The fixed point property for $(c)$-mappings and unbounded sets

TL;DR

This paper establishes that in a real Hilbert space, a closed convex set has the fixed point property for (c)-mappings if and only if it is bounded, with additional convergence results for iterative methods.

Contribution

It proves the equivalence between boundedness and the fixed point property for (c)-mappings in Hilbert spaces, providing new insights into fixed point theory.

Findings

Boundedness is necessary and sufficient for fixed point property for (c)-mappings.

Convergence results for iterative methods are derived.

Unbounded sets lack the fixed point property for (c)-mappings.

Abstract

We prove that a closed convex subset $C$ of a real Hilbert space $X$ has the fixed point property for $(c)$-mappings if and only if $C$ is bounded. Some convergence results about the iterations are obtained.