Covering Dimension and Nonlinear Equations

For a set S in a Banach space, we denote by dim(S) its covering dimension [1, p. 42]. Recall that, when S is a convex set, the covering dimension of S coincides with the algebraic dimension of S, this latter being understood as ∞ if it is not finite [1, p. 57]. Also, S and conv(S) will denote the closure and the convex hull of S, respectively. In [3], we proved what follows. dim({x ∈ X : Φ(x) = Ψ(x)}) ≥ dim(Φ −1 (0)). In the present paper, we improve Theorem A by establishing the following result. Now, fix any bounded open convex set A in X such that B X 0, ρ δ ⊆ A. Put K = Ψ(A). Since Ψ is completely continuous, K is compact. Fix any positive integer n such that n ≤ dim(Φ −1 (0)). Also, fix z ∈ K. Thus, Φ −1 (z) ∩ A is a convex set of dimension at least n. Choose n + 1 affinely independent points u z,1 ,. .. , u z,n+1 in Φ −1 (z) ∩ A. By the open mapping theorem again, the operator Φ is open, and so, successively, the multifunctions y → Φ −1 (y), y → Φ −1 (y) ∩ A, and y → Φ −1 (y) ∩ A are lower semicontinuous. Then, applying the classical Michael theorem [2, P. 98]