The approximate solution of the first kind operator equation in locally convex spaces by discrepancy method (Q1184863)

We consider the first kind operator equation \[ Ax= y_ 0,\quad x\in X,\quad y_ 0\in R(A)\subset Y,\tag{1} \] where \(X\) is a locally convex space, \(Y\) is a separated locally convex space, and \(A\) is a linear operator from \(X\) into \(Y\) which is weakly continuous and such that there exists the inverse operator \(A^{-1}\) defined on the range \(R(A)\) of \(A\). Given a filter base of \(y_ 0\) consisting of closed convex subsets \(\{V_ \delta\}\), we want to establish approximate solutions \(x_ \delta\) of equation (1) such that \(x_ \delta\to x_ 0\), where \(x_ 0\) is an exact solution of (1), i.e. \(x_ 0= A^{-1} y_ 0\). For given \(X\), \(Y\), \(A\), \(y_ 0\), we associate each filter base \(\{V_ \delta\}\) with the sequence of the approximate solutions \(\{x_ \delta\}\) and denote the above problem by \(\alpha[X,Y,A,x_ 0,\{V_ \delta(y_ 0)\}]\). If the approximate solutions \(x_ \delta\) are established by a discrepancy method and \(x_ \delta\to x_ 0\), then we say that this discrepancy method stabilizes the problem. In this paper, we give conditions under which a discrepancy method stabilizes any problem \(\alpha[X,Y,A,x_ 0,\{V_ \delta(y_ 0)\}]\).