10

1. PROBLEMS ON THE HALF-AXIS

where B(Dt) denotes the vector of the operators Bi(Dt), ... , 5

m +

j(Dt) , / is a

given function on R

+

, and g is a given vector from C m + J . We seek a function u

on M+ and a vector u = (u\, ... ,uj) such that u is a solution of the differential

equation (1.1.2), and the pair (u,u) satisfies the boundary conditions (1.1.3) which

can be written in the coordinate form as

J

Bk(Dt)u{t)\t^oJrY^Ck^u3 = 9k, k = 1, ... ,ra + J.

3 = 1

REMARK

1.1.1. Here and in the following we will not make a distinction be-

tween column and row vectors. In (1.1.3) u and g are considered as column vectors.

1.1.2. The Green formula and the formally adjoint problem. In order

to define the formally adjoint problem to (1.1.2), (1.1.3), we use a modification of

the classical Green formula.

First we consider the case jik 2m. Let

2ra

j=0

be the formally adjoint operator to L. Furthermore, let V denote the vector

(1.1.4) p = ( i , A , . . . , A

2 m

"

1

) -

Then the operator B(Dt) can be written in the form

(1.1.5) B(Dt) = Q-V

(here V is considered as a column vector), where the elements of the (ra + J) x 2m

matrix

Q = \Qk,j)

V ,J /ifc

lkm+J, lj2ra

are defined by the coefficients of the operators B^ as follows:

I 0

for j = 1, ... ,/x*. -hi ,

qk*

^

n

for j / i

f c

+ l.

THEOREM

1.1.1. The following Green formula is satisfied for all infinitely dif-

ferentiable functions u, v on M+ with compact support and all vectors u £ C

J

,

v E C

m + J

:

oo

(1.1.6) f Lu -vdt + (B(Dt)u\t=o + Cu, v)Cm+J

0

oo

= fu.L+v~dt+ ((Vu)(0), P(Dt)v\t=o + Q* v)Q2m + fe, C*t;)c, .

o

ifere P(Dt) denotes the vector with the components

2m-j

(1.1.7) Pj{Dt) = -i J2

*J+sDst

j = l,...,2m ,

and Q*, C* are the adjoint matrices to Q and C, respectively.