Boundary value problems for ordinary differential
equations on the half-axis
This chapter deals with boundary value problems for linear ordinary differential
equations of even order 2ra with constant coefficients on the interval (0,+00). It
prepares the treatment of boundary value problems for partial differential operators
in the half-space and in bounded domains of
We introduce the notion of regu-
larity and show that it is necessary and sufficient for the unique solvability of the
boundary value problem in Sobolev spaces of arbitrary integer order. Furthermore,
we study the connections between the formally adjoint boundary value problem
and the adjoint (in the functional analytic sense) problem.
1.1. The boundary value problem and its formally adjoint
In the beginning of the first section we describe the class of boundary value
problems on R
= (0,-hoc) which are considered in Chapter 1. While in classical
boundary value problems only an unknown function u on the semi-axis has to be
found, the boundary value problems considered here contain also an additional
unknown vector u CJ. We present a Green formula for these problems. This
formula allows to introduce a formally adjoint problem which has the same form as
the starting problem.
1.1.1. Formulation of the problem. Let
(1.1.1) L ( A) = $ ; ^ '
be a linear differential operator of order 2ra with constant coefficients aj , where
&2m 7^ 0. Here Dt denotes the derivative Dt = —idt = —id/dt. Furthermore, let
Bk{Dt) =
(fc = 1, .. . , m + J) be linear differential operators of order //& , and let
& = (Ck,j)
\ /l/cm+J, ljJ
be a constant (ra + J) x J-matrix. Here /i^ are integer numbers. We allow /ik to be
negative. In this case the operator Bk is assumed to be identically equal to zero.
We consider the problem
(1.1.2) L(Dt)u(t) = f(t), * 0 ,
(1.1.3) B(Dt)u{t)\t=0 + Cu = 9,
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