The main goal of this research is to achieve fairer assessment of students’ knowledge and skills in tests in view of the fact that learners with similar abilities are often assigned different test grades if their achievements are borderline cases between two grades. As an example, a student who has scored 29 points out of 60 in a test will fail, while another one with 30 or 31 points will pass, which can be viewed as unjust. This problem is addressed by using fuzzy logic and different fuzzy functions to determine whether students with borderline test scores are assigned the higher or the lower grade of the two. Furthermore, although changing the grades of individual students, the authors endeavor to preserve the overall score of all test takers statistically unchanged.

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 present a possible kind of generalization of the notion of ordered pairs of cyclic maps and coupled fixed points and its application in modelling of equilibrium in oligopoly markets. We have obtained sufficient conditions for the existence and uniqueness of coupled fixed in complete metric spaces. We illustrate one possible application of the results by building 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 unified formal method.We show that whenever the response functions of the two players are identical, then the equilibrium will be attained at equal levels of production and equal prices. 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 equilibrium and convexity condition of the payoff function can be skipped.

We generalize the notion of best proximity points in the context of modular function spaces. We have found sufficient conditions for the existence and uniqueness of best proximity points for cyclic maps in modular function spaces. We present an application of the main result for cyclic integral operators in Orlicz function spaces, endowed with an Orlicz function modular.