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We enrich the known results about coupled fixed points and coupled best proximity points. We generalize the notion of ordered pairs of cyclic contraction maps and we obtain sufficient conditions for the existence and uniqueness of best proximity points. We get a priori and a posteriori error estimates for the coupled fixed points and for the coupled best proximity points, provided that the underlying Banach space has modulus of convexity of power type in the case of best proximity points, obtained by sequences of successive iterations. We illustrate the main result with an example. [ABSTRACT FROM AUTHOR]

This article presents a hybrid approach to assessing students' foreign language proficiency in a cyber–physical educational environment. It focuses on the advantages of the integrated assessment of student knowledge by considering the impact of automatic assessment, learners' independent work, and their achievements to date. An assessment approach is described using the mathematical theory of fuzzy functions, which are employed to ensure the fair evaluation of students. The largest possible number of students whose reevaluation of test results will not affect the overall performance of the student group is automatically determined. The study also models the assessment process in the cyber–physical educational environment through the formal semantics of calculus of context-aware ambients (CCAs). [ABSTRACT FROM AUTHOR]

We have found a sufficient condition in order that the Kottman constant to be equal to the Riesz angle for Köthe sequence spaces. We have found the ball packing constant in weighted Orlicz sequence spaces, endowed with Luxemburg or p-Amemiya norm. We have calculated the Riesz angle for Musielak-Orlicz, Nakano, weighted Orlicz, Orlicz, Orlicz-Lorentz, Lorentz and Cesaro sequence spaces. [ABSTRACT FROM AUTHOR]

We find a priori and a posteriori error estimates of the best proximity point for the Picard iteration associated to a cyclic contraction map, which is defined on a uniformly convex Banach space with modulus of convexity of power type. [ABSTRACT FROM AUTHOR]

We generalize Ekeland's Variational Principle for cyclic maps. We present applications of this version of the variational principle for proving of existence and uniqueness of best proximity points for different classes of cyclic maps. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the present work we consider the coefficient (correction coefficient), which compliments Coulomb's Law in the case of electrostatic interaction between two charged conducting spheres with equal radii and charges. It is proved that the correction coefficient is smaller than one, when the ratio of the radii to the distance between their centers is smaller than 2/5. We obtained a formula for calculating the correction coefficient with an arbitrary precision with the help of partial sums. [ABSTRACT FROM AUTHOR]

We present a new approach to the teaching of Synthetic Geometry in schools and universities with the help of DGS. The introduction of new elements into DGS helps to optimize the teaching process. These new elements are infinite points in the extended Euclidean plane and the "Swap finite & infinite points" function. We give examples of the usage of these new elements in Projectivity, Homology, Conic Sections, Plane Sections of Polyhedra, and in the application of Pappus' Theorem and Desargues' Theorem. These new features increase the benefits of DGS in teaching and learning Geometry. We optimize the education process by saving time involved in drawing, generalizing large groups of problems, stimulating and helping investigations, and forming a creative style of thinking. [ABSTRACT FROM AUTHOR]

We present a condition that guarantees the existence and uniqueness of fixed (or best proximity) points in complete metric space (or uniformly convex Banach spaces) for a wide class of cyclic maps, called p–cyclic summing maps. These results generalize some known results from fixed point theory. We find a priori and a posteriori error estimates of the fixed (or best proximity) point for the Picard iteration associated with the investigated class of maps, provided that the modulus of convexity of the underlying space is of power type. We illustrate the results with some applications and examples. [ABSTRACT FROM AUTHOR]

A cyclic map with a contractive type of condition called p-cyclic orbital M-Kcontraction is introduced in a partial metric space. Sufficient conditions for the existence and uniqueness of fixed points and the best proximity points for these maps in complete partial metric spaces are obtained. Furthermore, a necessary and sufficient condition for the completeness of partial metric spaces is given. The results are illustrated with an example. [ABSTRACT FROM AUTHOR]

We have obtained a new class of ordered pairs of multivalued maps that have pairs of coupled fixed points. We illustrate the main result with two examples that cover a wide range of models. We apply the main result in models in duopoly markets to get a market equilibrium and in aquatic ecosystems, also to get an equilibrium. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]