Generalized viscosity implicit rules for solving quasi-inclusion problems of accretive operators in Banach spaces

Abstract

The generalized viscosity implicit rules for solving quasi-inclusion problems of accretive operators in Banach spaces are established. The strong convergence theorems of the rules to a solution of quasi-inclusion problems of accretive operators are proved under certain assumptions imposed on the sequences of parameters. The results presented in this paper extend and improve the main results of Refs. (Moudafi, J Math Anal Appl. 2000;241:46–55; Xu et al., Fixed Point Theory Appl. 2015;2015:41; López et al., Abstr Appl Anal. 2012;2012; Cholamjiak, Numer Algor. DOI:10.1007/s11075-015-0030-6.). Moreover, some applications to monotone variational inequalities, convex minimization problem and convexly constrained linear inverse problem are presented.

Keywords:

• The generalized viscosity implicit rules
• accretive operator
• maximal monotone operator
• Banach space
• splitting method
• forward–backward algorithm

Acknowledgements

The authors would like to express their thanks to the Referees for their helpful comments and advices.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Natural Science Foundation of Center for General Education, China Medical University, Taichung, Taiwan; the Natural Science Foundation Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, 807, Taiwan.