[Paper Review] A fixed point theorem for closed-graphed decomposable-valued correspondences

Idione Meneghel, Rabee Tourky|arXiv (Cornell University)|Jun 17, 2013

Advanced Graph Theory Research13 references3 citations

TL;DR

This paper establishes a fixed point theorem for closed-graphed, decomposable-valued correspondences on atomless measure spaces, extending prior results by relaxing compactness requirements. The key contribution is a new fixed point existence result for decomposable mappings with sequentially closed graphs, which improves upon Cellina et al.'s theorem by removing the need for norm compactness in the range, relying instead on $μ$-uniform compactness and metrizability in pointwise convergence topology.

ABSTRACT

We prove a fixed point theorem for closed-graphed, decomposable-valued correspondences whose domain and range is a decomposable set of functions from an atomless measure space to a topological space. One consequence is an improvement of the fixed point theorem in Cellina, Colombo, and Fonda (1986).

Motivation & Objective

• To establish a fixed point theorem for closed-graphed, decomposable-valued correspondences in the context of atomless measure spaces.
• To generalize Cellina et al.'s fixed point result by replacing norm compactness with $μ$-uniform compactness in the range.
• To demonstrate that compact and metrizable subsets in the topology of pointwise convergence admit the $μ$-fixed point property.
• To provide a foundation for fixed point analysis in strategic models with purely subjective uncertainty and Bayesian games.

Proposed method

• The proof constructs a sequentially pointwise continuous mapping $\theta$ from $L([0,1],C)$ to $L(S,T)$, where $C$ is the Cantor set, to transfer topological properties.
• It uses a maximal chain of measurable sets in the atomless measure space to define a measurable function $r(s)$ that assigns a real number in $[0,1]$ to each point in $S$, enabling a lifting of functions.
• The construction ensures that functions differing on a countable set are mapped to functions differing on a $μ$-null set, preserving essential equivalence.
• It leverages the Eilenberg-Montgomery fixed point theorem by showing that the inverse image of a decomposable set under $\theta$ is also decomposable and that the resulting space is a compact absolute retract.
• The proof establishes that the set of monotone functions in $L(S,T)$ with the topology of pointwise convergence forms a Peano continuum, ensuring local connectedness and absolute retract properties.
• It applies the fixed point theorem to the correspondence $\tilde{Q}$ on the inverse image space, concluding the existence of a fixed point via the Eilenberg-Montgomery theorem.

Experimental results

Research questions

• RQ1 Can a fixed point theorem be established for decomposable-valued correspondences without requiring norm compactness in the range?
• RQ2 Under what topological and measure-theoretic conditions does a closed-graphed, decomposable-valued correspondence admit a fixed point?
• RQ3 Can compact and metrizable subsets in the topology of pointwise convergence be embedded into a sequentially compact set with the $μ$-fixed point property?
• RQ4 How does the structure of decomposable sets in $L_1(\mu, L)$ interact with sequential closedness and fixed point existence?
• RQ5 What role does the topology of pointwise convergence play in enabling fixed point results for correspondences on function spaces?

Key findings

• A fixed point theorem is established for closed-graphed, decomposable-valued correspondences on atomless measure spaces, generalizing prior results by relaxing the need for norm compactness.
• The key result is that if a correspondence $B$ has a closed graph, is decomposable-valued, and satisfies $X \cap B(f) \neq \emptyset$ for all $f \in X$, where $X$ is $\mu$-uniformly compact, then $B$ has a fixed point.
• Compact and metrizable subsets of $L(S,T)$ in the topology of pointwise convergence can be embedded into a sequentially compact set $Y$ that has the $\mu$-fixed point property.
• The construction of $\theta$ ensures that sequential pointwise convergence is preserved, and that functions differing on a countable set are mapped to functions differing on a $\mu$-null set.
• The inverse image space $((\mathcal{G} \cap \mathcal{M})^{\leftarrow}, \delta)$ is shown to be a Peano continuum and thus a compact absolute retract, enabling the application of the Eilenberg-Montgomery fixed point theorem.
• The correspondence $\tilde{Q}$ on $\mathcal{Z}^{\leftarrow}$ is shown to have a closed graph and thus admits a fixed point, concluding the proof of Theorem 2.2.