130 NGUYEN VAN LOI, ZHENHAI LIU AND VALERI OBUKHOVSKII
[L
2
(I, H)] we denote the spaces of all continuous [respectively, square summable]
functions u: I → H with usual norms
kuk
C
= max
t∈I
ku(t)k
H
and kuk
2
=
Z
T
0
ku(t)k
2
H
dt
!
1
2
.
A closed ball of radius r centered at 0 in C(I, H) is denoted by B
C
(0, r). Consider the
space of all absolutely continuous functions u : I → H whose derivatives belong to
L
2
(I, H). It is known (see, e.g. [2]) that this space can be identified with the Sobolev
space W
1,2
(I, H) endowed with the norm
kuk
W
=
kuk
2
2
+ ku
0
k
2
2
1
2
.
The embedding W
1,2
(I, H) → C(I, H) is continuous, and for every n ≥ 1 the
space W
1,2
(I, H
n
) is compactly embedded in C(I, H
n
). By W
1,2
T
(I, H) we denote the
subspace of all functions x ∈ W
1,2
(I, H) satisfying the boundary condition x(0) =
x(T ).
Let n ∈ N, and : W
1,2
T
(I, H
n
) → L
2
(I, H
n
) be a linear Fredholm operator of
index zero. Then there exist the projections (see, e.g. [6]):
C
n
: W
1,2
T
(I, H
n
) → W
1,2
T
(I, H
n
)
and
Q
n
: L
2
(I, H
n
) → L
2
(I, H
n
)
such that Im C
n
= Ker and Ker Q
n
= Im . If the operator
C
n
: dom ∩ Ker C
n
→ Im
is defined as the restriction of on dom ∩ Ker C
n
, then
C
n
is a linear isomorphism
and we can define the operator K
C
n
: Im → dom , K
C
n
=
−1
C
n
. Now, set Coker =
L
2
(I, H
n
)/Im ; and let Π
n
: L
2
(I, H
n
) → Coker be the canonical projection
Π
n
(z) = z + Im
and Λ
n
: Coker → Ker be the linear continuous isomorphism. Then the equation
x = y, y ∈ L
2
(I, H
n
)
is equivalent to
(i − C
n
)x = (Λ
n
Π
n
+ K
C
n
,Q
n
)y,
where K
C
n
,Q
n
: L
2
(I, H
n
) → W
1,2
T
(I, H
n
) is given as
K
C
n
,Q
n
= K
C
n
(i − Q
n
).
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