Fixed Point Theorems for Multivalued Contractive Mapping in Fuzzy Partial Metric Spaces

This study develops the theoretical framework of Fuzzy Partial Metric Spaces (FPMS) and explores their role in fixed-point theory. We begin by presenting precise definitions of FPMS along with the notions of lower semi-continuous and upper semi-continuous functions, which are fundamental in analyzing continuity and convergence in fuzzy settings. The paper further introduces the collection of compact subsets and non-empty subsets within FPMS, establishing the structural basis for subsequent results. Several preliminary lemmas are proved to strengthen the foundation of the theory. A key contribution is the formulation of the Hausdorff fuzzy partial metric, which extends the classical Hausdorff metric to fuzzy partial contexts. Building on this, we define multivalued contractive mappings in complete FPMS and investigate their properties. Using upper semi-continuous functions, we establish fixed-point theorems for such mappings, ensuring the existence of solutions under contractive conditions. These results generalize classical fixed-point principles and provide new insights into the behavior of multivalued mappings in uncertain environments. Illustrative examples are included to demonstrate the applicability of the theorems, highlighting their relevance in mathematical modeling, optimization, and systems influenced by fuzziness. The findings contribute to the advancement of fixed-point theory by integrating fuzzy logic, partial metrics, and multivalued analysis into a unified framework.