Proximal Normal Structure: Solving the Fixed Point Problem for Disjoint Sets

TL;DR

In the realm of functional analysis, the existence of a fixed point ($Tx=x$) typically requires the mapping to act upon a single set. But what happens when a mapping $T$ jumps between two disjoint sets $A$ and $B$? This paper by Eldred, Kirk, and Veeramani introduces Proximal Normal Structure, a geometric property that guarantees the existence of a "Best Proximity Point"—a point that minimizes the error between $x$ and $Tx$ when a solution to $Tx=x$ is mathematically impossible.

Background & Motivation: Beyond Nonexpansive Mappings

A mapping is nonexpansive if it doesn't increase the distance between any two points. While the Kirk's Fixed Point Theorem (1965) settled the existence of fixed points for such maps on weakly compact convex sets, it assumed the mapping stayed within the same set.

The authors tackle relatively nonexpansive mappings, where: $\Vert Tx-Ty\Vert \le \Vert x-y\Vert extforallx\in A,y\in B$ The challenge arises when $A\cap B=\varnothing$. You can't have $Tx=x$ if $x\in A$ and $Tx\in B$. Instead, we look for $x$ such that the distance $\Vert x-Tx\Vert$ is the absolute minimum possible distance between the two sets, denoted as $extdist(A,B)$.

Methodology: The Geometry of Proximal Normal Structure

The core innovation is the Proximal Normal Structure.

1. Defining the Structure

A pair $(K_1,K_2)$ has this structure if for any smaller, closed, convex proximal pair $(H_1,H_2)$, there exist points $x_1,x_2$ that are not "extreme" in terms of their distance to the opposing set. Specifically: $\delta (x_1,H_2)<\delta (H_1,H_2)$ This ensures we can always "shrink" our candidate sets toward a solution.

2. The Existence Theorems

The paper presents two landmark results:

• Theorem 2.1: If $T:A\cup BoA\cup B$ swaps the sets ($T(A)\subseteq B,T(B)\subseteq A$), there is a point $x$ such that $\Vert x-Tx\Vert =extdist(A,B)$.
• Theorem 2.2: If $T$ stays within its set but is relatively nonexpansive across sets, and the space is strictly convex, then there exist fixed points $x_0\in A,y_0\in B$ which are also at the minimal distance from each other.

Uniform Convexity and Iterative Methods

One of the most practical contributions is the proof that Uniformly Convex Banach spaces (like Hilbert spaces or $L^p$ spaces for $1<p<\infty$) automatically possess this proximal normal structure.

Furthermore, the authors prove that a Krasnosel’skĭı iteration: $x_{n+1}=\frac{x_n+T{x}_n}{2}$ converges to the fixed point (if $T(A)$ is compact), providing a computational path to find these optimal points.

Deep Insight: Why It Works in Hilbert Spaces

In a Hilbert space, the geometry is much "cleaner." The paper demonstrates that in this setting, the metric projection $P$ is relatively nonexpansive. The authors use the Pythagorean Theorem to show that relative nonexpansivity actually implies global nonexpansivity on $A\cup B$ once the sets are restricted to their proximal points ($A_0,B_0$).

Critical Analysis & Conclusion

Takeaway

This paper provides the rigorous mathematical foundation for Best Proximity Point Theory. It extends fixed point theory to scenarios where constraints are mutually exclusive, a common occurrence in optimization and economic equilibrium modeling.

Limitations

• The second theorem requires Strict Convexity, which excludes spaces like $L^1$ or $L^{\infty }$.
• The convergence of the iteration requires the mapping to be compact ($T(A)$ lies in a compact set), which may not always hold in infinite-dimensional settings without additional assumptions.

Future Work

The introduction of proximal normal structure opens the door to studying "Cyclic Mappings" and complex dynamical systems where a particle moves between different manifolds or regions of a state space.