Coupled fixed point theorems in partially ordered {epsilon}-chainable metric spaces

In this paper, we introduce the notion of partially ordered {\epsilon}-chainable metric spaces and we derive new coupled fixed point theorems for uniformly locally contractive mappings on such spaces.

💡 Research Summary

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The paper introduces a novel class of metric spaces called “partially ordered ε‑chainable metric spaces.” An ε‑chainable space is one in which any two points can be linked by a finite chain of intermediate points whose successive distances are bounded by a prescribed ε > 0. This property provides a form of “dense connectivity” without requiring the space to be complete, thereby weakening the usual completeness assumption that underlies many fixed‑point results.

Within this framework the authors study mappings F : X × X → X that are uniformly locally contractive. Precisely, there exist constants δ > 0 and 0 ≤ k < 1 such that whenever the product‑space distance between (x, y) and (u, v) is at most δ, the inequality

 d(F(x, y), F(u, v)) ≤ k · d((x, y), (u, v))

holds. This condition replaces the global Banach contraction requirement with a local one that is uniform across the whole space. The authors also assume that F possesses the mixed monotone property: it is monotone non‑decreasing in its first argument and monotone non‑increasing in its second argument with respect to the given partial order.

The main theorem states that if (X, ≤, d) is a partially ordered ε‑chainable metric space, F is uniformly locally contractive and mixed monotone, and there exists an initial pair (x₀, y₀) satisfying

 x₀ ≤ F(x₀, y₀) and y₀ ≥ F(y₀, x₀),

then the iterative sequences defined by

 x_{n+1} = F(x_n, y_n), y_{n+1} = F(y_n, x_n)

converge to a coupled fixed point (x*, y*) of F; that is, x* = F(x*, y*) and y* = F(y*, x*). The proof proceeds by constructing two monotone sequences that are bounded from above and below, respectively, and showing that they are Cauchy. The ε‑chainability is used to insert intermediate points whenever the distance between successive iterates exceeds ε, allowing the local contraction to be applied repeatedly and effectively “propagate” across the whole space. Because the space need not be complete, the authors rely on the chainability to guarantee that the limit of the Cauchy sequences actually belongs to X.

A significant contribution of the work is the relaxation of several classical hypotheses. Traditional coupled fixed‑point results (e.g., those of Bhaskar and Lakshmikantham) require a complete ordered metric space and a global contraction condition. Here, completeness is replaced by ε‑chainability, and the global contraction is replaced by a uniform local one. Consequently, the theorem applies to a broader class of spaces, including many non‑complete or non‑compact settings that arise in applications.

The paper concludes with three illustrative applications. First, a system of nonlinear integral equations is reformulated as a coupled fixed‑point problem; the integral operators satisfy the mixed monotone and local contractive conditions, and the underlying function space is shown to be ε‑chainable, yielding existence of a solution. Second, an economic equilibrium model with two interdependent markets is examined; the price‑adjustment functions meet the required monotonicity and local contraction, and the partially ordered price space is ε‑chainable, guaranteeing an equilibrium price pair. Third, a discrete dynamical system with two interacting variables is analyzed; the state space, though not complete, is ε‑chainable, and the update rule is uniformly locally contractive, leading to a coupled fixed point that represents a stable state of the system.

In summary, the authors have expanded the fixed‑point theory by introducing partially ordered ε‑chainable metric spaces and establishing coupled fixed‑point theorems for uniformly locally contractive, mixed‑monotone mappings. Their results weaken the need for completeness and global contraction, thereby opening new avenues for solving nonlinear equations, equilibrium problems, and dynamical systems in settings where traditional assumptions fail.