---
title: Best Proximity Point Theorems
url: https://www.emergentmind.com/topics/best-proximity-point-theorems
type: topic
---

# Best Proximity Point Theorems

Best proximity point theorems provide a generalization of classical fixed point theory to settings where mappings act between non-intersecting subsets of a metric or Banach space. The primary objective in this theory is to guarantee the existence, uniqueness, and iterative construction of points in one subset which, under the action of a specified mapping, attain the minimal possible distance to an image in another subset, thus optimizing processes where genuine fixed points cannot exist.

## 1. Fundamental Concepts and Definitions

The central objects in best proximity theory are pairs of nonempty subsets $(A,B)$ of a metric space $(X,d)$ where $A \cap B = \varnothing$ (or more generally, $A$ and $B$ may be arbitrary). The critical metric is the minimal gap,
\[
\mathrm{dist}(A,B) = \inf\{ d(a,b) : a \in A, b \in B \}.
\]
Given a mapping $T : A \to B$, a point $x^* \in A$ is a best proximity point if $d(x^*, T(x^*)) = \mathrm{dist}(A,B)$. For cyclic self-maps $T: A \cup B \to A \cup B$ with $T(A) \subset B$ and $T(B) \subset A$, the problem is to find $x^* \in A$ (or $y^* \in B$) such that $d(x^*, T(x^*)) = \mathrm{dist}(A,B)$.

Generalizations involve multivalued mappings, order structures, higher cyclicity (e.g., $p$-cyclic), coupled pairs, and extensions to geometric settings such as $CAT_p(0)$ metric spaces and ultrametric spaces. Key subclasses of mappings include cyclic contractions, orbital contractions, relatively nonexpansive maps, asymptotically nonexpansive, and various general contractive types (proximal, Kannan, Geraghty-type, $F$-contractive, almost $\Theta$-contractive, and others).

## 2. Cyclic and Orbital Contraction Theorems

The classical best proximity point framework for cyclic contractions is governed by inequalities of the form
\[
d(Tx, Ty) \leq \eta d(x, y) + (1-\eta)\mathrm{dist}(A,B), \quad x \in A, y \in B, \quad \eta \in (0,1),
\]
as in Eldred–Veeramani's theorem. The recent extension via cyclic orbital contraction mappings replaces the contraction on direct arguments by a supremum over orbital distances along bounded orbits:
\[
d(\Xi\varsigma,\, \Xi\vartheta)
\leq
\eta \sup\{\, d(\Xi^i\varsigma,\Xi^j\varsigma),\, d(\Xi^k\vartheta, \Xi^\ell\vartheta),\, d(\Xi^p\varsigma, \Xi^q\vartheta)\}
+ (1-\eta)\, \mathrm{dist}(\Omega, \Delta),
\]
where $\Xi$ alternates between $\Omega$ and $\Delta$ and induces bounded orbits [2512.12361].

### Main Existence and Uniqueness Results:

Let $(\mathfrak X, d)$ be a complete $CAT_p(0)$ space or a uniformly convex Banach space and $\Omega$, $\Delta$ nonempty, closed and convex. If $\Xi$ is a cyclic orbital contraction:
- There exists a unique best proximity point $\varsigma^* \in \Omega$,
- $\varsigma^*$ is the unique fixed point of $\Xi^2$ in $\Omega$,
- The iterates $\Xi^{2n}\varsigma$ converge to $\varsigma^*$ for any $\varsigma \in \Omega$,
- $\Xi \varsigma^*$ is a best proximity point in $\Delta$, and orbits starting from $\Delta$ converge to it [2512.12361].

Proof techniques combine iterative contraction estimates with geometry-specific lemmas (thin-triangle in $CAT_p(0)$, modulus-of-convexity in Banach spaces) to establish Cauchy properties of the constructed sequences and derive convergence.

## 3. Multivalued, Ordered, and Relatively Nonexpansive Generalizations

Multivalued cyclic self-mappings $T: \bigcup A_i \to 2^X \setminus\{\varnothing\}$ (e.g., $p$-cyclic mappings between strips) are analyzed via Hausdorff contractive inequalities:
\[
H(Tx, Ty) \leq k_i d(x, y) + (1-k_i)D_i,
\]
where $D_i = \mathrm{dist}(A_i, A_{i+1})$, and $k = \prod_i k_i < 1$ [1212.0145]. The existence and uniqueness of best proximity points (and sequences converging to them) is ensured under suitable boundedness, closedness, monotonicity/order, and convexity, with further stability and uniqueness results in uniformly convex Banach spaces.

For relatively nonexpansive mappings (not necessarily contractive), proximal normal structure is a sufficient (but not necessary) condition for existence in reflexive Busemann-convex, CAT(0), or uniformly convex geodesic spaces:
\[
d(Tx, Ty) \leq d(x, y), \quad x \in A, y \in B
\]
implies existence of best proximity pairs under proximal normal structure, or even without it in some cases [1306.0358].

