In this paper the authors consider multivalued maps, or better set valued maps, and define upper and lower semicontinuity for these kind of maps. A set valued maps in which in particular the authors are interested in the following situation: given two functions \(f: A \times \Lambda \to\mathbb{R}\) and \(g: A \times A \times \Lambda \to \mathbb{R}\), where \(A\) is a nonempty compact and convex subset of a topological vector space, they consider \[ C(\lambda):=\{x\in A\mid f(x,\lambda)\geq 0\} \] and \[ S(\lambda):=\{x\in C(\lambda)\mid g(x,y,\lambda)\leq 0\,\forall y\in C (\lambda)\}. \] The authors define (EP) the problem of showing that, given, \(f, g, C\), the set \(S (\lambda)\) is not empty.
The authors prove some properties of the function \(C\) and use them to give sufficient conditions for \(S\) to be upper or lower semicontinuous. Then they give a notion of well posedness under some suitable perturbations and prove that problem (EP) is well posed in this sense.