Best Proximity Point Theorems

• Best proximity point theorems extend fixed point theory by ensuring existence and uniqueness of points in disjoint sets that achieve the minimal distance under specified mappings.
• They employ cyclic, orbital, and multivalued contraction methods in diverse settings such as uniformly convex Banach spaces, CAT_p(0) spaces, and ultrametric spaces.
• The theory leverages geometric properties like convexity and proximal normal structure to guarantee iterative convergence and stability in applications.

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 (Kumar et al., 13 Dec 2025).

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 (Kumar et al., 13 Dec 2025).

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 $k = \prod_i k_i < 1$ (Sen, 2012). 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 (Leon et al., 2013).

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 (Fernández-León, 2012, Som, 2021).

• $p$-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 (Som, 2021).

• 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 (Kumar et al., 30 Mar 2025).
• Geraghty-type: Contractive factor $\beta(t)$ is variable, only requiring $\beta(t)\to 1 \implies t\to 0$, broadening the contraction class (Fogh et al., 2 Oct 2025).
• $F$-contractive ($F\in \mathcal{F}$ Wardowski): General nonlinear contraction inequalities lead to best proximity points via reduction to Hardy–Rogers-type fixed point results (Som, 2021).
• Almost-$\Theta$-contraction: Nonlinear distortion via $\Theta$ ensures decay of distances in the iteration and application to continuous or multivalued mappings (Hussain et al., 2018).

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 $$d(x^*, T(x^*, y^*)) = d(y^*, T(y^*, x^*)) = \mathrm{dist}(A,B)$$ (Gupta et al., 2019, Kumar et al., 30 Mar 2025). 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 (Norouzian et al., 2019).

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 (Kumar et al., 13 Dec 2025, Leon et al., 2013).
• 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 (Chaira et al., 2021, Chaira et al., 2021).
• 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 (Som et al., 2020).

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 (Zhelinski et al., 2023, Rajesh et al., 2016).
• Proximal normal structure: Sufficient to guarantee existence but not necessary, with counterexamples available (Leon et al., 2013).

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) (Kumar et al., 13 Dec 2025, Zhelinski et al., 2023, Norouzian et al., 2019, Som et al., 2020).

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.