Difference-differential Equations with Fredholm Operator in the Main ...

has only the trivial solution. Notice that this condition will be satisfied if the diameter of
the domain ω will be sufficiently small. It is not difficult to show that ˜L is continuously
invertible. In fact A is a compact operator, Γ is a bounded operator, so AΓ is a compact
operator. So the problem
˜Lv = (I − Q)f(x, y)
with the ordinary Dirichlet conditions
v| x=x0 = B(I − P )φ 0
v| ∂1 = Bφ 1
can be reduced to the equation of the second kind with the compact operator. (In the case
when E 1 = E 2 = R n it was shown in [2]). So the operator ˜L is Fredholm operator, and
also N( ˜L) = {0}. Therefore it is continuously invertible. Based on lemma 1 coefficients
of the vector C(x, y) are defined from the split recurrent sequence of linear differential
equations of the first order with the boundary condition
(C, Φ)| x=x0 = P φ 0 (y).
So the conditions of theorem 3 for problem (16), (17), (18) are satisfied and it has unique
solution in the class C l,α (¯ω).
Remark. If operator B has a complete A 1 - Jordan set, A 1 , A 2 are compact operators,
a ij ξ i ξ j ≥ ν ∑ r
1 ξi 2 , ν − const > 0,
L 0 ( ∂ r∑
∂x ) = a ij (x) +
∂x i ∂x j
1
∂ 2
r∑
1
a i (x) ∂
∂x i
+ a(x),
L 1 ( ∂
∂x ) =
∂
∂x 1
+
r∑
2
x ∈ Ω ⊂ R r ,
then similar results can be obtained for the equation
with the boundary conditions (fig.2):
b i (x) ∂
∂x i
,
L 0 ( ∂
∂x )Bu + L 1( ∂
∂x )A 1u + A 2 u = f(x), x ∈ ω,
(I − P )u| ∂ω = φ 0 (x), P Φ 0 = 0,
P u| x1 =x 0 1 = φ 1(x 2 , . . . , x r ) (I − P )Φ 1 = 0.
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