Fixed point theorems for set valued functions



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This dissertation considers fixed point theorems for set valued functions, F : X -> X. The first part of this dissertation extends the class of those set valued functions which induce homomorphisms h : H[lowered *] (X) H[lowered *] (X) that satisfy the Lefschetz Fixed Point Theorem. An example is given of a collection of set valued maps on a 2-cell which are fixed point free. Moreover, the class of spaces X for which F induces h : H[lowered *] (X) -> H[lowered *] (X) is extended from polyhedra to ANR's and NR[lowered delta]'s. It is also shown that the inverse limit of spaces with the fixed point property for set valued maps has the fixed point property. Finally, the notion of contractive set valued functions is introduced and investigated for fixed point theorems.