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

ABSTRACT

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.

Keywords:

• Forward-backward algorithm
• accretive operator
• Banach space
• strong convergence

2010 Mathematics Subject Classifications:

• 46B20
• 46N10
• 47H09

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Y. Wang was supported in part by National Natural Science Foundation of China with grant no.11901171, and Foundation of Henan Educational Committee with grant no. 16A110017. F. Wang was supported in part by National Natural Science Foundation of China with grant no.11971216. H. Zhang was supported in part by National Natural Science Foundation of China with grant no. 11701154.