Perimetric Proximal Contractions

• Perimetric proximal contractions are a geometric framework that generalizes classical fixed point theory by controlling the combined distance (perimeter) of multiple points under mapping iterations.
• They extend traditional contraction mappings, such as Banach, Kannan, and Chatterjea types, to multi-point settings, resulting in novel fixed point and best proximity point theorems.
• The approach offers concrete criteria (e.g., for quadrilaterals and triangles) that facilitate convergence proofs and have potential applications in solving integral and differential equations.

Perimetric proximal contractions are a geometric generalization of classical contraction principles in metric fixed point theory, designed to control the "perimeter" formed by multiple points under repeated application of a mapping. These concepts extend traditional two-point contractive conditions to multi-point settings, revealing new structures for fixed point and best proximity point phenomena. Perimetric proximal contractions have recently been formulated both for fixed point problems—using quadrilaterals or triangles—and for best proximity point problems in which the mapping’s image falls outside its domain, yielding new existence and finiteness results for solutions (Banerjee et al., 2024, Garai et al., 31 Jan 2026).

1. Formal Definitions and Main Types

Let $(X, d)$ be a metric space, with $A, B \subset X$ nonempty (typically disjoint) subsets.

1.1 Perimetric Contraction on Quadrilaterals

A mapping $T: X \to X$ is a perimetric contraction on quadrilaterals if there exists $\alpha \in [0, 1)$ so that for all distinct $p, q, r, s \in X$

$P(Tp, Tq, Tr, Ts) \leq \alpha P(p, q, r, s),$

where

$$P(x, y, z, w) = d(x, y) + d(y, z) + d(z, w) + d(w, x).$$

This shrinks the perimeter of every quadrilateral by at least the factor $\alpha$ (Banerjee et al., 2024).

1.2 Kannan- and Chatterjea-Type Perimetric Contractions

Four-point analogues of Kannan and Chatterjea contractions are defined:

• Kannan-type: Exists $\beta \in [0,2)$ such that

$$P(Tp, Tq, Tr, Ts) \leq \beta \left[ d(p,Tp) + d(q,Tq) + d(r,Tr) + d(s,Ts) \right].$$

• Chatterjea-type: Exists $\gamma \in [0,2)$ such that

$$P(Tp, Tq, Tr, Ts) \leq \gamma \sum_{u \neq v} d(u,Tv), \quad \text{for } u,v \in \{p,q,r,s\}.$$

These broaden the scope beyond strict perimetric contraction, capturing further classes of non-expansive-type mappings (Banerjee et al., 2024).

1.3 Perimetric Proximal Contractions

Let $T: A \to B$, define:

• $d(A,B) = \inf\{ d(a, b): a\in A,\, b\in B \}$
• $$A_0 = \{ x \in A: d(x, y) = d(A,B) \text{ for some } y \in B \}$$
• $B_0$ similarly.

First Kind: $T$ is a perimetric proximal contraction of the first kind if for $\alpha \in [0,1)$: For every $u_1, u_2, u_3, x_1, x_2, x_3 \in A$ with $x_1, x_2, x_3$ pairwise distinct,

$$\left( d(u_i, T x_i) = d(A,B)\ \forall i \right)\ \Longrightarrow\ \sum_{\text{cyc}}d(u_i, u_{i+1}) \leq \alpha \sum_{\text{cyc}}d(x_i, x_{i+1}).$$

Second Kind: $T$ is a perimetric proximal contraction of the second kind if for $\alpha \in [0,1)$: For every such 6-tuple, with $T x_1, T x_2, T x_3$ pairwise distinct, and $d(u_i, T x_i) = d(A,B)$,

$$\sum_{\text{cyc}}d(Tu_i, Tu_{i+1}) \leq \alpha \sum_{\text{cyc}}d(Tx_i, Tx_{i+1}).$$

(Garai et al., 31 Jan 2026)

Additionally, Condition $\Lambda$ ensures absence of distinct points $x \ne y$ in $A$ with $d(x,Ty) = d(y,Tx) = d(A,B)$.

2. Fixed Point and Best Proximity Point Theorems

2.1 Fixed Point Theorems in Quadrilaterals

Let $(X, d)$ be complete, $T: X \to X$ a perimetric contraction on quadrilaterals as above for $\alpha \in [0,1)$. Then:

• $T$ has a fixed point if and only if it has no periodic points of prime period 2 or 3.
• There are at most three fixed points (uniqueness if further constraints hold).
• The method of proof involves: generating an orbit $x_{n+1} = T x_n$, showing the perimeter $P(x_n, x_{n+1}, x_{n+2}, x_{n+3})$ contracts geometrically, and using Cauchy estimates and completeness to guarantee convergence and fixedness (Banerjee et al., 2024).

