Metric regularity, fixed points and some associated problems of variational analysis (Q490042)

Let \(X\) be a metric space and let \[ A:X\rightrightarrows X \eqno(1) \] be a set-valued mapping whose graph is complete in the product metric of \(X\times X\). For the mapping (1), under certain conditions (nonlocal regularity etc.), the existence of a fixed-point is proved, i.e. Fix\(A\neq \varnothing\). This theorem is a generalization of known theorems in this direction. Let \[ F:X\rightrightarrows Y, \;G:Y\rightrightarrows X\eqno(2) \] be set-valued mappings with closed graphs, where \(Y\) is a metric space. The elements of \[ \text{Fix}(F,G)=\{(x,y):y\in F(x),\;x\in G(y)\} \] are called double fixed points of the mapping pair \((F,G)\). For the mappings (2) it is proved that Fix\((F,G)\neq \varnothing.\) Moreover, two types of iteration procedures are proposed for the construction of a fixed-point and many important examples are considered.