[Paper Review] Existence of fixed points for a particular multifunction

Biagio Ricceri|arXiv (Cornell University)|Jan 25, 2010

Fixed Point Theorems Analysis2 references3 citations

TL;DR

This paper establishes the existence of a fixed point for a specific multifunction derived from compact mappings on the unit sphere of an infinite-dimensional reflexive Banach space with the Kadec-Klee property. By applying the Fan-Kakutani fixed point theorem to a composition of upper semicontinuous multifunctions and continuous retractions, it proves that there exists a point $\hat{x} \in S$ such that $f(\hat{x})(\hat{x}) = \|f(\hat{x})\|_{E^*}$, under the condition $\inf_{x\in S}\|f(x)\|_{E^*} > 0$.

ABSTRACT

We prove a fixed point theorem for a particular multifunction from the unit sphere of a reflexive Banach space with the Kadec-Klee property into itself.

Motivation & Objective

• To establish conditions under which a continuous compact mapping $f: S \to E^*$ on the unit sphere of a reflexive Banach space admits a point $\hat{x} \in S$ such that $f(\hat{x})(\hat{x}) = \|f(\hat{x})\|_{E^*}$.
• To investigate the necessity of the Kadec-Klee property in ensuring the existence of such a point.
• To demonstrate that the combination of reflexivity, compactness of $f$, and the infimum condition $\inf_{x\in S}\|f(x)\|_{E^*} > 0$ guarantees a solution.
• To explore the role of upper semicontinuity and convexity in the multifunction framework via the Fan-Kakutani theorem.
• To propose a broader problem on whether the Kadec-Klee property is necessary for the fixed point property under the stated conditions.

Proposed method

• Define the multifunction $\Phi_f(x) = \{y \in S : f(x)(y) = \|f(x)\|_{E^*}\}$, transforming the problem into finding a fixed point of $\Phi_f$.
• Construct the multifunction $\Psi: E^* \setminus \{0\} \to 2^S$ by $\Psi(\varphi) = \{x \in S : \varphi(x) = \|\varphi\|_{E^*}\}$, and prove its upper semicontinuity using weak convergence and the Kadec-Klee property.
• Use the Eberlein–S̆mulyan theorem to show that weakly compact subsets of $S$ are norm-compact under the Kadec-Klee property, ensuring $\Psi(\varphi)$ is compact for $\varphi \neq 0$.
• Utilize a retraction $\omega: B \to S$ with $\omega(x) = x$ for $x \in S$, where $B$ is the closed unit ball, to define a composition $G(x) = \Psi(f(\omega(x)))$ on a compact convex set $Y$.
• Apply the Fan-Kakutani fixed point theorem to the upper semicontinuous multifunction $G: Y \to 2^Y$ with non-empty, closed, convex values to obtain a fixed point $\hat{x} \in Y$ with $\hat{x} \in G(\hat{x})$.
• Leverage the fact that $\hat{x} \in S$ implies $\omega(\hat{x}) = \hat{x}$, so $\hat{x} \in \Phi_f(\hat{x})$, thus satisfying the desired equality.

Experimental results

Research questions

• RQ1 Under what conditions on a reflexive Banach space $E$ does every compact mapping $f: S \to E^*$ with $\inf_{x\in S}\|f(x)\|_{E^*} > 0$ admit a point $\hat{x} \in S$ such that $f(\hat{x})(\hat{x}) = \|f(\hat{x})\|_{E^*}$?
• RQ2 Is the Kadec-Klee property necessary for the existence of such a fixed point when $E$ is infinite-dimensional and reflexive?
• RQ3 Can the fixed point property be guaranteed without compactness of $f$, even if $E$ is reflexive and has the Kadec-Klee property?
• RQ4 What role does the upper semicontinuity of the norm-attaining multifunction $\Psi$ play in the existence proof?
• RQ5 Does the existence of a retraction $\omega: B \to S$ with $\omega(x) = x$ on $S$ enable the application of fixed point theorems in this setting?

Key findings

• The paper proves that if $E$ is an infinite-dimensional reflexive real Banach space with the Kadec-Klee property, and $f: S \to E^*$ is compact with $\inf_{x\in S}\|f(x)\|_{E^*} > 0$, then there exists $\hat{x} \in S$ such that $f(\hat{x})(\hat{x}) = \|f(\hat{x})\|_{E^*}$.
• The multifunction $\Psi: E^* \setminus \{0\} \to 2^S$ defined by $\Psi(\varphi) = \{x \in S : \varphi(x) = \|\varphi\|_{E^*}\}$ is upper semicontinuous, a key step in the proof.
• The set $\Psi(K)$, where $K = \overline{f(S)}$ is compact and does not contain zero, is compact in $S$, due to the Kadec-Klee property and reflexivity.
• The composition $G(x) = \Psi(f(\omega(x)))$ defines an upper semicontinuous multifunction with non-empty, closed, convex values from a compact convex set $Y$ into itself.
• The Fan-Kakutani fixed point theorem guarantees a fixed point $\hat{x} \in Y$ with $\hat{x} \in G(\hat{x})$, and since $\hat{x} \in S$, it satisfies $\hat{x} \in \Phi_f(\hat{x})$.
• The paper provides counterexamples showing that the result fails if either the infinite-dimensionality of $E$ or the compactness of $f$ is removed, and also shows that condition (1) is necessary.