Cyclic Orbital Contraction Mappings

• Cyclic Orbital Contraction Mappings are generalizations of classical cyclic contractions that employ orbit-based conditions across disjoint sets.
• They guarantee convergence and best proximity points in settings such as CAT_p(0) spaces and uniformly convex Banach spaces using both geometric and analytic techniques.
• The methodology leverages bounded orbits, even-subsequence convergence, and diagonal lemmas to ensure existence, uniqueness, and stability of best proximity points.

A cyclic orbital contraction mapping is a fundamental generalization of the classical cyclic contraction mapping paradigm in metric fixed-point theory. Rather than imposing contractivity solely on successive iterates between two disjoint closed convex sets, cyclic orbital contraction (COC) mappings employ an orbit-based contractive condition, accounting for the global trajectory of iterates. This concept yields significant advances in the analysis of best proximity points—points in one set whose images under the map achieve minimal distance to the other set—even when no true fixed point exists. The theory applies both in $CAT_p(0)$ spaces, which encompass Hilbert spaces, $\ell_p$ spaces ($p\ge2$), and manifolds of nonpositive curvature, as well as uniformly convex Banach spaces, connecting geometric and analytic fixed-point techniques.

1. Definition and Basic Properties

Let $(X,d)$ be a metric space with nonempty subsets $\Omega,\,\Delta\subset X$ and $$\operatorname{dist}(\Omega,\Delta) = \inf\{d(x,y):x\in\Omega,\,y\in\Delta\}$$. A mapping $T:\Omega\cup\Delta\to\Omega\cup\Delta$ is a cyclic map if $T(\Omega)\subset\Delta$ and $T(\Delta)\subset\Omega$. The orbit of $x$ under $T$ is $O_T(x)=\{T^n x: n\ge0\}$.

Cyclic Orbital Contraction Mapping. $T$ is a COC mapping if:

• (Bounded Orbit) $O_T(x)$ is bounded for all $x\in \Omega\cup\Delta$;
• (Cyclicity) $T(\Omega)\subset \Delta$, $T(\Delta)\subset \Omega$;
• (Orbital Contraction) There exists $\eta\in(0,1)$ such that for all $\sigma\in\Omega,\,\tau\in\Delta$,

$$d(T\sigma,T\tau) \le \eta\;\sup_{*} \{\cdots\} + (1-\eta)\operatorname{dist}(\Omega,\Delta)$$

where the supremum runs over odd and even differences in the orbits (specifically, odd-difference indices $i-j\equiv1\!\pmod2$, etc.). If the supremum is replaced by $d(\sigma,\tau)$, this reduces to the Eldred–Veeramani cyclic contraction form.

This generalization allows contractivity to leverage the global orbit structure, extending beyond pointwise or synchronous/asynchronous differences considered in (Pacurar, 2022).

2. Geometric Framework: $CAT_p(0)$ Spaces

A $CAT_p(0)$ space is a complete geodesic metric space $(X,d)$ in which all geodesic triangles are "thinner" than their model triangles in the Banach space $\ell_p$, $p\ge2$. More precisely, given $x,y,z\in X$ and corresponding points $\bar{u},\bar{v}$ in the model triangle in $\ell_p$, every pair of points $u,v$ on the geodesic triangle in $X$ satisfies $d(u,v)\le \|\bar{u}-\bar{v}\|_p$.

Key properties include:

• Convexity of Distance: For $0\le\lambda\le1$, $$d((1-\lambda)x\oplus\lambda y,(1-\lambda)x'\oplus\lambda y')^p \le (1-\lambda)d(x,x')^p + \lambda d(y,y')^p$$.
• Completeness and Convexity: Closed convex subsets in $CAT_p(0)$ spaces behave analogously to those in Hilbert spaces.
• Geodesic Uniqueness: Every pair of points can be joined by a unique constant-speed geodesic.

This geometric structure is crucial for quantitative iterative arguments required in the existence and uniqueness theorems for COC mappings (Kumar et al., 13 Dec 2025).

3. Best Proximity Point Theorems

Denote by $\sigma^*\in\Omega$ a best proximity point if $$d(\sigma^*,T\sigma^*) = \operatorname{dist}(\Omega,\Delta)$$. For COC mappings in $CAT_p(0)$ spaces, the following results hold (Kumar et al., 13 Dec 2025):

• Existence: There exists $\sigma^*\in\Omega$ such that $$d(\sigma^*,T\sigma^*) = \operatorname{dist}(\Omega,\Delta)$$.
• Uniqueness: $\sigma^*$ is the unique fixed point of $T^2$ in $\Omega$, and $T\sigma^*\in\Delta$ is the unique best proximity point in $\Delta$.
• Iterative Convergence: The even iterates $T^{2n}\sigma\to\sigma^*$ for any $\sigma\in\Omega$, and $T^{2n}\tau\to T\sigma^*$ for any $\tau\in\Delta$.

These results generalize the classical cyclic Banach contraction principle and best proximity point theorems (Sen, 2012, Sen, 2012). Notably, the convergence arguments exploit thin-triangle criteria and diagonal Cauchy-type lemmas in place of the Banach modulus of convexity.

