In the paper we construct a topological fixed-point theory for a class of s
et-valued maps which appears in natural way in boundary value problems for
differential inclusions. Our construction is based upon the notion of (U, V
)-approximation in the sense of Ben-El-Mechaiekh and Deguire. As applicatio
ns we consider initial-value problems for nonlinear evolution inclusions of
the type
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where the operator Lambda satisfies various monotonicity assumptions and F
is an upper semi-continuous set-valued perturbation.