We investigated the existence and uniqueness of coupled best proximity points for some cyclic and semi-cyclic maps in a reflexive Banach space. We found sufficient conditions, ensuring the existence of coupled best proximity points in reflexive Banach spaces and some convexity types of conditions, ensuring uniqueness of the coupled best proximity points in strictly convex Banach spaces. We illustrate the results with examples and we present an application of one of the theorems in the modeling of duopoly markets, to have an existence of market equilibrium. We show that, in general, the iterative sequences can have chaotic behavior.

In the study we explore an oligopoly market for equilibrium and stability based on statistical data with the help of response functions rather than payoff maximization. To achieve this, we extend the concept of coupled fixed points to triple fixed points. We propose a new model that leads to generalized triple fixed points. We present a possible application of the generalized tripled fixed point model to the study of market equilibrium in an oligopolistic market dominated by three major competitors. The task of maximizing the payout functions of the three players is modified by the concept of generalized tripled fixed points of response functions. The presented model for generalized tripled fixed points of response functions is equivalent to Cournot payoff maximization, provided that the market price function and the three players’ cost functions are differentiable. Furthermore, we demonstrate that the contractive condition corresponds to the second-order constraints in payoff maximization. Moreover, the model under consideration is stable in the sense that it ensures the stability of the consecutive production process, as opposed to the payoff maximization model with which the market equilibrium may not be stable. A possible gap in the applications of the classical technique for maximization of the payoff functions is that the price function in the market may not be known, and any approximation of it may lead to the solution of a task different from the one generated by the market. We use empirical data from Bulgaria’s beer market to illustrate the created model. The statistical data gives fair information on how the players react without knowing the price function, their cost function, or their aims towards a specific market. We present two models based on the real data and their approximations, respectively. The two models, although different, show similar behavior in terms of time and the stability of the market equilibrium. Thus, the notion of response functions and tripled fixed points seems to present a justified way of modeling market processes in oligopoly markets when searching whether the market has reached equilibrium and if this equilibrium is unique and stable in time

We investigate the dynamics of tripodal markets using the response functions, which is a continuation of recent research in the field. Instead of investigating the optimization problem of finding the levels of production that maximize the payoff functions of the participants in an oligopolistic market, based on the available statistical data on market presence, we construct a model of the reaction of the participants. This approach allows, in the absence of information about the cost functions of producers and the demand and utility functions of consumers, to construct a model that is statistically reliable and answers the questions about the levels at which the market has reached equilibrium and whether it is sustainable. On the other hand, any external impact, such as changes in the regulations or the behavior of small market participants, is implicitly included in the response functions. The additional analysis confirms that there are no dependencies, even of a nonlinear type, in the constructed models that are not included. Stability and equilibrium are investigated in the proposed models. The statistical performance measurements for the constructed models are calculated, and their credibility is tested. The models demonstrate high statistical performance and adequacy.

We establish adequate conditions for the existence and uniqueness of solutions to systems of two matrix equations when the unknown matrices are raised to a power $k\in [-1,1]\setminus \left\{0\right\}$. The findings from coupled fixed points for ordered pairs of maps are used. Numerical examples are provided to illustrate the results shown. Some of the known results are the consequence of our acquisitions.

We generalize the notion of coupled fixed (or best proximity) points for cyclic ordered pairs of maps to p-cyclic ordered pairs of maps. We find sufficient conditions for the existence and uniqueness of the coupled fixed (or best proximity) points. We illustrate the results with an example that covers a wide class of maps.

We unify a known technique used for fixed points and coupled, tripled and N-tupled fixed points for weak monotone maps, i.e., maps that exhibit monotone properties for each of their variables. We weaken the classical contractive condition in partially ordered metric spaces by requiring it to hold only for a sequence of successive iterations, generated by the considered map, provided that it is a monotone one. We show that some known results are a direct consequence of the main result. The introduced technique shows that the partial order in the constructed Cartesian space is induced by both the partial order in the considered metric space and by the monotone properties of the investigated maps. We illustrate the main result, which is applied to solve a nonlinear matrix equation, following key ideas from Berzig, Duan & Samet. We present an illustrative example. We comment that a similar approach can be used to solve systems of nonlinear matrix equations.