  Motivated by a class of nonlinear imaging inverse problems, for instance,
multispectral computed tomography (MSCT), this paper studies the convergence
theory of the nonlinear Kaczmarz method (NKM) for solving the system of
nonlinear equations with component-wise convex mapping, namely, the function
corresponding to each equation being convex. However, such kind of nonlinear
mapping may not satisfy the commonly used component-wise tangential cone
condition (TCC). For this purpose, we propose a novel condition named relative
gradient discrepancy condition (RGDC), and make use of it to prove the
convergence and even the convergence rate of the NKM with several general index
selection strategies, where these strategies include cyclic strategy and
maximum residual strategy. Particularly, we investigate the application of the
NKM for solving nonlinear systems in MSCT image reconstruction. We prove that
the nonlinear mapping in this context fulfills the proposed RGDC rather than
the component-wise TCC, and provide a global convergence of the NKM based on
the previously obtained results. Numerical experiments further illustrate the
numerical convergence of the NKM for MSCT image reconstruction.
