On the dynamic solution of an operator inverse problem (Q1286376)

For a given family of linear operators \( B(x): U \to V^*, \;x \in V \), and an operator \( B_1(\cdot): V \to V^* \) where \( U \) and \( V \) are Banach spaces and \( V^* \) denotes the dual space of \( V \), the following equation is considered \[ B(x)u - B_1(x) = f.\tag{1} \] Here \( f \in V^* \) is given, and \( x = x(u) \in V \) in equation (1) depends on \( u \). It is supposed that equation (1) possesses a solution \( u \) in a given set \( P \subset U \), and on the other hand \( x(u) \) is approximately known only, i.e., \( \xi_h \in V \) and \( h > 0 \) are available such that \( \| \xi_h - x(u) \| \leq h \) is satisfied. Some applications are presented, and a Tikhonov-type method is considered for equation (1) which on further conditions on the spaces and operators is shown to be regularizing when a specific a priori parameter choice is applied.