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Relation theoretic contractions and their applications in b-metric like spaces

Tomar Anita, Joshi Meena, Padaliya S. K.

Journal of Applied Analysis

2022

Vol. 28, nr 2

295--309

We introduce a relational generalized Meir-Keeler contraction and a relational generalized Meir-Keeler contraction with rational terms in non-complete relational b-metric like spaces in order to establish non-unique fixed point results for a discontinuous single-valued map. Also, we provide an illustrative example to demonstrate that a relational generalized Meir-Keeler contraction with rational terms in a relational b-metric like space admits discontinuity at the fixed point. Thereby,we provide a novel explanation via a binary relation to the question of the existence of a contractive map admitting a fixed point at the point of discontinuity. Finally, we give applications to solve an initial value problem and a non-linear matrix equation which demonstrate the usability and effectiveness of our results.

Approximation Operators, Binary Relation and Basis Algebra in L-fuzzy Rough Sets

Wu Z., Li T., Qin K., Ruan D.

Fundamenta Informaticae

2011

Vol. 111, nr 1

47-63

Approximation operators play a vital role in rough set theory. Their three elements, namely, binary relation in the universe, basis algebra and properties, are fundamental in the study of approximation operators. In this paper, the interrelations among the three elements of approximation operators in L-fuzzy rough sets are discussed under the constructive approach, the axiomatic approach and the basis algebra choosing approach respectively. In the constructive approach, the properties of the approximation operators depend on the basis algebra and the binary relation. In the axiomatic approach, the induced binary relation is influenced by the axiom set and the basis algebra. In the basis algebra choosing approach, the basis algebra is constructed by properties of approximation operators and specific binary relations.