This paper deals with the concepts of upper and lower (τ 1 , τ 2 )-precontinuous multifunctions. Some characterizations of upper and lower (τ 1 , τ 2 )-precontinuous multifunctions are investigated. The relationships between upper and lower (τ 1 , τ 2 )-precontinuous multifunctions and the other types of continuity are discussed.Keywords: τ 1 τ 2 -preopen, upper (τ 1 , τ 2 )-precontinuous multifunction, lower (τ 1 , τ 2 )-precontinuous multifunction.

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 main goal of this article is to introduce the concepts of upper and lower (τ1, τ2)α-continuous multifunctions. Characterizations of upper and lower (τ1, τ2)α-continuous multifunctions are investigated. The relationships between upper and lower (τ1, τ2)α-continuous multifunctions and the other types of continuity are discussed.

This article deals with the concepts of Λp-sets and (Λ, p)-closed sets which are defined by utilizing the notions of preopen sets and preclosed sets. We also introduce and characterize some new low separation axioms. Characterizations of Λp-R0 spaces are given. Moreover, we introduce the concept of weakly (Λ, p)-continuous functions. In particular, several characterizations of weakly (Λ, p)-continuous functions are established.

A new class of multifunctions, called upper (lower) α(µ X , µ Y )-continuous multifunctions, has been defined and studied. Some characterizations and several properties concerning upper (lower) α(µ X , µ Y )-continuous multifunctions are obtained.