An iterative viscosity approximation method for the split common fixed-point problem

ABSTRACT

The split common fixed point problem (SCFPP) for the demicontractive operators is studied. An iterative viscosity approximation method (VAM) for SCFPP is introduced. It is shown that the sequence generated by VAM is strongly convergent to a solution of SCFPP under certain conditions, and this solution is identified to be the unique solution of some variational inequality. The main result of this paper extends and improves some results of Yao et al. [Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theory Appl. 2014;183], Boikanyo [A strongly convergent algorithm for the split common fixed point problem. Appl Math Comput. 2015;265:844–853] and Cui–Wang [Iterative methods for the split common fixed point problem in Hilbert spaces. Fixed Point Theory Appl. 2014;78].

Keywords:

• Split common fixed-point problem
• demicontractive operator
• viscosity approximation method
• strong convergence

2010 Mathematics Subject Classifications:

• 47J25
• 47H45

Acknowledgments

This work was carried out by the three authors in collaboration. Moreover, the three authors have read and approved the final manuscript. The authors thank especially one of the anonymous referees for his/her constructive remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Science Foundation of China [No. 12031003 and No. 11771347], the China Postdoctoral Science Foundation [No. 2019M652837], Natural Science Foundation of Shaanxi Provincial Department of Education [No. 18JK0344].