotonic, that is, (Au, u) 0 for all u
(2) there is a sequence of expanding subspaces Vn c D(A) whose union is dense in V and which are such that A: Vn -> V* is bounded for each n;
(3) if 6, is the restriction operator of the functional y E V * to Vt.') then there is a constant 1 such that
mull liOnAUll E Va ).
Theorem 3. Suppose that, in addition to conditions (1)-(3), the operator A is coercive. Then (5) has at least one solution for any f e V*.
Proof. The operator Q :D(A)-> V*, where Qu Au +Au -f, is
monotonic. There is a number ro (depending on in such that (Qu, u) 0 (u E D (A), Iluil = r 0) . The operator Qn = ang : Vn Vn *
satisfies the conditions of Theorem 1. Let Qu n =0, liun 11 ,-5 ro. Since the operators &A are uniformly bounded, the sequence 3nAun, and so the sequence Aug, is bounded. Hence, since A is a closed operator, there is at least one u E d(A) for which un, —u and Aun, Au. Then it is necessary to proceed as in Theorem 1 (using Remark 1). This proves Theorem 3.