In this paper we build 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 a unified formal method. In contrast to the restrictive theoretical constructs of duopoly equilibrium, our study is able to account for real-world limitations like minimal sustainable production levels and exclusive access to certain resources. We prove and demonstrate that by using carefully constructed response functions it is possible to build and calibrate a model that reflects different competitive strategies used in extremely concentrated markets. 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 equilibria and convexity condition of the payoff function can be skipped.

We have found a new type of a contractive condition that ensures the existence and uniqueness of fixed points and best proximity points in uniformly convex Banach spaces. We provide some examples to validate our results. These results generalize some known results from fixed point theory.

We generalize the cyclic orbital Meir–Keeler contractions, which were introduced by S. Karpagam and S. Agrawal in the context of p-summing maps. We found sufficient conditions for these new type of maps, that ensure the existence and uniqueness of fixed points in complete metric spaces, when the distances between the sets are zero, and the existence and uniqueness of best proximity points in uniformly convex Banach spaces.

Let (X, d) be a metric space, and A1, A2,. .. , Ap be nonempty subsets of X. We introduce a self map T on X, called p-cyclic orbital contraction map on the union of A1, A2,. .. , Ap, and obtain a unique best proximity point of T , that is, a point x ∈ ∪ p i=1 Ai such that d(x, T x) = dist(Ai, Ai+1), 1 i p, where dist(Ai, Ai+1) = inf{d(x, y): x ∈ Ai, y ∈ Ai+1}.

In this manuscript, we introduce the concept of Ω -class of self mappings on a metric space and a notion of p-cyclic complete metric space for a natural number ( p ≥ 2 ) . We not only give sufficient conditions for the existence of best proximity points for the Ω -class self-mappings that are defined on p-cyclic complete metric space, but also discuss the convergence of best proximity points for those mappings.