A novel adaptive method for operator inclusions

Abstract

A novel splitting algorithm for solving operator inclusion with the sum of the maximal monotone operator and the monotone Lipschitz continuous operator in the Banach space is proposed and studied. The proposed algorithm is an adaptive variant of the forward-reflected-backward algorithm, where the rule used to update the step size does not require knowledge of the Lipschitz constant of the operator. For operator inclusions in 2-uniformly convex and uniformly smooth Banach space, the theorem on the weak convergence of the method is proved.