Single-valued Accretive Operators; Applications; Some Open Questions
Set-valued accretive operators in Banach spaces have been extensively studied for several decades under various continuity assumptions. In the first part of this chapter we establish a recent incisive finding that every set-valued lower semi-continuous accretive mapping defined on a normed space is, indeed, single-valued on the interior of its domain. No reference to the well-known Michael's Selection Theorem is needed. In Section 23.3, this result is used to extend known theorems concerning the existence of zeros for such operators, as well as, showing existence of solutions for variational inclusions. In Section 23.4, we make some general comments on some fixed point theorems; the rest of the chapter is devoted to some examples of accretive operators; examples of nonexpansive retracts; open problems; and some suggestions for further reading.
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