Regularization of operator equations with \(B\)-symmetric and \(B\)-positive operators in Banach spaces (Q736183)

The paper is concerned with a family of regularization methods for linear operator equations \(Ax=y_0\) with a \(B\)-symmetric and \(B\)-positive operator \(A\in L(X,Y)\) acting from a uniformly convex Banach space into an arbitrary Banach space, where \(B \in L(X,Y^*)\). An operator \(A\) is called \(B\)-symmetric (resp., \(B\)-positive) if \(\langle A x_1, B x_2\rangle=\langle A x_2, B x_1\rangle\) for all \(x_1,x_2 \in X\) (resp., \(\langle A x, B x \rangle \geq 0\) for all \(x \in X\) and \(\langle A x, B x \rangle =0\) iff \(Ax=0\)). The methods involve the regularized problem \(\min_{x \in X} \{ \langle Ax, Bx \rangle- 2 \langle y_{\delta}, Bx \rangle +\alpha \| x -x^0 \|^2 \}\), where \(\| y_{\delta}-y_0 \| \leq \delta\) and \(\alpha=\alpha(\delta)\). Convergence theorems for regularized solutions and their discrete approximations are established.