Multivalued Fixed Point Results and Homotopy Theorem in Hausdorff-Type Fuzzy Partial Metric Spaces

We introduce a Hausdorff-type fuzzy partial metric in the family of compact nonempty subsets of a fuzzy partial metric space (FPMS) and investigate its basic properties. The construction allows the usual set-valued Hausdorff approach to be formulated in a setting in which both fuzziness and nonzero self-distances are present. Using this structure, we establish a FP theorem for multivalued fuzzy (η)-contractive mappings on 1-complete FPMSs. The proof is based on the construction of an iterative sequence, the derivation of a geometric estimate through the function (η), and the completeness of the underlying space. An example is provided to verify the hypotheses of the main FP theorem and to illustrate the resulting FP. We further study a parameterized multivalued mapping and prove a homotopy FP theorem under suitable contractive, parameter-continuity, boundary, and local stability conditions. The result shows that the existence of a FP is preserved between the two endpoints of the parameter interval. These results extend the study of multivalued FP problems from partial metric and fuzzy metric settings to the combined framework of FPMSs.