Regularity and Stability for a Convex Feasibility Problem

Let us consider two sequences of closed convex sets {An} and {Bn} converging with respect to the Attouch-Wets convergence to A and B, respectively. Given a starting point a0, we consider the sequences of points obtained by projecting onto the “perturbed” sets, i.e., the sequences {an} and {bn} defined inductively by $b_{n}=P_{B_{n}}(a_{n-1})$ b n = P B n ( a n − 1 ) and $a_{n}=P_{A_{n}}(b_{n})$ a n = P A n ( b n ) . Suppose that A ∩ B is bounded, we prove that if the couple (A,B) is (boundedly) regular then the couple (A,B) is d-stable, i.e., for each {an} and {bn} as above we have dist(an,A ∩ B) → 0 and dist(bn,A ∩ B) → 0. Similar results are obtained also in the case A ∩ B = ∅, considering the set of best approximation pairs instead of A ∩ B.

Solving a Split Feasibility Problem by the Strong Convergence of Two Projection Algorithms in Hilbert Spaces

The goal of this manuscript is to establish strong convergence theorems for inertial shrinking projection and CQ algorithms to solve a split convex feasibility problem in real Hilbert spaces. Finally, numerical examples were obtained to discuss the performance and effectiveness of our algorithms and compare the proposed algorithms with the previous shrinking projection, hybrid projection, and inertial forward-backward methods.