Errata to ‘Localized Normal Maps and the Stability of Variational Inclusions’ (Set-Valued Analysis 12 (2004), 259–274)

Abstract

PROPOSITION 3.1. Assume the hypotheses of Theorem 2.1. Choose w0 2 W0 and take S 1⁄4 U0 fw0g. Define multifunctions X: U0 ! X0 and Z : U0 ! Z0 by requiring that ðu; xÞ belong to the graph of X if and only if ðu;w0; xÞ 2 MS, and that ðu; zÞ belong to the graph of Z if and only if ðu;w0; zÞ 2 NS, where MS and NS are defined by (7). Then one of the multifunctions X and Z is inner semicontinuous if and only if the other is. Proof. Let G be an open subset of X0 and suppose that Xðu0Þ meets G. Then there is x0 2 G such that ðu0;w0; x0Þ 2 MS. Let z0 1⁄4 ðu0;w0; x0Þ; then ðu0; w0; z0Þ 2 NS and therefore z0 2 Zðu0Þ; moreover, ðu0; z0Þ 1⁄4 x0. As is continuous, we can find an open neighborhood H of z0 and a neighborhood U 00 of u0 with U 00 U0 and such that if ðu; zÞ 2 U00 H then ðu; zÞ 2 G. As Z is inner semicontinuous, by shrinking U00 further if necessary we can ensure that if u 2 U00 then ZðuÞ meets H. Choose u 2 U00, and find z in ZðuÞ \ H. The triple ðu; w0; zÞ is then in NS, so the point ðu;w0; xÞ :1⁄4 ðu;w0; zÞ belongs to MS, and x 1⁄4 ðu; zÞ. As ðu; zÞ 2 U00 H we have x 1⁄4 ðu; zÞ 2 G; however, as ðu;w0; xÞ 2 MS we also have x 2 XðuÞ. Therefore XðuÞ meets G, so X is inner semicontinuous at u0. A parallel argument gives the proof in the other direction. I