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Remark 3.18 Assume that • : H ’] ’ ∞, +∞] is a lower-semicontinuous
proper convex function such that Range(‚•) = H which is equivalent to

lim {•(x) ’ (x, y)} = +∞, ∀y ∈ H,
x ’∞


see [4], pp.41. In particular, by taking y = f we deduce that the hypothesis
(62) is veri¬ed.

Remark 3.19 Assume that • is coercive

(‚•(x), x ’ x0 )
∀x0 ∈ D(•),
lim = +∞,
x
’∞
x


which is equivalent to lim x ’∞ •(x) = +∞ (see [4], pp.42). Then Range(•) =
x
H because the previous condition is satis¬ed: lim x ’∞ {•(x) ’ (x, y)} = +∞,
for all y ∈ H and therefore (62) is veri¬ed.

Theorem 3.20 Suppose that • : H ’] ’ ∞, +∞] is a lower-semicontinuous
proper convex function and f ∈ W 1,1 (]0, T [; H) such that

lim {•(x) ’ (x, f )} = +∞, (71)
x ’∞


and every level subset {x ∈ H; •(x) + x 2 ¤ M } is compact. Then equation
(61) has T -periodic solutions x ∈ C([0, T ]; H) © W 1,∞ (]0, T [; H) which satisfy

d+
x(t) + (‚•x(t) ’ f (t))—¦ = 0,
x(t) ∈ D(‚•), ∀ t ∈ [0, T ], ∀ t ∈ [0, T ],
dt
where (‚• ’ f )—¦ denote the minimal section of ‚• ’ f .

Proof Since W 1,1 (]0, T [; H) ‚ L2 (]0, T [; H) the previous theorem applies.
Consider x ∈ C([0, T ]; H) © W 1,2 (]0, T [; H) a T -periodic solution for (61). Since
x L2 (]0,T [;H) ¤ f L2 (]0,T [;H) it follows that there is t ∈]0, T [ such that x is
1
di¬erentiable in t and x (t ) ¤ √T f L2 (]0,T [;H) . By standard calculation
we ¬nd that:
t
1 1 1
(x(t + h) ’ x(t)) ¤ (x(t + h) ’ x(t )) + (f („ + h) ’ f („ ) d„,
h h h
t

1
and therefore sup0¤t¤T, h>0 h (x(t + h) ’ x(t)) ¤ C which implies that x ∈
W 1,∞ (]0, T [; H). Making use of the inequality
t+h
1 1
(x(t + h) ’ x(t), x(t) ’ u) ¤ (f („ ) ’ v, x(„ ) ’ u) d„, 0 ¤ t < t + h ¤ T,
h h t
Mihai Bostan 41


which holds for every [u, v] ∈ ‚• we deduce that x(t) ∈ D(‚•) for all t ∈
1
[0, T ] and the weak closure of the set { h (x(t + h) ’ x(t)), h > 0} belongs to
f (t) ’ ‚•x(t), ∀t ∈ [0, T ]. On the other hand by writing
t+h
x(t + h) ’ u ¤ x(t) ’ u + f („ ) ’ v d„, 0 ¤ t < t + h ¤ T,
t

for u = x(t) and v ∈ ‚•x(t) we ¬nd that
1
(‚•x(t) ’ f (t))—¦ ¤ w ’ lim
(x(t + h) ’ x(t))
h 0h
1
¤ lim sup (x(t + h) ’ x(t))
h
h0
(‚•x(t) ’ f (t))—¦ .
¤
d+
1
This shows that limh 0 h (x(t + h) ’ x(t)) = exists for every t ∈ [0, T ]
dt x(t)
and coincides with ’(‚•x(t) ’ f (t))—¦ .


References
[1] V. Barbu, Nonlinear Semigroups and Di¬erential Equations in Banach
Spaces, Noordho¬ (1976).
[2] M. Bostan, Solutions p´riodiques des ´quations d™´volution, C. R. Acad.
e e e
´
Sci. Paris, S´r. I Math. t.332, pp. 1-4, Equations d´riv´es partielles, (2001).
e ee
[3] M. Bostan, Almost periodic solutions for evolution equations, article in
preparation.
[4] H. Brezis, Op´rateurs maximaux monotones et semi-groupes de contractions
e
dans les espaces de Hilbert, Noth-Holland, Lecture Notes no. 5 (1972).
[5] H. Brezis, A Haraux, Image d™une somme d™op´rateurs monotones et ap-
e
plications, Israel J. Math. 23 (1976), 2, pp. 165-186.
[6] H. Brezis, Analyse fonctionnelle, Masson, (1998).
´
[7] A. Haraux, Equations d™´volution non lin´aires: solutions born´es
e e e
p´riodiques, Ann. Inst. Fourier 28 (1978), 2, pp. 202-220.
e

Mihai Bostan
Universit´ de Franche-Comt´
e e
16 route de Gray F-25030
Besan¸on Cedex, France
c
mbostan@math.univ-fcomte.fr

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