Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Banach spaces

In this paper, we combine the subgradient extragradient method with the Halpern method for finding a solution of a variational inequality involving a monotone Lipschitz mapping in Banach spaces. By using the generalized projection operator and the Lyapunov functional introduced by Alber, we prove a strong convergence theorem. We also consider the problem of finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of a relatively nonexpansive mapping. Our results improve some well-known results in Banach spaces or Hilbert spaces.