# Coupled Fixed Points for Hardy–Rogers Type of Maps and Their Applications in the Investigations of Market Equilibrium in Duopoly Markets for Non-Differentiable, Nonlinear Response Functions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**Definition**

**1**

**Theorem**

**1**

- 1.
- there is a unique fixed point $\xi \in X$ of T and, moreover, for any initial guess ${x}_{0}\in X$, the iterated sequence ${x}_{n}=T{x}_{n-1}$ for $n=1,2,\cdots $ converges to the fixed point ξ;
- 2.
- there holds a priori error estimates $\rho (\xi ,{x}_{n})\le \frac{{k}^{n}}{1-k}\rho ({x}_{0},{x}_{1})$;
- 3.
- there holds a posteriori error estimate $\rho (\xi ,{x}_{n})\le \frac{k}{1-k}\rho ({x}_{n-1},{x}_{n})$;
- 4.
- the rate of convergence is $\rho (\xi ,{x}_{n})\le k\rho (\xi ,{x}_{n-1})$;

**Definition**

**2**

## 3. Main Result

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**2.**

- 1.
- there is a unique coupled fixed point $(\xi ,\eta )\in {X}_{1}\times {X}_{2}$ of $({F}_{1},{F}_{2})$ and, moreover, for any initial guess $({x}_{0},{y}_{0})\in x$ the iterated sequences ${x}_{n}={F}_{1}({x}_{n-1},{y}_{n-1})$ and ${y}_{n}={F}_{2}({x}_{n-1},{y}_{n-1})$ for $n=1,2,\cdots $ converge to the coupled fixed point $(\xi ,\eta )$;
- 2.
- there holds a priori error estimate$${d}_{1}(\xi ,{x}_{n})+{d}_{2}(\eta ,{y}_{n})\le \frac{{k}^{n}}{1-k}({d}_{1}({x}_{0},{x}_{1})+{d}_{2}({y}_{0},{y}_{1}))$$;
- 3.
- there holds a posteriori error estimate$${d}_{1}(\xi ,{x}_{n})+{d}_{2}(\eta ,{y}_{n})\le \frac{k}{1-k}({d}_{1}({x}_{n},{x}_{n-1})+{d}_{2}({y}_{n},{y}_{n-1}))$$;
- 4.
- the rate of convergence is ${d}_{1}(\xi ,{x}_{n})+{d}_{2}(\eta ,{y}_{n})\le k({d}_{1}(\xi ,{x}_{n-1})+{d}_{2}(\eta ,{y}_{n-1}))$

**Remark**

**1.**

**Proof.**

## 4. Applications of of the Main Result

#### 4.1. Generalization of Some Known Results about Coupled Fixed Points and Corollaries

**Theorem**

**3.**

**Remark**

**2.**

#### 4.2. Application in the Investigation of Market Equilibrium in Duopoly Markets

#### 4.2.1. The Basic Model

#### 4.2.2. Connection between the Second-Order Conditions and the Contraction-Type Conditions

**Assumption**

**1.**

- 1.
- The two player are producing homogeneous goods that are perfect substitutes.
- 2.
- The first player can produce quantities from the set ${X}_{1}$, and the second one can produce quantities from the set ${X}_{2}$, where ${X}_{1}$ and ${X}_{2}$ are closed, nonempty subsets of a complete metric space $(X,d)$.
- 3.
- Let there be a closed subset $D\subseteq {X}_{1}\times {X}_{2}$ and maps ${F}_{i}:D\to {X}_{i}$, $i=1,2$, so that:$$({F}_{1}(x,y),{F}_{2}(x,y))\subseteq D$$for every $(x,y)\in D$, are the response functions for Players One and Two, respectively.
- 4.
- Let $\alpha <1$, so that the inequality:$$d({F}_{1}(x,y),{F}_{1}(u,v))+d({F}_{2}(x,y),{F}_{2}(u,v))\le \alpha (d(x,u)+d(y,v))$$holds for all $(x,y),(u,v)\in {X}_{1}\times {X}_{2}$.

