Positivity 12 (2008), 667–676
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040667-10, published online March 12, 2008
Existence of nonzero solutions for a class
of generalized variational inequalities
Abstract. The existence of nonzero solutions for a class of generalized vari-
ational inequalities is studied by topological degree theory for multi-valued
mappings in ﬁnite dimensional spaces and reﬂexive Banach space. One of the
mappings concerned here is nonlinear with coercive or monotone and other is
set-contractive or upper semi-continuous. Under some suitable assumptions,
some existence theorems of nonzero solutions for this generalized variational
inequalities are obtained.
Mathematics Subject Classiﬁcation (2000). Primary 49J40, Secondary 47H10.
Keywords. Variational inequality, ﬁxed point index, nonzero solutions,
Variational inequality theory as an important part of nonlinear analysis has become
a kind of very powerful tool of the current mathematical technology. It has been
applied intensively to mechanics, diﬀerential equation, cybernetics, quantitative
economics, optimization theory and nonlinear programming, etc.(see [1–4]).
In virtue of minimax theorem of Ky Fan and KKM technique, variational
inequalities, generalized variational inequalities and generalized quasi-variational
inequalities were studied intensively in the last 30 years with topological method,
variational method, semi-ordering method and ﬁxed point method ([1–4]).
Recently, the existence of nonzero solutions for variational inequalities, as another
important topic of variational inequality theory, has been discussed by ﬁxed point
index approach in the literatures ([5–7]).
Let X be a real Banach space, X
its dual and (·, ·) the pair (q, u):=q(u)
and X. Suppose that K is a nonempty closed convex subset of X.
In this paper, we discuss the existence of nonzero solutions for another class of
generalized variational inequalities involving multi-valued mappings as follows:
This work was supported by the Young Talent Foundation of Zhejiang Gongshang University
and the Foundation of Department of Education of Zhejiang Province No. 20070628.