cartesian product of two sets X × Y, denote by A ( x ) and A − $^{1}$( y ) the respective sections { y ∈ Y : ( x,y ) ∈ A } and { x ∈ X : ( x,y ) ∈ A } ; denote by A − 1 the subset { ( y,x ) : ( x,y ) ∈ A } . Definition 8 A von Neumann relation is a subset A of a cartesian product X × Y, where X and Y are subsets of topological vector spaces, satisfying: (i) for every x ∈ X, the section A ( x ) is convex and non-empty; (ii) for every y ∈ Y, the section A − $^{1}$( y ) is open in X and X \ A − $^{1}$( y ) is convex. Denote by N ( X,Y ) the class of von Neumann relations in X × Y and by N − $^{1}$( X,Y ) := { A : X −→ 2 Y : A − 1 ∈ N ( Y,X ) } . Note that von Neumann relations are particular cases of F $^{∗}$− maps ( appli- cations de Ky Fan ) introduced in [3]. Theorem 9 (Fixed Point for N − maps) Let E be a t.v.s.,  ̸ = Y ⊆ X ⊆ E with X convex, and let A ∈ N ( X,Y ) . If there exist a compact subset K of X and a compact convex subset D of Y such that for every x ∈ X \ K, A ( x ) ∩ D =  , then A has a fixed point, i.e., (ˆ x, ˆ x ) ∈ A for some ˆ x ∈ X. Proof. Define Γ : Y −→ 2 X as Γ( y ) := Y \ A − $^{1}$( y ) ,y ∈ Y. Clearly, Γ has closed and convex values. Also, obviously, A ( x ) = Y \ Γ − $^{1}$( x ) , for any x ∈ X. One readily verifies that the compactness condition in Theorem 9 is equiv- alent to the compactness condition in Theorem 6. Indeed, ( x / ∈ K = ⇒ ∃ y ∈ A ( x ) ∩ D ) ⇐⇒ (( ∀ y ∈ D,y / ∈ A ( x )) = ⇒ x ∈ K ) ⇐⇒ ( ⋂ y ∈ $_{D}$Γ( y ) ⊆ K ) . The intersection ⋂ y ∈ $_{D}$Γ( y ) being closed in K which is compact, is also compact. For any subset Y 0 of D, it also holds ⋂ y ∈ Y 0 Γ( y ) is a compact subset of K. The fact that all sections A ( x ) ,x ∈ X, are non-empty rules out the the- sis of Theorem 6 (indeed, ( A ( x ) =  , ∀ x ∈ X ) ⇐⇒ ⋂ y ∈ Y Γ( y ) =  ) . There- fore Γ cannot be a KKM map, i.e., there exist { y 1 ,...,y$_{n}$ } ⊆ Y and ˆ y ∈ 9