Optim Lett (2018) 12:287–299
Weak sharp solutions of mixed variational inequalities
in Banach spaces
· Man He
Received: 17 April 2016 / Accepted: 30 January 2017 / Published online: 4 February 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract In this paper, using the approximate duality mapping, we introduce the
deﬁnition of weak sharpness of the solution set to a mixed variational inequality in
Banach spaces. In terms of the primal gap function associated to the mixed variational
inequality, we give several characterizations of the weak sharpness.
Keywords Variational inequality · Weak sharpness · Primal gap function ·
Mathematics Subject Classiﬁcation 49J52 · 90C26 · 58E35
be the n dimensional Euclidean space, C be a nonempty closed convex subset
and F : R
→ R be a mapping. A variational inequality problem (VIP) is to
ﬁnd an element ¯x ∈ C such that
F( ¯x), x −¯x≥0, ∀ x ∈ C.
The solution set of (VIP) is denoted by
Many researchers studied variational inequality since it has important applications
in mathematical programming, partial differential equations and optimal control. The
reader can refer to the reference  for more knowledge about variational inequality.
Department of Mathematics, Yunnan University, Kunming 650091, PR China