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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 $X$, $J:X\to {X}^{*}$ be a duality map and $T:D\subset K\to {X}^{*}$ a PM-map. The authors consider the problem of finding $y\in D$ such that

$〈Jy-Ty,y-x〉\le 0\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}x\in K·$

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:
 49J40 Variational methods including variational inequalities 47H10 Fixed point theorems for nonlinear operators on topological linear spaces