A fixed point theorem for measurable selection valued correspondences induced by upper Caratheodory correspondences

We show that any measurable selection valued correspondence induced by the composition of an m-tuple of real-valued Caratheodory functions with an upper Caratheodory (uC) correspondence has fixed points if the underlying uC correspondence in the composition contains a continuum valued uC sub-correspondence. Moreover, this composition of the m-tuple of real-valued Caratheodory functions with the continuum valued uC sub-correspondence induces a measurable selection valued sub-correspondence that is weak star upper semicontinuous