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Generalized $θ$-Parametric Metric Spaces: Fixed Point Theorems and Applications to Fractional Economic Models

Abhishikta Das, Hemanta Kalita, Mohammad Sajid, T. Bag

TL;DR

The paper develops generalized $\theta$-parametric metric spaces by combining a $\mathcal{B}$-action with a Suzuki-type contraction framework, unifying $\theta$-metric and parametric distance notions. It establishes convergence and topological structure, and proves a Suzuki-type fixed point theorem with Suzuki-Banach and Suzuki-Kannan corollaries, supported by illustrative examples. The framework is then applied to Caputo fractional differential equations modeling economic growth, yielding existence and uniqueness results in a function space setting. This work links abstract generalized metric theory to memory-influenced economic dynamics and fractional calculus applications.

Abstract

The objective of this manuscript is to introduce and develop the concept of a generalized $θ$-parametric metric space-a novel extension that enriches the modern metric fixed point theory. We study of its fundamental properties, including convergence and Cauchy sequences that establishes a solid theoretical foundation. A significant highlight of our work is the formulation of Suzuki-type fixed point theorem within this framework which extends classical results in a meaningful way. To demonstrate the depth and applicability of our findings, we construct non-trivial examples that illustrate the behavior of key concepts. Moreover, as a practical application, we apply our main theorem to analyze an economic growth model, demonstrating its utility in solving fractional differential equations that arise in dynamic economic systems.

Generalized $θ$-Parametric Metric Spaces: Fixed Point Theorems and Applications to Fractional Economic Models

TL;DR

The paper develops generalized -parametric metric spaces by combining a -action with a Suzuki-type contraction framework, unifying -metric and parametric distance notions. It establishes convergence and topological structure, and proves a Suzuki-type fixed point theorem with Suzuki-Banach and Suzuki-Kannan corollaries, supported by illustrative examples. The framework is then applied to Caputo fractional differential equations modeling economic growth, yielding existence and uniqueness results in a function space setting. This work links abstract generalized metric theory to memory-influenced economic dynamics and fractional calculus applications.

Abstract

The objective of this manuscript is to introduce and develop the concept of a generalized -parametric metric space-a novel extension that enriches the modern metric fixed point theory. We study of its fundamental properties, including convergence and Cauchy sequences that establishes a solid theoretical foundation. A significant highlight of our work is the formulation of Suzuki-type fixed point theorem within this framework which extends classical results in a meaningful way. To demonstrate the depth and applicability of our findings, we construct non-trivial examples that illustrate the behavior of key concepts. Moreover, as a practical application, we apply our main theorem to analyze an economic growth model, demonstrating its utility in solving fractional differential equations that arise in dynamic economic systems.
Paper Structure (9 sections, 17 theorems, 79 equations)

This paper contains 9 sections, 17 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Introduction to generalized $\theta$-parametric metric space
  4. Basic properties of generalized $\theta$-parametric metric space
  5. Topological properties of generalized $\theta$-parametric metric space:
  6. Convergence of sequence in generalized $\theta$-parametric metric space:
  7. Suzuki's type fixed point theorem and its application
  8. Suzuki's type fixed point theorem
  9. Application to Economic Growth Modeling via Caputo Fractional Differential Approach

Key Result

Proposition 2.16

8 If $\mathcal{P}$ is a generalized parametric metric space on $\mathcal{X}$. Then $\mathcal{P} ( a, \xi, .)$ is non increasing function, for all $a, \xi \in \mathcal{X}$.

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 61 more