In this paper we generalize Hardy–Rogers maps in the context of coupled fixed points. We comment on the symmetry of some of the coefficients involved in the Hardy–Rogers condition, and thus, we deduce a simpler formula. We generalize, with the help of the obtained main theorem, some known results about existence and uniqueness of market equilibrium in duopoly markets. As a consequence, we ascertain that the equilibrium production should be equal for both market participants provided that they have symmetric response functions. With the help of the main theorem, we investigate and enrich some recent results regarding market equilibrium in duopoly markets. We define a generalized response function that includes production and surpluses. Finally, we illustrate a possible application of the main result in the investigation of market equilibrium when the payoff functions are non-differentiable.

We investigate the connections between UC and UC$^*$ properties for ordered pairs of subsets $(A,B)$ in metric spaces, which are involved in the study of existence and uniqueness of best proximity points. We show that the UC property and the UC$^*$ property lead to one and the same corollaries, when iterated sequences, generated by cyclic maps, are investigated. We introduce some new notions: bounded UC (BUC) property and uniformly convex set about a function $\phi$. We prove that these new notions are generalizations of the UC property and that both of them are sufficient to ensure existence and uniqueness of best proximity points. We show that these two new notions are different from a uniform convexity and even from a strict convexity. If we consider the underlying space to be a Banach space, we find a sufficient condition which ensures that from the UC property follows the uniform convexity of the underlying Banach space. We illustrate the new notions with examples. We present an example of a cyclic contraction $T$ in a space, which is not even strictly convex and the ordered pair $(A,B)$ does not have the UC property, but has the BUC property and thus there is a unique best proximity point of $T$ in $A$.

We develop an empirical model of the market for mobile services in the USA based on providers’ response functions. Guided by a duopoly model, we obtain our empirical response functions from an approximation of quarterly response data on smartphone subscriptions by sigmoid functions of time. The robustness analysis suggests that our model fits the data well and outperforms the regression model. Further, we demonstrate that our empirical response functions satisfy the conditions for semi-cyclic contractions which guarantee the existence, uniqueness and stability of long-run equilibrium.

We investigate fixed points for p cyclic maps by introducing a new notion of p–cyclic infimum summing maps and a generalized best proximity point for p–cyclic maps. The idea generalizes some results about best proximity points in order to widen the class of sets and maps for which we can ensure the existence and uniqueness of best proximity points. The replacement of the classical notions of best proximity points and distance between the consecutive set arises from the well-known group traveling salesman problem and presents a different approach to solving it. We illustrate the new result with a map that does not satisfy the known results about best proximity maps for p–cyclic maps.