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This paper aims to introduce the new concept of rational type fuzzy-contraction mappings in fuzzy metric spaces. We prove some fixed point results under the rational type fuzzy-contraction conditions in fuzzy metric spaces with illustrative examples to support our results. This new concept will play a very important role in the theory of fuzzy fixed point results and can be generalized for different contractive type mappings in the context of fuzzy metric spaces. Moreover, we present an application of a nonlinear integral type equation to get the existing result for a unique solution to support our work.

The role of multipolar uncertain statistics cannot be unheeded while confronting daily life problems on well-founded basis. Fusion (aggregation) of a number of input values in multipolar form into a sole multipolar output value is an essential tool not merely of physics or mathematics but also of widely held problems of economics, commerce and trade, engineering, social sciences, decision-making problems, life sciences, and many more. The problem of aggregation is very wide-ranging and fascinating, in general. We use, in this article, Pythagorean fuzzy numbers (PFNs) in multipolar form to contrive imprecise information. We introduce Pythagorean m-polar fuzzy weighted averaging (PmFWA), Pythagorean m-polar fuzzy weighted geometric (PmFWG), symmetric Pythagorean m-polar fuzzy weighted averaging (SPmFWA), and symmetric Pythagorean m-polar fuzzy weighted geometric (SPmFWG) operators for aggregating uncertain data. Finally, we present a practical example to illustrate the application of the proposed operators and to demonstrate its practicality and effectiveness towards investment strategic decision making.

The aim of this paper is to introduce the notion of nearly prime submodules as a generalization of prime submodules. We investigate some of their basic properties and point out the similarities between these submodules and the prime submodules. We also indicate some applications of nearly prime submodules. These applications show how nearly prime submodules control the structure of modules and they recover earlier relative theorems.