Journal of the
Korean Mathematical Society JKMS

In this paper we are concerned with theoretical properties of gap functions for the extended variational inequality problem (EVI) in a general Banach space. We will present a correction of a recent result of Chen et. al. without assuming the convexity of the given function $\Omega$. Using this correction, we will discuss the continuity and the differentiablity of a gap function, and compute its gradient formula under two particular situations in a general Banach space. Our results can be regarded as infinite dimensional generalizations of the well-known results of Fukushima, and Zhu and Marcotte with some modifications.

Keywords: variational inequalities, gap functions, G\^ateaux differentiable, the Clarke generalized gradient, nonconvex programming

MSC numbers: 49J40, 47J20, 49J50, 52A41, 90J26