# On the reverse of the Krasnoselskii-Browder boundary inequality

## Abstract

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.

## Authors

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

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## Paper coordinates

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

## PDF

##### Journal

Studia Universitatis Babes-Bolyai Mathematica

##### Publisher Name

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

1221-8103

2065-9490

ZB: 828.47055.