Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems

ABSTRACT

In this paper, we propose and study new inertial viscosity Tseng's extragradient algorithms with self-adaptive step size to solve the variational inequality problem (VIP) and the fixed point problem (FPP) in Hilbert spaces. Our proposed methods involve a projection onto a half-space and self-adaptive step size. We prove that the sequence generated by our proposed methods converges strongly to a common solution of the VIP and FPP of an infinite family of strict pseudo-contractive mappings in Hilbert spaces under some mild assumptions when the underlying operator is monotone and Lipschitz continuous. Furthermore, we apply our results to find a common solution of VIP and zero-point problem (ZPP) for an infinite family of maximal monotone operators. Finally, we provide some numerical experiments of the proposed methods in comparison with other existing methods in the literature.

KEYWORDS:

• Minimization problem
• quasi-pseudocontractive mappings
• Lipschitzian
• fixed point problem
• iterative scheme

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 47H09
• 47H10
• 49J20
• 49J40

Acknowledgments

The authors sincerely thank the editor and anonymous referees for their careful reading, constructive comments, and fruitful suggestions that substantially improved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author acknowledges with thanks the scholarship and financial support from the University of KwaZulu-Natal (UKZN) Doctoral Scholarship. The third author is supported by the National Research Foundation (NRF) of South Africa, Incentive Funding for Rated Researchers [grant number 119903].