On the second mixed boundary value problems for linear equations with generalized right invertible operators (Q1399076)

There are given necessary and sufficient conditions for the problem \[ Q[V]x:= \sum^M_{m=0} \sum^N_{n=0} V^m A_{mn} V^nx= y,\quad y\in X, \] \[ F_ix= y_i\qquad (i= 0,\dots, k-1), \] \[ F_i V^i x= y_i\qquad (i= k,\dots,M+ N-1), \] where \(y\), \(y_i\in \text{ker }V\) are given, and \(M,N,K\in\mathbb{N}\), \(K< M+ N\), \(A_{mn}\in L_0(X)\), \(A_{MN}= \Gamma\), \(A_{mn}X_{M+N-n}\subset X_m\), \(X_i:= \text{dom }V^i\subset X\) and \(F_0,\dots, F_{M+N-1}\) are right initial operators of the generalized right invertible operator \(V\in L(X)\) having the so-called \(C(W)\)-property, to have solutions. In particular, there are given conditions for the existence of a unique solution. Note that the \(C(W)\)-property (introduced by the reviewer) here can be stated as follows: Given a generalized right invertible operator \(V\) with a generalized right inverse \(W\) and a right initial operator \(F_0\). Then \(F_0\) has the \(C(W)\)-property if and only if there exist scalars \(d_k\) such that \(F_0W^kz= d_kz\) for all \(z\in \text{ker }V\), \(k\in\mathbb{N}\).