Extended Brouwer's Fixed-Point Theorem

• Extended Brouwer's Fixed-Point Theorem is a generalization that establishes approximate fixed point properties in abstract topological spaces with relaxed compactness and continuity requirements.
• It uses finite-dimensional approximations and combinatorial sequence analysis to extend classical fixed point results from normed finite-dimensional spaces to broader functional-analytic contexts.
• The theorem offers practical insights for optimization, economics, and game theory by ensuring approximate fixed points through weak sequential methods and topological relaxations.

The Extended Brouwer's Fixed-point Theorem encompasses a collection of modern generalizations, structural relaxations, and computable/constructive refinements of Brouwer's classical fixed point result for continuous self-maps on convex, compact subsets of finite-dimensional spaces. These extensions address abstract topological vector spaces, weak/approximate versions of fixed points, parametrized and combinatorial settings, and various functional-analytic, algebraic, and topological situations extending the theorem’s reach well beyond normed finite-dimensional domains.

1. Approximate Fixed Point Property in Abstract Spaces

In the context of an arbitrary Hausdorff topological vector space $X$ and a subset $Z$ of its topological dual $X^*$, the $\sigma(X, Z)$-approximate fixed point property (AFPP) replaces exact fixed points with sequences approximating fixedness in the weak sense. Concretely, a bounded, closed convex subset $C \subset X$ has the $o(X, Z)$-AFPP if for every continuous $f: C \to C$, there exists a sequence $\{x_n\}\subset C$ such that

$\forall x^* \in Z,\quad x^*(x_n - f(x_n)) \to 0.$

This framework interpolates between classical fixed point theorems in Banach spaces and much weaker topologies, allowing approximate fixed points when strong compactness or metrizability are unavailable—particularly relevant in infinite-dimensional or nonmetrizable settings (Barroso et al., 2011).

2. Use and Adaptation of Brouwer’s Fixed-point Theorem

Brouwer's finite-dimensional result is repurposed as a technical core through finite-dimensional approximation. The core steps, as formalized in (Barroso et al., 2011), are:

• Project the image of $C$ under a continuous linear map into a finite-dimensional subspace.
• Cover this projection by finitely many small neighborhoods ("perturbation sets").
• Isolate a finite-dimensional compact convex subset $K \subset C$ where $f$ is suitably continuous with respect to the $\sigma(X, Z)$-topology.
• Apply Brouwer's theorem on $K$, yield a fixed point for a related map, and infer that $0$ is in the closure of $\{x - f(x): x\in C\}$ in the weak topology.

This reduction facilitates approximate fixed points in highly abstract settings, generalizing previous arguments for Banach and locally convex spaces.

3. Principal Results and Structural Dichotomies

The approximate fixed point property is characterized under three mutually exclusive hypotheses:

1. Separability of $Z$ in strong topology: When $Z$ is separable (for uniform convergence on bounded sets), for every $f$ continuous on $C$, there is a sequence $\{z_n\}$ such that $z_n - f(z_n)\to 0$ in the $\sigma(X, Z)$-topology.
2. Metrizable locally convex $X$, $Z = X^*$: In metrizable locally convex spaces, the presence of an $\ell_1$-sequence in $C$ acts as an obstruction; the absence of such sequences is equivalent to the weak–AFPP on all nonempty closed convex subsets of $C$.
3. Nonmetrizable $X$ admitting a compatible metrizable topology: If $X$ allows a finer, compatible metrizable locally convex topology, weak–AFPP in $C$ is equivalent to the property that every sequence in $C$ admits a weakly Cauchy subsequence.

This trichotomy generalizes and unifies diverse scenarios covering classical Banach, Fréchet, and locally convex topologies—with explicit logical dependencies between combinatorial sequence structures and fixed point approximability.

4. Fréchet–Urysohn Property and Sequential Reduction

Sequential mechanisms are essential for extracting explicit approximation schemes. The set $C-C$ is shown in (Barroso et al., 2011) to be Fréchet–Urysohn in the $\sigma(X, Z)$- or weak topology: any point in its closure can be reached by a sequence from $C-C$. This property is crucial, as it allows:

• Passage from general net-based convergence (in nonmetrizable spaces) to sequential arguments,
• Extraction of sequences $\{x_n\}$ from abstract closure and continuity arguments, ensuring constructive formation of approximate fixed point sequences.

This is vital for both foundational and computational reasons since nets are generally not algorithmically accessible.

5. Role of Rosenthal’s $\ell_1$-Theorem: Absence of $\ell_1$-Sequences

An adapted version of Rosenthal’s classical $\ell_1$-theorem enables structural classification of bounded sequences. In a metrizable locally convex space, every bounded sequence either contains a weakly Cauchy subsequence or a subsequence equivalent to the unit vector basis of $\ell_1$ (an $\ell_1$-sequence).

Formally, for any sequence $\{x_i\}$ constituting an $\ell_1$-sequence (per Definition 3.1), there exists a continuous seminorm $p$ and $M > 0$ such that

$\sum_{i=1}^n p(x_i) \geq M \sum_{i=1}^n |a_i|$

for all choices of scalars $a_i$. Absence of such $\ell_1$-sequences ensures the extraction of weakly Cauchy subsequences, which, combined with Fréchet–Urysohn's property, allows construction of approximate fixed point sequences. Thus, the presence of $\ell_1$-type configuration serves as a complete obstruction to the AFPP in the metrizable case.

6. Generalization, Flexibility, and Applications

The analytic and topological machinery of (Barroso et al., 2011) substantially enlarges the reach of fixed point theory:

• Extends from Banach to arbitrary Hausdorff TVS and metrizable/nonmetrizable locally convex spaces.
• Frees the strictly compact (or closed) map requirements in classical theorems, substituting the weak/approximate property and permitting noncompact, "large" contexts.
• Provides necessary and sufficient criteria for approximate fixed point properties, including in highly non-classical (e.g., non-Hausdorff, nonnormable) topological vector spaces.
• Enables practical applications in optimization, economics, and game theory where truly compactness is too restrictive and weak sequential properties are realistic.
• Embeds combinatorial, sequence-based tools as analytic invariants to diagnose the existence/lack of fixed points.

The main LaTeX formulas capturing the approach are: $$\forall x^* \in Z,\quad \lim_{n\to\infty} x^*(x_n - f(x_n)) = 0$$ and

$\sum_{i=1}^n p(x_i) \geq M \sum_{i=1}^n |a_i| \quad (\text{characterizing absence of %%%%47%%%%-sequences}).$

In summary, the Extended Brouwer's Fixed-point Theorem in this sense structurally links topological, functional-analytic, and combinatorial properties relevant to fixed point approximability in very general classes of spaces, enabling new existence and approximation results that generalize, unify, and extend much of classical fixed point theory (Barroso et al., 2011).