Variational principles for supinf problems with constraints (Q1980270)

This paper studies the supinf problem \[ \sup_{x\in X}\inf_{y\in Kx}f(x,y) \] where \(X\), \(Y\) are completely regular topological spaces, \(K:X\rightrightarrows Y\) is a set-valued mapping and \(f:X\times Y\rightarrow \left[ -\infty,\infty \right] \) is an extended real-valued function. The question studied is the existence of solutions, in case \(f(x,y)\) is replaced by a perturbation \(f(x,y)+p(x)+q(y)\), \(\left( p,q\right) \in C(X)\times C(Y)\), or \(f(x,y)+u(x,y)\), \(u\in C(X\times Y)\). Conditions are given under which the set of perturbations \((p,q)\) under which the problem has a solution, is dense in \(C(X)\times C(Y)\). An analogous result is given for the density of the set of perturbations \(u\) in \(C(X,Y)\). Further, the results are applied to the weak Stackelberg problem. For the entire collection see [Zbl 1445.53003].