Weak, strong and linear convergence of the CQ-method via the regularity of Landweber operators

We consider the split convex feasibility problem in a fixed point setting. Motivated by the well-known CQ-method of Byrne [Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 2002;18:441-453], we define an abstract Landweber transform which applies to more general operators than the metric projection. We call the result of this transform a Landweber operator. It turns out that the Landweber transform preserves many interesting properties. For example, the Landweber transform of a (quasi/firmly) nonexpansive mapping is again (quasi/firmly) nonexpansive. Moreover, the Landweber transform of a (weakly/linearly) regular mapping is again (weakly/linearly) regular. The preservation of regularity is important because it leads to (weak/linear) convergence of many CQ-type methods.