4. Extensions in Uniformly Convex Banach Spaces

Uniform convexity in Banach spaces is defined by a modulus of convexity $\delta(\varepsilon)>0$ so that if $\|x\|,\|y\|\le 1$ and $\|x-y\|\ge\varepsilon$, then $\|(x+y)/2\| \le 1-\delta(\varepsilon)$. In this analytic setting, the framework for COC mappings adapts as follows (Kumar et al., 13 Dec 2025, Sen, 2012, Sen, 2012):

• The contractivity is established via Clarkson-type inequalities and convexity of the norm.
• The existence and uniqueness of best proximity points for COC mappings hold, with the sequence $T^{2n}x$ (for $x\in\Omega$) converging to the unique $\sigma^*\in\Omega$ satisfying $$d(\sigma^*,T\sigma^*) = \operatorname{dist}(\Omega,\Delta)$$.
• The proof substitutes thin-triangle and projection lemmas with Banach-space convexity arguments, utilizing the behavior of the modulus $\delta$ to force gap contractions in the sequence.

The fundamental property is that uniform convexity ensures not only existence and uniqueness but also strong (norm) convergence of iterates.

5. Proof Techniques and Central Lemmas

The fixed-point and best-proximity results are derived from an interplay of orbital contraction estimates and geometric or analytic convergence criteria. The proof structure includes:

• Orbital Iteration: Starting with $\sigma_0\in\Omega$, set $\sigma_{n+1}=T\sigma_n$. Use the contraction estimate to show $$d(\sigma_n,\sigma_{n+1})\to \operatorname{dist}(\Omega,\Delta)$$.
• Even-Subsequence Cauchy: Employ successive contraction bounds to demonstrate that $d(\sigma_{2n},\sigma_{2n+2})\to0$, ensuring the even subsequence is Cauchy.
• Diagonal Lemmas: Apply thin-triangle or diagonal Cauchy lemmata (or, in Banach spaces, corresponding convexity inequalities) to ensure full Cauchy convergence.
• Limit Point Characterization: The limiting point $\sigma^*$ satisfies $$d(\sigma^*,T\sigma^*)=\operatorname{dist}(\Omega,\Delta)$$, and is the unique $T^2$-fixed point.
• Global Convergence: By cyclical structure, the convergence holds for any starting point in $\Omega\cup\Delta$, preserving the structure of orbits and best proximity pairs (Kumar et al., 13 Dec 2025, Sen, 2012).

6. Examples and Special Cases

A concrete instance illustrating the power of COC mappings, from (Kumar et al., 13 Dec 2025), is as follows:

• Let $X=\mathbb{R}^2$ with the Euclidean norm.
• Define $\Omega=\{(-1,-a): a\in[-\tfrac12,\tfrac12]\}$ and $\Delta=\{(1,-b): b\in[-\tfrac12,\tfrac12]\}$.
• Define $T$ on $\Omega\cup\Delta$ by:

$$T(x_1,x_2) = \begin{cases} (x_1,-x_2/2)+(2,0) &\text{if } (x_1,x_2)\in\Omega,\ (x_1,-x_2/3)-(2,0) &\text{if } (x_1,x_2)\in\Delta. \end{cases}$$

• It is readily checked that $\operatorname{dist}(\Omega,\Delta)=2$, the orbital contraction holds with $\eta=0.95$, and $T$ is not a classical cyclic contraction. The unique best proximity point is $(-1,0)\in\Omega$ with $T(-1,0)=(1,0)\in\Delta$.

This illustrates that the COC mapping framework strictly generalizes previous cyclic contraction theories, capturing mappings excluded from the classical setting.

7. Role in Broader Fixed-Point Theory and Applications

COC mappings unify and extend several stratifications in fixed-point theory:

• Classical cyclic contractions [cf.\ Kirk–Srinivasan–Veeramani, 2003; (Pacurar, 2022)] are recovered as special cases by restricting the supremum in the orbital contraction to initial points.
• Multi-set (k-cyclic) mappings allow orbits to traverse more than two blocks, possibly with different residence times and contractive structures (Sen, 2012, Sen, 2012).
• Pseudo-contractive and intermediate-sense inequalities further generalize contractivity, incorporating iteration-dependent coefficients, mixed terms, and perturbative error sequences, but the COC framework offers geometric convergence guarantees under minimal assumptions (Sen, 2012, Sen, 2012).

The existence, uniqueness, and algorithmic convergence of best proximity points for COC mappings have ramifications for:

• Alternating projection algorithms,
• Equilibrium problem solvers involving nonintersecting constraint sets,
• Numerical PDE schemes with boundary alternation,
• Optimization in spaces lacking intersection of feasible sets.

A plausible implication is that the COC paradigm provides an optimal orbit-centric toolset for convergence analysis in both geometric (Riemannian or $CAT_p(0)$) and analytic (Banach space) contexts, especially where classical synchronous or Banach-type contractions fail or are overly restrictive (Kumar et al., 13 Dec 2025, Pacurar, 2022).