A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods

• Published in: Journal of optimization theory and applications Vol. 185; no. 3; pp. 744 - 766
• Main Authors: Jolaoso, Lateef Olakunle, Taiwo, Adeolu, Alakoya, Timilehin Opeyemi, Mewomo, Oluwatosin Temitope
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2020; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Banach spaces; Calculus of Variations and Optimal Control; Optimization; Engineering; Estimates; Inequalities; Iterative methods; Mathematical analysis; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Operators (mathematics); Optimization; Theory of Computation; Projection method; Fixed point problem; 65K15; Variational inequality; 90C33; Iterative method; Extragradient method; Banach space; 47J25; 65J15
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-020-01672-3

Several iterative methods have been proposed in the literature for solving the variational inequalities in Hilbert or Banach spaces, where the underlying operator
A
is monotone and Lipschitz continuous. However, there are very few methods known for solving the variational inequalities, when the Lipschitz continuity of
A
is dispensed with. In this article, we introduce a projection-type algorithm for finding a common solution of the variational inequalities and fixed point problem in a reflexive Banach space, where
A
is pseudo-monotone and not necessarily Lipschitz continuous. Also, we present an application of our result to approximating solution of pseudo-monotone equilibrium problem in a reflexive Banach space. Finally, we present some numerical examples to illustrate the performance of our method as well as comparing it with related method in the literature.