Contraction map sets with an external factor and weakly fixed points

In this paper I introduce the property CBD which is a more convenient variant of the UC property and show one of the possible relationships between them, I also extend the concept of a fixed point, introducing the concept of a weak fixation of a point about a sequence. I introduce contraction map sets with an external factor and formulate a theorem for them, on which the main focus of this article falls

On the UC and UC* properties and the existence of best proximity points in metric spaces

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 is included into the UC property. We introduce some new notions: bounded UC (BUC) property and uniformly convex set about a function. We prove that these new notions are generalizations of the $UC$ property and that both of them are sufficient for 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 it 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) has not the UC property, but has the $BUC$ property and thus there is a unique best proximity point of T in A.Comment: 22 page, 2 figure

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

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