On approximation theorems and fixed point theorems for non-self-mappings in infinite dimensional Banach spaces (Q1342361)

The author proves some results concerning the minimal displacement property of a 1-set-contraction \(f\) defined on a nonnecessarily convex subset of an infinite-dimensional normed space \(X\) and such that the field \(I-f\) is proper. The typical result states that, for a strict-contraction \(f\) of an annulus \(A=\{x\in X: r\leq|x|\leq R\}\), there is \(u\in A\) such that \(|u-f(u)|= d(f(u),A)\) provided \(|f(x)|\geq r\) for all \(|x|=r\). Some other results of the same nature yielding fixed point theorems and valid for 1-set-contractions are stated and proved. The author remarks that these results have purely infinite-dimensional character and provides a counterexample; however, he does not attempt to explain the nature of this restriction which, by the way, is rather clear.