Calculus rules for derivatives of multimaps
In the paper under review, the authors give calculus rules for derivatives of the compositions of two multifunctions. Let \(X\), \(Y\) and \(Z\) be normed vector spaces and \(F : X \rightrightarrows Y\) and \(G : Y \rightrightarrows Z\) be multimaps. The composition \((G \circ F) : X \rightrightarrows Z\) is defined by \((G \circ F) (x) = \bigcup_{y \in F(x)} G(y)\). For a point \(z= (x,y) \in \mathrm{cl} (\mathrm{gph}(F))\), the contingent derivative of \(F\) at \(z\) is the multimap \(DF(x,y) : X \rightrightarrows Y\) whose graph is the contingent cone to \(\mathrm{gph}(F)\) at \(z\) in \(X \times Y\), that is, \(\mathrm{gph} (DF(x,y)) = \lim \sup _{t \to +0} \frac{1}{t} (\mathrm{gph}(F) -z)\), where \(\mathrm{gph}(F)\) stands for the graph \(\{(x,y) \in X \times Y : y \in F(x) \}\) of \(F\), and \(\mathrm{cl}(A)\) for the closure of a subset \(A\) in a space. The multimap \(G ^{-1} : Z \rightrightarrows Y\) is defined by \(G^{-1} (z) = \{ y \in Y : z \in G(y)\}\). Define \(C : X \times Z \rightrightarrows Y\) by \(C(x,z) = F(x) \cap G^{-1}(y)\). The authors prove the following: Suppose \(Y\) is finite dimensional. Given \((x,z) \in \mathrm{gph}(G \circ F)\) such that \(C\) is directionally semi-compact at \((x,z)\) and \(DC((x,z) , y) (0,0) = \{0\}\) holds for every \(y \in \mathrm{cl}(C (x,z))\), then one has \[ D(G \circ F)(x,z) \subset \bigcup_{y \in \mathrm{cl}(C (x,z))} DG( y,z) \circ DF(x,y). \] If moreover, \(DF(x,y)(u) \cap DG(z,y)^{-1}(w) \subset DC((x,z),y)(u,w)\) holds for every \((u,w) \in X \times Z\) and \(y \in \mathrm{cl}(C (x,z))\), then the above inclusion is equality. Similar results for incident derivatives of multifunctions and calculus rules for sums of multimaps are also proved.
contingent derivative: incident derivative
scientific article; zbMATH DE number 5593754