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Page 1

Bounded and Almost Periodic Solutions of
Nonlinear Operator Differential Equations

Page 2

Mathematics and Its Applications (Soviet Series)

Managing Editor:

M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board:

A. A. KIRILLOV, MGU, Moscow, U.S.S.R.
Yu. 1. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R.
S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R.
M. C. POL YVANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.

Volume 55

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106 A.A. PANKOY

Bf,I'(R, E) CBf,'1(R, E).

However, regarding u(t) as a function with values in H, we have, by Proposition
3.12,

for all 1 ~s < 00. We note also that if q = 2, Y = 1 and f, I ES P' (R; E'), then u
belongs to S P (R; E) n CAP(R; H) and u' EBS P (R; E) n L 00 (R; H). Since u
satisfies (5.11), we additionally have u' EBP(R; E) (see the discussion at the end of
nO. 3.3).

6. Singular perturbation.

We consider (formally) the following inequality with a small parameter fL>O in
front of the derivative:

(p.u' (t) + A (t)u(t) - f(t), v - u(t» + q>(V)- q>(u(t» ;;;. 0, \tv E V,

for almost all t ER. If fL=O, then the problem turns into the stationary inequality

(A (t)u(t)-f(t), v-u(t»+q>(v)-q>(u(t»;;;. 0, \tVEV, (6.10>

where t ER plays the role of parameter. According to nO. 1.1, a weak setting of the
problem corresponding to (6.1 1') is stated in the following way. Let the conditions
of nO. l.1 (in particular, inequalities (1.3) and (1.5) and condition (c» be satisfied.
Given f EIJ{~ (R; V,), find a function u = u I' EIJ{oc (R; V) n C(R; H) such that

I,

![(uv'+Au-j,v-u)+</>(v)-</>(u)]dt ~ -II v-u 12 1:;,
I,

for any test function v. The corresponding limit inequality is

I,

j[(Au- j, v -u)+q>(v)-q>(u)]dt ;;;. 0, tl ~ t2,
I,

for any v EIJ{oc(R; V). This is clearly equivalent to (6.10)' It is easy to see that
under suitable conditions all previous results are applicable to (6.21') for any fixed

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CHAPTER 3 107

1-'->0 (it is sufficient to divide (6.2,,) by 1-'-.
The usual stationary theory (see, for example, [44, ll8]) is applicable to (6.20).

We present here the corresponding results in a suitable form.

PROPOSITION 6.1. Assume that condition (c) is satisfied and that inequalities (1.3)
and (1.5) hold. If f ELf;c (R; V'), then there is a solution U ELfoc (R; V) of inequality
(6.20) · Moreover, iff EBSI' (R; V'), then U EBSI(R; V).
Proof By general results [44, 118], for any fixed N>O problem (6.20) has a solu-

tion u=uNCU(-N, N; V) and

II U I I LP(-N,N;V) ",;;; CllfIILP'(-N,N;V), (6.3)

where C(x), x;;;'O, depends only on constants involved in (1.3) and (1.5). Moreover,
estimate (6.3) is valid for any solution of (6.20) on the interval (-N, N). It is easy
to see that UN I (a,b) is a solution of (6.20) on (a, b) C( - N, N) and is bounded in
LP(a, b; V). Hence passing a subsequence, if necessary, we may assume that
limuN=UO exists weakly in !,foc(R; V). Results in [142] imply that Uo is a solution
of (6.20) on R (but it is not difficult to see this immediately). The second assertion
easily follows from the estimate (6.3). 0

REMARK 6.2. If all operators A (t) are strictly monotone, then the solution con-
structed is unique. Assume now that condition (capv,p) is satisfied and that ine-
quality (3.1) holds with E= V. If fESP'(R; V), then we may consider a family of
inequalities,

J

f[(A (s + t)us(t) - f(s + t), u(t) - usCt» + q,(v(t» - tp(us(t»]dt ;;;. 0,
o

where sERB. Any such inequality has a unique solution US ELP(O, 1; V), and

us(t) = uo(s + t), s ER, t E(O, 1).

Results in [142) imply that UsOEU(O, 1; V) depends continuously on sERB and,
consequently, Uo ESP(R; V). Using results in [142) it can also be shown that
fECAP(R; V) implies Uo ECAP(R; V).

Now we consider the behaviour of bounded solutions, u,,' of (6.2 ,,) as JL~O.
Under the conditions of Theorem 2.7 such solutions are uniquely determined.

THEOREM 6.3. Assume that condition (c) is satisfied, inequalities (1.3) and (1.5)
hold, and one of inequalities (3.1) or (3.2) hold with q;;;'2 Then the family

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220 A.A.PANKOV

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Page 230

Subject Index

almost periodic function in the sence of
Stepanov 25

almost periodic impulse measure 200
Besicovitch almost periodic function 11
Bohner transform 24
Bohner-Fejer operators 13
Bohr almost periodic function 7
Bohr compactification 5
character 6
coercive operator 52
compact hull 5
composition operator 49
coupled parabolic equations 147
evolution variational inequality 56
€-almost period 7
Fourier-Bohr transform 16
frequency group of a.p. function 16
group of characters 5
impulse measure 198
lower semicontinuous function 54
mean value 10
min-max procedure 92
monotone operator 50
nonlinear Schrodinger equation 144
positive boundary condition 135
proper function 54
p-coercive operator 159
p-monotone operator 159
relative Bohr compactification 21
relatively dense set 7
semi continuous operator 50
semi-permiable thin wall problem 119
singularly perturbed 106
variational inequality
spectrum of a.p. function 16

Stepavon almost periodic function 25
strictly hyperbolic operator 181
symmetric hyperbolic system 134
uniformly almost periodic function 7
variational inequality with shifted

variable 119
weak solution of evolution variational

inequality 58
variational inequality weakly almost

periodic function 28