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

However, in all papers mentioned above the problems were considered, which could be
interpreted as ordinary differential equations in Banach spaces. On the other hand, there
are a lot of systems of partial difference differential equations with a noninvertible operator
in the main part, which can not be interpreted in this way. So, in papers [6],[9], [10],
significantly more complex degenerated partial differential equations in Banach spaces
are considered. In this paper we develop the similar results for difference-differential
equations.
Let us consider the following equation
Λu ≡ L 0 Bu + L 1 A 1 u + · · · + L q A q u = f(x), (1)
where B and A i , i = 1, q are closed linear operators with the dense domains from E 1 to
E 2 ; E 1 , E 2 are Banach spaces and D(B) ⊆ D(A i ), i = 1, q, x ∈ Ω ⊂ R r , B is Fredholm
operator with dimN(B) = dimN(B ∗ ) = n, R(B) = R(B), f(x) : Ω ⊂ R r → E 2 is a
sufficiently smooth function; linear operators L i act on the sets of abstract functions,
defined on x ∈ Ω ⊂ R r with the values in Banach spaces E 2 (E 1 ) and satisfy the following
conditions:
1)D(L 0 ) ⊂ D(L 1 ) ⊂ D(L i ), i = 2, . . . , q;
2)L 0 Bu(x) = BL 0 u(x), L i A i u(x) = A i L i u(x) on u(x) ∈ D(B) ∩ D(L 0 ).
Conditions 1) and 2) are carried out, for example, for difference- differential operators
of the following form
L i ( ∂
∂x , ∆) = ∑
|k|≤q i
a i k (x)Dk + ∑
|k|≤q i
b i k (x)∆k ,
where
D k =
∂ k
∂x k ,
1
1 . . . ∂x kr
r
∆ k u =
k 1 ∑
i 1 =0
k r
∑
. . . (−1) |k|−|i| C i 1
k 1
. . . C ir
k r
u(x 1 + i 1 h 1 , . . . , x r + i r h r ),
i r=0
q 0 > q 1 > q 2 ≥ . . . ≥ q q , a i k (x), bi k (x) : Ω ⊂ Rr −→ R 1 . In what follows L i will denote this
concrete form of operators. It will be more convenient not to specify concretely D(L 0 ).
The investigation of singular equation (1) is reduced to regular problems, i.e. to
equations solvable with respect to the main part L 0 . A special decompositions of the
Banach spaces E 1 and E 2 in accordance with the generalized Jordan structure of the
operator coefficients B, A i , i = 1, q are used. This reduction makes it possible to pose
boundary value problems for (1).
2 P, Q-commutability of Linear Operators in Case of
Fredholm Operator
Let E 1 = M 1 ⊕ N 1 , E 2 = M 2 ⊕ N 2 , P be a projector on M 1 along N 1 , Q be a projector on
M 2 along N 2 , A be a linear closed operator from E 1 to E 2 , D(A) = E 1 , A ∈ {A 1 , . . . , A q }.
2