Let
F
1
=
0. It is easy to check that the norm of
F
1
in L
2
(Ω)
can be calculated
as
F
1
2
=
F
0
−
λ
1
ϕ
α
1
, F
0
−
λ
1
ϕ
α
1
=
=
F
0
2
−
2
λ
1
F
0
,
ϕ
α
1
+
λ
2
1
ϕ
α
1
,
ϕ
α
1
=
F
0
2
−
λ
2
1
.
So long as
F
1
=
0, the Lemma 1 and Theorem 2 are applicable for
F
1
as
well. Having
F
0
replaced by
F
1
in (8), one gets a new operator
Φ
1
and,
correspondingly, new system of equations (8’).
By Theorem 2, it possesses at least one positive eigenvalue
λ
2
and
associated eigenfunctions
α
ϕ
α
2
of (8) with
F
replaced by
F
1
. Similarly,
we have
F
2
=
F
1
−
λ
2
n
α
=1
ϕ
α
2
=
F
0
−
λ
1
n
α
=1
ϕ
α
1
−
λ
2
n
α
=1
ϕ
α
2
,
with its norm
F
2
2
=
F
1
2
−
λ
2
2
=
F
0
2
−
λ
2
1
−
λ
2
2
.
The process can be continued inductively, if
F
N
=
0,
F
N
=
F
N
−
1
−
λ
N
n
α
=1
ϕ
α
N
=
F
0
−
N
β
=1
λ
β
n
α
=1
ϕ
α
β
,
(10)
and
F
N
2
=
F
N
−
1
2
−
λ
2
N
=
F
0
2
−
N
β
=1
λ
2
β
.
(11)
Further, each
F
N
generates its own gradient operator
Φ
N
= (Φ
N
1
, . . . ,
Φ
Nn
)
,
Φ
Ni
α
=
i
ϕ
α
N
+1
=
Ω
i
F
N
(
x
)
α
=
i
ϕ
α
N
+1
(
x
α
)
d
Ω
i
=
λ
N
+1
ϕ
i
N
+1
(
x
i
)
,
i
= 1
,
2
,
· · ·
, n.
(12)
If the process continues infinitely, (10) becomes an infinite series. Now we
prove that the following equality holds as
N
→ ∞
in the norm of
L
2
(Ω)
,
F
0
(
x
1
, x
2
,
· · ·
, x
n
) =
∞
β
=1
λ
β
n
α
=1
ϕ
α
β
(
x
α
)
.
(13)
40
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
