### On some classes of operator inclusions with lower semicontinuous nonlinearities

DOI: http://dx.doi.org/10.12775/TMNA.2001.010

#### Abstract

We consider a class of multimaps which are the composition of a superposition

multioperator ${\mathcal P}_F$ 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 applied to semilinear inclusions,

inclusions with $m$-accretive operators and time-dependent subdifferentials,

nonlinear evolution inclusions and integral inclusions in Banach spaces.

multioperator ${\mathcal P}_F$ 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 applied to semilinear inclusions,

inclusions with $m$-accretive operators and time-dependent subdifferentials,

nonlinear evolution inclusions and integral inclusions in Banach spaces.

#### Keywords

Multivalued map; topological degree; measure of noncompactness; differential inclusion

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