New Continuity Structures for Fuzzy Multifunctions via υ-Fuzzy γ-Open Sets and Fuzzy Ideals

This study introduces several new classes of continuity for fuzzy multifunctions based on υ-fuzzy γ-open sets, namely fuzzy upper γ-continuity (F+γ-continuity), fuzzy lower γ-continuity (F−γ-continuity), fuzzy upper almost γ-continuity (F+Aγ-continuity), fuzzy lower almost γ-continuity (F−Aγ-continuity), fuzzy upper weakly γ-continuity (F+Wγ-continuity), and fuzzy lower weakly γ-continuity (F−Wγ-continuity). The fundamental properties and characterizations of these continuity notions are systematically investigated, and their relationships with existing concepts on fuzzy topological spaces in the sense of Šostak are rigorously analyzed. Furthermore, it is shown that F+F−γ-continuity ⟹F+F−Aγ-continuity ⟹F+F−Wγ-continuity, while the converse implications do not necessarily hold. As an application, the proposed continuity concepts are employed to generalize, extend, and unify several well-known results in fuzzy topology. In addition, new continuity classes for fuzzy multifunctions based on fuzzy ideals, namely F+F−γI-continuity, F+F−AγI-continuity, and F+F−WγI-continuity, are defined and examined. A collection of illustrative examples is provided to clarify the distinctions among these notions and to demonstrate their nontrivial nature. The results not only refine and broaden classical theoretical frameworks but also pave the way for further investigations into the structural and topological properties of fuzzy multifunctions.