Geometric inequalities for solving variational inequality problems in certain Banach spaces

Abstract

In this paper, we develop some new geometric inequalities in p-uniformly convex and uniformly smooth real Banach spaces with p > 1. We use the inequalities as tools to obtain the strong convergence of the sequence generated by a subsgradient method to a solution that solves fixed point and variational inequality problems. Furthermore, the convergence theorem established can be applicable in, for example, L p (Ω), where Ω ⊂ R is bounded set and l p (R) for p ∈ (2, ∞). Finally, numerical implementations of the proposed method in the real Banach space L 5 ([−1, 1]) are presented.