Inverse problem for elliptic equation in a Banach space with Bitsadze-Samarsky boundary value conditions (Q2848390)

The paper deals with the inverse problem for elliptic differential equations in a Banach space NEWLINE\[NEWLINEu''=Au +p,\quad 0 \leq t \leq T,NEWLINE\]NEWLINE where \(A\) is a positive operator with either Dirichlet conditions \(u(0)=x\), \(u(T)=\sum_{i=1}^n k_i u(\tau_i) +y\), \(u(\tau)=z\) or von Neumann conditions. The inverse problem consists in determining \(u(t)\) and \(p\) given \(x\), \(y\) and \(z\). For the Dirichlet problem, it is proved that the inverse problem has a unique solution provided the characteristic function, defined in terms of the real coefficients \(\{k_i\}\), does not vanish in the right half-plane. Further, in the case of self-adjoint \(A\), the existence and uniqueness of the solution of the inverse problem is proved under the conditions \(k_i\geq 0\) and \(\sum_{i=1}^n k_i\leq 1\). A similar analysis is given for the von Neumann problem.