Tangency conditions for multivalued mappings (Q1915627)

The author discusses some conditions for two multivalued mappings \(F,G: X\rightrightarrows Y\) from a topological space \(X\) into a normed space \(Y\), guaranteeing the lower semicontinuity of their intersection. In fact, if \(F,G\) are convex valued near \(x_0\in X\) and satisfy an interiority condition \([\exists \alpha>0\), \(\exists s>0\) such that \(sB_Y \subseteq F(x)\cap \alpha B_Y-G(x) \cap\alpha B_Y\) for \(x\) in a neighbourhood of \(x_0]\) then \(F,G\) also satisfy a tangency condition at \(x_0\) \([\forall y_0\in \overline {F(x_0)} \cap \overline {G(x_0)}\) \(\exists r>0\), \(\exists a >0\) such that \(d(y,F(x)\cap G(x)) \leq ad(y,F(x))\) for \(x\) in a neighbourhood of \(x_0\), \(y\in y_0+ rB_Y\), \(d(y,F(x)) <r\), \((x,y)\in \text{Graph} (G)]\), which implies \(\liminf_{x\to x_0} (F(x)\cap G(x)) = \liminf_{x\to x_0} F(x)\cap \liminf_{x\to x_0} G(x)\). Some variants of these notions are also discussed and an application to the epi-upper semicontinuity of the sum of convex-valued mappings is given.