Résumé:
In this master thesis, we study the following abstract second order problem
u
′′(x) + Au(x) − λu(x) = f(x), x ∈ (0, +∞)
u
′
(0) − Hu(0) − µu(0) = d0,
(1)
posed in the unbounded domain R+, such that A and H are two closed operators in a complex
Banach space X, d0 is an auxiliary element given in X, and f is a function in L
p
(R+, X).
λ and µ are two complex parameters belong to the set Ωφ0,φ1,r defined as :
Ωφ0,φ1,r = {(λ, µ) ∈ Sφ0 × Sφ1
: |λ| ≥ r > 0 and |µ|
2
|λ|
≥ r > 0},
where Sφ0
, Sφ1 are defined by (1.2).
For some large positive number r0, We study the existence and uniqueness of the
classical solution in a complex Banach space X of type UMD of the problem (1), with the
regular second condition
u(+∞) = 0.
By using semigroups theory [22], and fractional powers of operators [2] and [23],
we prove that
u ∈ W2.p (0, +∞; X)
T
L
p
(0, +∞; D(A)) and u(0) ∈ D(H).
