Multivalued extension of the Caristi-type theorem in semi-metric spaces and its numerical simulation

We present a novel extension of the Caristi-type fixed point theorem recently established by Zubelevich (2025) for single-valued mappings on complete semi-metric spaces to the multivalued setting. Specifically, we prove that, in a complete semi-metric space R, if a set-valued mapping F:R→2R satisfies a generalized Caristi-type inequality involving a family of lower semi-continuous, bounded-from-below functions and a semi-metric structure, then F admits a fixed point. Our approach constructs a suitable selection from the multivalued map and applies Zubelevich’s partial order method in conjunction with Zorn’s lemma to ensure the existence of a fixed point. Furthermore, we establish a multivalued counterpart to Zubelevich’s noncompactness theorem: if one of the associated potential functions fails to attain its minimum, then the fixed point set of F is necessarily noncompact. These results provide the first known multivalued Caristi-type fixed point framework for semi-metric spaces, unifying and generalizing prior work in both metric and topological vector space settings. The results are further validated by a numerical simulation showing convergence of iterates under deterministic selections.