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This article presents the notions of extended b-gauge space (U,Qφ;Ω)\left(U,{Q}_{\varphi ;\Omega }) and extended Jφ;Ω{{\mathcal{J}}}_{\varphi ;\Omega }-families of generalized extended pseudo-b-distances on UU. Furthermore, we look at these extended Jφ;Ω{{\mathcal{J}}}_{\varphi ;\Omega }-families on UU and define the extended Jφ;Ω{{\mathcal{J}}}_{\varphi ;\Omega }-sequential completeness. We also look into some fixed and periodic point theorems for set-valued mappings in the new space with a graph that does not meet the completeness condition of the space. Additionally, this article includes some examples to explain the corresponding results and highlights some important consequences of our obtained results.

In this paper, our aim is to make a novel interpretation of relation between (N̅,p,q) method and P-convergence for double sequences of fuzzy numbers. In accordance with this aim, we derive some Tauberian conditions, controlling O-oscillatory behavior of a double sequence of fuzzy numbers in certain senses, from (N̅,p,q) summability to P- convergence with some restrictions on the weight sequences. As special cases, we indicate that O-condition of Hardy type with respect to (Pm) and Qn are Tauberian conditions for (N̅,p,q) summability under some additional conditions. In the sequel, we prove a fuzzy Korovkin-type approximation theorem by using the (N̅,p,q) summability method for fuzzy positive linear operators. [ABSTRACT FROM AUTHOR]

In this paper, we have extended the concepts of I-statistical limit superior and I-statistical limit inferior to I-statistical φ-limit superior and I-statistical φ-limit inferior and studied some of their properties for sequence of real numbers. [ABSTRACT FROM AUTHOR]