zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Variational inequalities and fixed point theorems for PM maps. (English) Zbl 0942.49011

Let X be a Banach space, X * be the dual space of X, K be a closed convex subset in X, J:XX * be a duality map and T:DKX * a PM-map. The authors consider the problem of finding yD such that

Jy-Ty,y-x0forallxK·

They used Brezis results on minimax principle to obtain results on variational inequalities for J-T with T being a PM-map and establish an acute angle principle for a PM-map. They use the theory of variational inequalities for J-T where T is a PM-map, to give new applications on the existence of fixed points for generalized inward PM-maps in Hilbert spaces. They establish a new equivalence between variational inequalities and nearest points of maps. For generalized inward maps the equivalence between variational inequalities and fixed points of these maps is established. They used this relationship not only to study properties of PM-maps but also to derive a fixed point principle for non-self maps from variational inequalities. The results generalize and unify many earlier results with different and new methods. This paper also has applications of fixed point theory to homogeneous integral equations.

MSC:
49J40Variational methods including variational inequalities
47H10Fixed point theorems for nonlinear operators on topological linear spaces