**Example**

**1.**

**Example**

**2.**

#### 4.2.3. Comments on the Coefficients $\alpha $, $\beta $, $\gamma $, and $\delta $

**Example**

**3.**

#### 4.2.4. Some Applications on Newly Investigated Oligopoly Models

#### 4.2.5. A Generalized Response Function

**Assumption**

**2.**

- 1.
- The two players are producing homogeneous goods that are perfect substitutes.
- 2.
- The player i, $i=1,2$ can produce quantities from the set ${U}_{i}$, its set of the realized, on-the-market production as ${P}_{i}$, and the set of its surplus production is ${s}_{i}$, where $X={P}_{1}\times {s}_{1}$ and $Y={P}_{2}\times {s}_{2}$ are closed, nonempty subsets of a complete metric space $(Z,\rho )$.
- 3.
- Let there be a closed subset $D\subseteq X\times Y$ and maps ${F}_{1}:D\to X$ and ${F}_{2}:D\to Y$, such that $({F}_{1}(x,y),{F}_{2}(x,y))\subseteq D$ for every $(x,y)\in D$ is the generalized response function of the player and the market for Players One and Two, respectively.
- 4.
- Let $\alpha \in (0,1)$, so that the inequality:$$\rho ({F}_{1}(x,y),{F}_{1}(u,v))+\rho ({F}_{2}(x,y),{F}_{2}(u,v))\le \alpha (\rho (x,u)+\rho (y,v))$$holds for all $(x,y),(u,v)\in X\times Y$.

#### 4.2.6. Applications of Theorem 2 for Optimization of Non-Differentiable Payoff Functions and Examples

**Assumption**

**3.**

- 1.
- The two players are producing homogeneous goods that are perfect substitutes.
- 2.
- The first player can produce quantities from the set ${X}_{1}$, and the second one can produce quantities from the set ${X}_{2}$, where ${X}_{1}$ and ${X}_{2}$ are closed, nonempty subsets of a complete metric space $(X,d)$.
- 3.
- Let there be a closed subset $D\subseteq {X}_{1}\times {X}_{2}$ and maps ${F}_{i}:D\to {X}_{i}$, so that:$$({F}_{1}(x,y),{F}_{2}(x,y))\subseteq D$$for every $(x,y)\in D$ are the response functions for Players One and Two, respectively.
- 4.
- Let $\beta \in [0,1/2)$, so that the inequality:$$\begin{array}{ccc}{S}_{5}\hfill & =\hfill & \sum _{i=1}^{2}d({F}_{i}(x,y),{F}_{i}(u,v))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \le \hfill & \beta (d(x,{F}_{1}(x,y))+d(y,{F}_{2}(x,y))+d(u,{F}_{1}(u,v))+d(v,{F}_{2}(u,v)))\hfill \end{array}$$holds for all $(x,y),(u,v)\in D$.

**Example**

**4.**

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Graphic of the function $|F\left(x\right)-F\left(y\right)|-\frac{1}{7}\left(\right|x-F\left(x\right)|+|y-F\left(y\right)\left|\right)$.

**Figure 2.**Graphic of the function $|{F}_{2}\left(y\right)-{F}_{2}\left(v\right)|-\frac{1}{8}\left(\right|y-{F}_{2}\left(y\right)|+|v-{F}_{2}\left(v\right)\left|\right)$.

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

${x}_{n}$ | 20 | 29 | 24 | 17 | 60 | 0 | 100 |

${y}_{n}$ | 31 | 18 | 35 | 6 | 71 | 0 | 100 |

n | 0 | 1 | 2 | ⋯ | 2k | 2k + 1 |
---|---|---|---|---|---|---|

${x}_{n}$ | 20 | 30 | 20 | ⋯ | 20 | 30 |

${y}_{n}$ | 30 | 20 | 30 | ⋯ | 30 | 20 |

n | 0 | 1 | 2 | 3 | 4 | 5 | 10 | 21 | 50 | 51 | 120 | 121 | 599 | 600 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${x}_{n}$ | 10 | 37 | 12 | 35 | 13 | 33.7 | 16.8 | 30.8 | 21.1 | 26.9 | 22.64 | 25.43 | 24.07 | 24.05 |

${y}_{n}$ | 30 | 18 | 33 | 20 | 31 | 21.4 | 28.6 | 24.1 | 25.8 | 26.4 | 26.03 | 26.34 | 26.19 | 26.18 |

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Kabaivanov, S.; Zhelinski, V.; Zlatanov, B.
Coupled Fixed Points for Hardy–Rogers Type of Maps and Their Applications in the Investigations of Market Equilibrium in Duopoly Markets for Non-Differentiable, Nonlinear Response Functions. *Symmetry* **2022**, *14*, 605.
https://doi.org/10.3390/sym14030605

**AMA Style**

Kabaivanov S, Zhelinski V, Zlatanov B.
Coupled Fixed Points for Hardy–Rogers Type of Maps and Their Applications in the Investigations of Market Equilibrium in Duopoly Markets for Non-Differentiable, Nonlinear Response Functions. *Symmetry*. 2022; 14(3):605.
https://doi.org/10.3390/sym14030605

**Chicago/Turabian Style**

Kabaivanov, Stanimir, Vasil Zhelinski, and Boyan Zlatanov.
2022. "Coupled Fixed Points for Hardy–Rogers Type of Maps and Their Applications in the Investigations of Market Equilibrium in Duopoly Markets for Non-Differentiable, Nonlinear Response Functions" *Symmetry* 14, no. 3: 605.
https://doi.org/10.3390/sym14030605