  In recent years Landweber(-Kaczmarz) method has been proposed for solving
nonlinear ill-posed inverse problems in Banach spaces using general convex
penalty functions. The implementation of this method involves solving a
(nonsmooth) convex minimization problem at each iteration step and the existing
theory requires its exact resolution which in general is impossible in
practical applications. In this paper we propose a version of
Landweber-Kaczmarz method in Banach spaces in which the minimization problem
involved in each iteration step is solved inexactly. Based on the
$\varepsilon$-subdifferential calculus we give a convergence analysis of our
method. Furthermore, using Nesterov's strategy, we propose a possible
accelerated version of Landweber-Kaczmarz method. Numerical results on computed
tomography and parameter identification in partial differential equations are
provided to support our theoretical results and to demonstrate our accelerated
method.
