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By combining the results of Wardowski's cyclic contraction operators and admissible multi-valued mappings, the motif of $ \eta $-cyclic $ \left(\alpha _{\ast }, \beta _{\ast }\right) $-admissible type $ \digamma $-contraction multivalued mappings are presented. Moreover, some novel fixed point theorems for such mappings are proved in the context of $ M_{b} $-metric spaces. Also, two examples are given to clarify and strengthen our theoretical study. Finally, the existence of a solution of a pair of ordinary differential equations is discussed as an application.