On the reverse of the Krasnoselskii-Browder boundary inequality

Let $G$ be a bounded open subset of a Banach space $X$ with $0\in G$ and let $f$ be a map from $G$ into the dual $X$. The following Krasnoselskii-Browder boundary inequality $(x,f(x))\leq 0$ for all $x\in \partial G$ is for some types of maps sufficient for the existence of solutions $x\in G$ for equation $f(x)=0$.  This article deals with the reverse of the above inequality, namely $(x,f(x))\leq 0$ for all $x \in \partial G$.  We prove that if $X$ is an infinite-dimensional Hilbert space, $f=I-g$ where $I$ is the identity on $X$ and $g: G\rightarrow X$ is completely continuous, then the inequality $(x,f(x))\leq 0$ can not be true for all $x \in \partial G$.  Consequently two existence theorems proved in [4] have no content since their assumptions are never satisfied. We then ask if such a negative result holds true even for more general maps of monotone type. A partial answer is finally given.

Radu Precup
Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania

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R. Precup, On the reverse of the Krasnoselskii-Browder boundary inequality, Studia Univ. Babeş-Bolyai Math. 38 (1993) no. 2, 41-55.

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http://www.cs.ubbcluj.ro/~studia-m/old_issues/subbmath_1993_38_02.pdf

ZB: 828.47055.

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