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

?

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

https://ictp.acad.ro/wp-content/uploads/2022/08/1993a-Precup-On-the-reverse-of-the-Krasnoselskii-Browder-boundary-inequality.pdf

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

ZB: 828.47055.

Google Scholar Profile

[1] Browder, F E , Fixed point-theory and nonlinear problems, Bull Amer Math Soc (New Series) 9(1983), 1-39
[2] Browder, F E , The theory of degree of mapping for nonlinear mappings of monotone type, Nonlinear Partial Differential Equations and Applications (H Brezis, J L Lions eds ) VI, Pitman, 1984,165-177
[3] Deimling, К , Nonlinear functional analysis, Springer, 1985
[4] Lakshimikantham, V , Sun, Yong, A theorem on the existence of two fixed points, Jour Math. Phy Sci 25(1991), 281-286
[5] Precup, R., Generalized topological transversality and mappings of monotone type. Studia Universitatis Babeş-Bolyai, Mathematica 35, No 2(1990), 44-50