An extended inertial Halpern-type ball-relaxed CQ algorithm for multiple-sets split feasibility problem

In an infinite-dimensional real Hilbert space setting, we introduce a new self-adaptive inertial Halpern-type projection algorithm for solving multiple-sets split feasibility problem assuming that the involved convex subsets are level subsets of strongly convex functions by approximating the original convex subsets by a sequence of closed balls instead of half-spaces. Since the projection onto the closed ball has a closed form, the proposed algorithm is hence easy to implement. Moreover, we construct a new self-adaptive stepsize that is bounded away from zero and is independent of the operator norm. Under some mild assumptions and without the usual Lipschitz’s continuity of the gradient operator assumption, we establish and prove a strong convergence of the sequence generated by the proposed algorithm. Several numerical illustrations indicate that the proposed algorithm is computationally efficient and competes well with some existing algorithms in the literature. The proposed method is an improvement and generalization of many results in the literature.