The classical Banach contraction follows when the perimetric inequality is restricted to repeated points, yielding a standard contraction mapping.

2.2 Best Proximity Point Theorems

• For perimetric proximal contractions (first kind), assume:
  • $(X,d)$ is complete, $A, B$ closed, $A_0 \ne \varnothing$, $|A_0| \geq 3$, $T$ continuous as above, and Condition $\Lambda$.
  • Then $T$ has at least one best proximity point (i.e., $d(u,Tu) = d(A,B)$), and at most two such points (Garai et al., 31 Jan 2026).
• For the second kind: with $A$ approximatively compact w.r.t.\ $B$ and $T$ injective, same conclusion (existence, at most two best proximity points).

The proof mechanism involves constructing a sequence with attainers of the proximity bound and leveraging the perimetric perimeter decay to ensure Cauchyness.

2.3 Table: Summary of Perimetric Contraction Types

Variant	Domain	Contractive Structure	Maximum Distinct Solutions
Quadrilateral (fixed point)	Full space	Perimeter on 4 points shrinks by α	3
Triangular proximal, 1st kind	A→B	Preimage perimeter via proximity	2
Triangular proximal, 2nd kind	A→B	Image perimeter via proximity	2 (if injective)

3. Illustrative Examples

• Four-point orbit contraction: Let $X = \{x_0, x_1, \dots\} \cup \{x^*\}$, with an explicit metric where $T(x_n) = x_{n+1}$, $T(x^*) = x^*$, and the perimeter for any quadruple contracts by $\alpha = 3/4$. The sequence started at $x_0$ converges to the unique fixed point $x^*$ (Banerjee et al., 2024).
• Proximal, two best proximity points: Let $A = \{-3,0,3,4\}$, $B = [-2, -1] \cup [1,2]$, and

$$T(-3) = -2,\; T(0) = 1.5,\; T(3) = 2,\; T(4) = 1.$$

Then $d(A,B) = 1$, $A_0 = \{-3,0,3\}$, and $-3, 3$ are both best proximity points (Garai et al., 31 Jan 2026).

• Injective, unique best proximity point: $A = \{(x,1): -1 \leq x \leq 1\}$, $B = \{(x,0): -1 \leq x \leq 1\}$, $T(x,1) = (x/10, 0)$. Here, exactly one best proximity point exists: $(0,1)$.

4. Geometric and Analytical Structure

Perimetric contractions generalize the geometric contraction of figures (triangles, quadrilaterals, etc.) under a mapping, controlling the perimeter—a natural measure in many geometric and analytic settings. This perspective unifies and extends several existing fixed point frameworks (Banach, Kannan, Chatterjea) by interpreting them as degenerate cases (e.g., two-point contractions as quadrilaterals with repeated vertices).

A key analytic property is the translation of perimeter contraction into strong Cauchyness of iterates, allowing direct arguments for existence and sometimes finiteness of solutions, in contrast to traditional approaches focused on metric distances alone (Banerjee et al., 2024, Garai et al., 31 Jan 2026).

5. Relation to Classical Contractions and Generalizations

• The perimetric framework recovers Banach’s fixed point theorem, Kannan and Chatterjea contractions as special (two-point) cases under perimeter shrinking, supporting a unified theory.
• Extensions to higher-point perimetric contractions (pentagons, etc.) are direct, suggesting a hierarchy of multi-point contraction principles.
• In contrast to classical best proximity point results (which typically guarantee uniqueness or infinitely many solutions), perimetric proximal contractions inherently cap the number of solutions at two; this nonuniqueness bound is strictly new relative to earlier two-point contractive schemes (Garai et al., 31 Jan 2026).

6. Applications and Future Directions

Potential applications include iterative solution methods for integral and differential equations where a "perimeter-shrinking" condition is natural; for example, in the geometry of discretized solution sets or in optimization theory where best proximity points yield optimal approximate solutions.

Further research is aimed at:

• Developing perimetric proximal contractions for higher polygons and in more general distance structures, such as semimetric spaces or spaces with triangle functions.
• Combining perimetric conditions with simulation-function techniques, $F$-contractions, or other functional contractions to obtain new existence and uniqueness results.
• Investigating algorithmic realizations and rates of convergence in numerical approximation settings (Banerjee et al., 2024, Garai et al., 31 Jan 2026).

A plausible implication is the potential to systematically refine fixed point and best proximity theory by formulating contractive conditions capturing geometric information about sets of points well beyond traditional pairwise distances.