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

TL;DR

This paper establishes a sharp Hilbert-space characterization for the fixed point property of $(c)$-mappings: a nonempty closed convex subset $C$ of a real Hilbert space has the $(c)$-FPP if and only if $C$ is bounded. It further shows that every $(c)$-mapping is encompassed by the class of firmly nonexpansive mappings, and proves convergence results for Picard sequences under uniform convexity and symmetry assumptions. Additional analyses link the orbit behavior, fixed points, and the range of $(I-T)$, with illustrative examples highlighting the necessity of hypotheses. The work concludes with several open questions about $(c)$-FPP in broader spaces and structural properties of $(I-T)$, signaling directions for future research and potential extensions beyond Hilbert spaces.

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.

Theorems & Definitions (14)

• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Example 2.4
• Theorem 2.5
• Theorem 2.6
• Corollary 2.7
• Definition 2.8
• Theorem 2.9
• Example 2.10