Iteration families for which expansion implies collapse (Q816930)

Let \(X\) be a set and let \(A: X \to 2^{\mathbb R}\) be a set-valued function with non-empty values. Assume that for every \(s,t \in (0,+ \infty)\) and \(x,z \in X\) with \([A(x) + s + t] \cap A(z) \neq \emptyset\) there exists \(y \in X\) such that \([A(x) + s] \cap A(y) \neq \emptyset\) and \([A(y) + t] \cap A(z) \neq \emptyset\). Let \(e(t,x) := \sup \{[A(x) + s] \cap A(X) \neq \emptyset \}\) and \(F^{t}(x) := A^{-}(A(x) + e(t,x))\), where \(A^{-}(V) := \{x \in X: A(x) \cap V \neq \emptyset\}\) for \(V \subset R\). The main result is: If \(F^{t}[F^{s}(x)] \subset F^{t+s}(x)\) for \(x \in X\) and \(s,t \in (0,+ \infty)\), then the inclusion \(F^{t+s}(x) \subset F^{t}[F^{s}(x)]\) holds for \(x \in X\) and \(s,t \in (0,+ \infty)\).