# Viscosity approximations considering boundary point method for fixed-point and variational inequalities of quasi-nonexpansive mappings

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## Abstract

Existing viscosity approximation schemes have been extensively investigated to solve equilibrium problems, variational inequalities, and fixed-point problems, and most of which contain that contraction is a self-mapping defined on certain bounded closed convex subset *C* of Hilbert spaces *H* for standard viscosity approximation. In particular, if the zero point does not belong to *C*, the standard viscosity approximation cannot be applied to solve the minimum norm fixed point of some nonlinear operators. In this paper, we introduce three generalized viscosity approximation algorithms with boundary point method for quasi-nonexpansive mappings, which overcome this deficiency above. These three proposed algorithms have simple expressions; moreover, they are easy to implement in the actual computation process. Especially, we can find the minimum norm fixed point of quasi-nonexpansive mappings under contraction is a zero operator.

## Keywords

Quasi-nonexpansive mappings Minimum norm fixed point Viscosity approximation Strong convergence## Mathematics Subject Classification

90C25 90C30 47J25## Notes

### Acknowledgements

This work is supported by Humanity and Social Science Youth Foundation of Ministry of Education under Grant No. 18YJC630274, Start-up Funds for Doctors of Anhui University under Grant No. J01003282, Tianjin Science and Technology Development Strategy Research Plan under Grant No. 17ZLZXZF01060, and National Natural Science Foundation of China under Grant Nos.71431005, 71771168, 71701078.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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