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In this paper, we establish an extension of the well-known Contraction Mapping Principle for Set-Valued Mappings. We consider a larger class of weakly contractive mappings, which allows us to cover, extend and unify many assertions of fixed point theory. As an application, we derive two quantitative stability results for fixed points of weakly contractive maps.

An extended local principle of fixed points for weakly contractive set-valued mappings