Strong convergent inertial Tseng’s extragradient method for solving non-Lipschitz quasimonotone variational inequalities in Banach spaces

Abstract

. The class of quasimonotone variational inequalities is more general and applicable than the class of pseudomonotone and monotone variational inequalities. However, few results can be found in the literature on quasimonotone variational inequalities and currently results are mostly on weak convergent methods in the framework of Hilbert spaces. In this paper, we study the class of non-Lipschitz quasimonotone variational inequalities and the class of non-Lipschitz variational inequalities without monotonicity in the framework of Banach spaces. We propose a new inertial Tseng’s extragradient method and obtain some strong convergence results for the proposed algorithm under some mild conditions on the control parameters. While the cost operator is non-Lipschitz, our proposed method does not require any linesearch procedure but employs a more efficient and simple self-adaptive step sizes with known parameters. Finally, we present several numerical experiments to demonstrate the implementability of our proposed method.