lim
k
→∞
Φ
i
α
=
i
ψ
α
k
, ψ
i
k
L
2
(0
,
1)
=
= Φ
i
α
=
i
ϕ
α
, ϕ
i
L
2
(0
,
1)
=
λ, i
= 1
,
2
,
· · ·
, n.
Now we construct new functions,
η
i
k
(
x
i
) =
λψ
i
k
−
Φ
i
α
=
i
ψ
α
k
=
=
λψ
i
k
(
x
i
)
−
Ω
i
F
(
x
1
, x
2
,
· · ·
, x
n
)
ψ
1
k
(
x
1
)
· · ·
· · ·
ψ
ˆ
i
k
· · ·
ψ
n
k
(
x
n
)
dx
1
· · ·
dx
ˆ
i
· · ·
dx
n
, ψ
α
k
≤
1
, i
= 1
,
2
,
· · ·
, n.
An accurate calculation of the norm of
η
i
k
shows that Lemma 1 implies
also while
α
ψ
α
k
approaches weakly to its limit
α
ϕ
α
,
η
i
k
strongly tends
to zero. Namely,
lim
k
→∞
η
i
k
, η
i
k
= lim
k
→∞
λψ
i
k
−
Φ
i
(
α
=
i
ψ
α
k
)
2
= 0
, i
= 1
,
2
,
· · ·
, n.
This indicates, the sequence
{
ψ
α
k
, α
= 1
,
2
,
· · ·
, n, k
= 1
,
2
,
· · · }
has a
strong limit, denoted again by
{
ϕ
α
, α
= 1
,
2
, . . . , n
}
, which satisfies (8).
Taking inner product for (8’) with
ϕ
i
, we obtain finally,
Φ
i
α
= 1
α
=
i
ϕ
α
, ϕ
i
=
=
Ω
F
(
x
1
, x
2
,
· · ·
, x
n
)
ϕ
1
(
x
1
)
ϕ
2
(
x
2
)
· · ·
ϕ
n
(
x
n
)
dx
1
dx
2
· · ·
dx
n
=
λ,
(9)
that is,
α
ϕ
α
is a solution of (8), as claimed in the Theorem.
Let
λ
1
and
n
α
=1
ϕ
α
1
are the greatest eigenvalue and its associated
eigenfunctions, respectively. Construct new function
F
1
(
x
1
, x
2
,
· · ·
, x
n
) =
F
0
(
x
1
, x
2
,
· · ·
, x
n
)
−
λ
1
n
α
=1
ϕ
α
1
(
x
α
)
.
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
39