## 4. Variants: Proximal, Kannan, Geraghty, and $F$-Contractive Types

Several sharp contraction frameworks guarantee existence and often uniqueness of best proximity points:
- **Proximal contraction of the first kind:** For $T:A\to B$, $\alpha \in [0,1)$,
\[
d(u_1,u_2) \leq \alpha\,d(x_1,x_2)
\]
when $d(u_i,T x_i) = d(A,B)$, ensures unique best proximity point and geometric convergence of iterates [1207.4349][2107.01362].
- **$p$-proximal contraction:** For some $k\in (0,1)$,
\[
d(u_1,u_2) \leq k\bigl( d(x_1,x_2) + |d(u_1,x_1) - d(u_2,x_2)| \bigr)
\]
with $k<1/3$ producing contraction on $A_0$ and hence existence and uniqueness via Banach's principle [2107.01685].
- **Kannan-type or p-cyclic Kannan nonexpansive mappings:** Satisfy averaged contraction conditions on the images of pairs, with existence and sometimes uniqueness under weak compactness and convexity [2503.23548].
- **Geraghty-type:** Contractive factor $\beta(t)$ is variable, only requiring $\beta(t)\to 1 \implies t\to 0$, broadening the contraction class [2510.02406].
- **$F$-contractive ($F\in \mathcal{F}$ Wardowski):** General nonlinear contraction inequalities lead to best proximity points via reduction to Hardy–Rogers-type fixed point results [2108.04945].
- **Almost-$\Theta$-contraction:** Nonlinear distortion via $\Theta$ ensures decay of distances in the iteration and application to continuous or multivalued mappings [1810.07624].

## 5. Coupled, Tripartite, and Multistrip Best Proximity Results

Recent works analyze coupled best proximity point concepts for $T: (A\times B)\cup (B\times A) \to A\cup B$ under various cyclic or contractive conditions, guaranteeing pairs $(x^*, y^*)$ with $d(x^*, T(x^*, y^*)) = d(y^*, T(y^*, x^*)) = \mathrm{dist}(A,B)$ [1908.07225][2503.23548]. These frameworks accommodate $p$-cyclic $\phi$-contractions, p-cyclic Kannan mappings, and provide additional Ulam–Hyers stability.

Tripartite best proximity theory, established in generalized metric spaces (G-metric), addresses mappings acting on three strips $A, B, C$ with combinations of cyclic and noncyclic assignment rules, concrete convexity structure, and contractive inequalities to ensure existence and convergence of tripartite coincidence-best proximity points [1904.10309].

## 6. Extensions to Nonlinear Geometries and Topological Settings

Best proximity theory now encompasses settings beyond Banach and standard metric spaces:
- **$CAT_p(0)$ metric spaces ($p\ge2$):** Results mirror Banach space theory but rely on geometric comparison triangles and “thin-triangle” lemmas [2512.12361][1306.0358].
- **Ultrametric spaces:** Convexity is supplanted by spherical completeness and strong triangle inequality; minimal closed balls and invariance yield best proximity pairs even for nonexpansive and contractive maps [2104.06748][2109.05728].
- **Topological spaces with continuous gauges $g(x,y)$:** New notions of $g$-closed, $g$-sequentially compact sets facilitate Banach-type contraction and best proximity analysis in the absence of metric structure [2007.10852].

## 7. Convexity and UC-Type Properties; Open Directions

Convexity plays a central role:
- **Uniform convexity** (of space or just of sets $A$): ensures Cauchy property and uniqueness; however, generalized versions (UC, UC*, BUC properties) suffice in many cases [2303.05850][1611.02484].
- **Proximal normal structure:** Sufficient to guarantee existence but not necessary, with counterexamples available [1306.0358].

Current open directions include:
- Relaxing bounded-orbit and uniform convexity hypotheses via coercivity, pointwise contraction, or implicit relations;
- Generalizing to multivalued, set-valued, or hybrid cyclic orbital contractions;
- Extending best proximity principles to more general geometries (e.g., incomplete, nonpositively curved, G-metric, or topological spaces) [2512.12361][2303.05850][1904.10309][2007.10852].

## Summary Table: Key Theorem Settings

| Space Type                      | Mapping/Condition Type                    | Existence/Uniqueness Criteria                     |
|----------------------------------|-------------------------------------------|----------------------------------------------------|
| Complete $CAT_p(0)$/Banach      | Cyclic orbital contraction                | $\Xi$ bounded orbits, contraction sup estimate     |
| Uniformly convex Banach space    | p-cyclic, Hausdorff contractive, ordered  | Product contraction constant $<1$                  |
| Geodesic/metric/ultrametric      | Relatively nonexpansive, thin-structure   | Proximal normal structure, spherical completeness  |
| Banach/metric/topological        | Proximal/Kannan/Geraghty/$F$-contractive  | Contraction or dominance condition, completeness   |

Best proximity point theory thus unifies and extends fixed-point methods to diverse geometries, contraction classes, and mapping topologies, providing well-defined generalizations and constructive iterative schemes for minimization and optimization when genuine fixed points are excluded by structure.

Source: https://www.emergentmind.com/topics/best-proximity-point-theorems