Boundary value matrix problems and Drazin invertible operators

Abstract

Let $A$ and $B$ be given linear operators on Banach spaces $X$ and $Y$, we denote by $M_C$ the operator defined on $X \oplus Y$ by $M_{C}=
\begin{pmatrix}
A & C \\
0 & B%
\end{pmatrix}.$
In this paper, we study an abstract boundary
value matrix problems with a spectral parameter described by Drazin invertibile operators of the form $$
\begin{cases}
U_L=\lambda M_{C}w+F, & \\
\Gamma w=\Phi, &
\end{cases}%
$$
where $U_L , M_C$ are upper triangular operators matrices $(2\times 2)$ acting in Banach spaces, $\Gamma$ is boundary operator, $F$ and $\Phi $ are given vectors and $\lambda $ is a complex spectral parameter.
We introduce the
concept of initial boundary operators adapted to the Drazin invertibility and
we present a spectral approach for solving the problem. It can be shown that
the considered boundary value problems are uniquely solvable and that their
solutions are explicitly calculated. As an application we give an example to illustrate our results.