be arbitrary element in
L
(
i
)
2
(0
,
1)
,
t
∈
[0, 1],
λ
= 2
λ
. A straightforward
computation yields
D
i
f
= lim
t
→
0
1
t
f
α
=
i
ϕ
α
(
ϕ
i
+
th
i
−
−
f
n
α
=1
ϕ
α
= Φ
i
α
=
i
ϕ
α
−
λϕ
i
, h
i
L
(
i
)
2
(0
,
1)
= 0
.
Due to the arbitrariness of
h
i
, we have
Φ
i
α
=
i
ϕ
α
−
λϕ
i
= 0
, i
= 1
,
2
,
· · ·
n,
(8)
or, after unfolding,
λϕ
i
(
x
i
) = Φ
i
α
=
i
ϕ
α
=
=
Ω
i
F
(
x
1
,
· · ·
, x
n
)
ϕ
1
(
x
1
)
· · ·
ϕ
ˆ
i
(
x
i
)
· · ·
ϕ
n
(
x
n
)
dx
1
· · ·
· · ·
dx
ˆ
i
· · ·
dx
n
, i
= 1
,
2
,
· · ·
n,
(8’)
where
ˆ
i
means absence of ith coordinate,
Ω
i
is the (
n
−
1
) dimensional
unit cube without
x
i
.
The operator
Φ = (Φ
1
,
Φ
2
, . . . ,
Φ
n
)
generated by
G
-derivative and
defined in (8’) is called the gradient operator of functional (5). Now we
proceed to prove the following, the primary theorem as a start-point for
further investigation.
Theorem 2.
For any given
F
(
x
)
∈
L
2
(Ω)
,
F
(
x
) =
0, its gradient
operator
Φ
or, equivalently, the system of homogeneous integral equations
(8’) possesses at least one positive eigenvalue. The greatest eigenvalue
and its associated eigenfunctions satisfy (4), and are a solution of this
supremum problem.
Proof (abridged).
It is known that for any given
F
(
x
)
∈
L
2
(Ω)
,
all components of its gradient operator
Φ
i
, defined by (8), are compact
[6, 7], so the range
Φ
i
(
B
n
−
1
i
)
is a compact subset in
L
(
i
)
2
(0
,
1)
, here
B
n
−
1
=
α
=
i
ψ
α
(
x
α
)
, ψ
α
≤
1
. By Lemma 1, there exists a sequence
in
B
n
,
ψ
α
k
, weakly convergent to
α
ϕ
α
∈
B
n
as
k
→ ∞
so that
the following holds,
38
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
