We consider a class of multimaps which are the composition of a superposition multioperator P F generated by a nonconvex-valued almost lower semicontinuous nonlinearity F and an abstract solution operator S. We prove that under some suitable conditions such multimaps are condensing with respect to a special vector-valued measure of noncompactness and construct a topological degree theory for this class of multimaps yielding some fixed point principles. It is shown how abstract results can be

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... lied to semilinear inclusions, inclusions with m-accretive operators and time-dependent subdifferentials, nonlinear evolution inclusions and integral inclusions in Banach spaces. 2000 Mathematics Subject Classification. 47H11, 47H09, 47H04, 34G25, 34A60.