Strong convergence of viscosity forward-backward algorithm to the sum of two accretive operators in Banach space

In recent years, the Forward-Backward algorithm (FBA) received much attention due to its various applications in image recovery, signal processing, and machine learning. In this paper, we consider the FBA in the setting of Banach spaces that are uniformly convex and q-uniformly smooth. We introduce two viscosity FBA, and one of them with weakly contractive mapping, which generalizes many previous results on the viscosity approximation method with fixed contraction. Moreover, we establish their strong convergences under more general conditions.