12 1. PROBLEMS ON THE HALF-AXIS
DEFINITION
1.1.1. Assume that the Green formula (1.1.6) is valid. Then the
problem
(1.1.15) L+(Dt)v(t) = f{t) for£0 ,
(1.1.16) P(Dt) v(t)\t=o + Q*v = g, C*v = h
is said to be formally adjoint to (1.1.2), (1.1.3).
By the representation of the elements q^j of the matrix (J, the boundary con-
ditions (1.1.16) of the formally adjoint problem have the following form
771+J
Pj(Dt)v(t)\t=o+ ^2 bk,j-iVk= g3 , j = l, . . . , 2 m ,
M/cj- i
m-\-J
^ Ck,jVk = h3; , j = 1, . . . , J .
fe=l
The formally adjoint problem has the same structure as the starting problem. How-
ever, the number of the boundary conditions and of the unknowns is greater than
in (1.1.2), (1.1.3).
1.1.3. Boundary operators of higher order. Now we consider the bound-
ary value problem (1.1.2), (1.1.3) without the restriction ^ 2m on the orders of
the differential operators £?*.. Let K be an integer number such that
K
2m,
K
m a x ^ for k 1,... ,m + J,
and let V^ be the column vector with the components 1, Dt,... ,
D^~l.
Then the
vector B(Dt) can be written in the form
(1.1.17) B(Dt) =
QlK)-V(K\
where Q^ is a (m + J) x n matrix of complex numbers. Furthermore, according
to (1.1.11), we have
oo oo
(1.1.18) / Lu-vdx= fu -IT^vdx + ({V^u)(0), (PMv){0))£K ,
0 0
where P^ is the vector with the components Pi(Dt),.. , P2m{Dt),0,... ,0. We
introduce the (K 2m) x K, matrix
RM
[
a0 ai •••
a2m
0 ••• 0 ^
0 CLQ Q2m-1
a
2m ' ' ' 0
\ 0 0 a0 ai a2m /
Obviously, the vector p(^-
2 m
) L(Dt) has the representation
(1.1.19) £(*-2™) L ( A ) = ii(/c) VM
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