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  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.
R. Precup, On the reverse of the Krasnoselskii-Browder boundary inequality, Studia Univ. Babeş-Bolyai Math. 38 (1993) no. 2, 41-55.
Studia Universitatis Babes-Bolyai Mathematica
Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania
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