ABSTRACT
Let E be topological space (metric space, normed space or locally convex spaces). Denote by P(E) = {Y ⊂ E : Y 6= ∅}, Pcl(E) = {Y ∈ P(E) : Y is closed}, Pb(E) = {Y ∈ P(E) : Y is bounded}, Pcv(E) = {Y ∈ P(E) : Y is convex}, and Pcp(E) = {Y ∈ P(E) : Y is compact}. Let X and Y be two spaces and assume that for every point x ∈ X a nonempty closed subset ϕ(x) of Y is given (sometime we will assume only that ϕ(x) 6= ∅); in this case, we say that ϕ is a multivalued mapping from X to Y and write ϕ : X → P(Y ). More precisely a multivalued map ϕ : X → P(Y ) can be defined as a subset ϕ ⊂ X×Y such that the following condition is satisfied:
for all x ∈ X there exists y ∈ Y such that (x, y) ∈ ϕ. In what follows, the symbol ϕ : X → Y is reserved for single-valued mappings, i.e. ϕ(x) is a point of Y . Let ϕ : X → P(Y ) be a multivalued map. We associate with ϕ the graph (interchangeably, we use the notations Γ(ϕ) or Γϕ) of ϕ by putting:
Γ(ϕ) = {(x, y) ∈ X × Y | y ∈ ϕ(x)} as well as two natural projections pϕ : Γϕ → X , qϕ : Γ(ϕ)→ Y defined as follows: pϕ(x, y) = x and qϕ(x, y) = y, for every (x, y) ∈ Γ(ϕ).
