A variational principle, coupled fixed points and market equilibrium

Abstract

We present a possible kind of generalization of the notion of ordered pairs of cyclic maps and coupled fixed points and its application in modelling of equilibrium in oligopoly markets. We have obtained sufficient conditions for the existence and uniqueness of coupled fixed in complete metric spaces. We illustrate one possible application of the results by building a pragmatic model on competition in oligopoly markets. To achieve this goal, we use an approach based on studying the response functions of each market participant, thus making it possible to address both Cournot and Bertrand industrial structures with unified formal method.We show that whenever the response functions of the two players are identical, then the equilibrium will be attained at equal levels of production and equal prices. The response functions approach makes it also possible to take into consideration different barriers to entry. By fitting to the response functions rather than the profit maximization of the payoff functions problem we alter the classical optimization problem to a problem of coupled fixed points, which has the benefit that considering corner optimum, corner equilibrium and convexity condition of the payoff function can be skipped.

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

“…A similar condition to ( 17) is investigated in [17], where maps with the mixed monotone property are considered. In this case (9) holds only for part of the variables and therefore we can not take limits in (11).…”

“…In this case (9) holds only for part of the variables and therefore we can not take limits in (11). Thus the response functions from [17] may be not differentiable.…”

“…The idea of coupled fixed points was generalized for coupled best proximity points [29]. There are applications of coupled fixed points in different fields of mathematics: impulsive differential equations [3], integral equations [21], ordinary differential equations [20], periodic boundary value problem [9], fractional equations [23], nonlinear matrix equations [27] in other sciences: economics [8,16], aquatic ecosystems [11], dynamic programming [5], which are just the most recent investigation that are dealing with coupled fixed points.…”

Existence of Coupled Best Proximity Points of p-Cyclic Contractions

“…It turns out that the last 10 years there is a great interest on coupled fixed points, both in fundamental results and their applications [2][3][4][5]. We would like to mention a new kind of applications in the theory of equilibrium in duopoly markets [6,7].…”

“…The notion of best proximity points [9] actually generalizes the notion of cyclic maps from [8], as far as if A ∩ B = ∅, then any best proximity point is a fixed point, too. It turns out that best proximity points are interesting not only as a pure mathematical results, but also as a possibility for a new approach in solving of different types of problems [2][3][4][5][6][7].…”