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Zbl 0634.47048
Morales, Claudio
Zeros for strongly accretive set-valued mappings. (English)
[J] Commentat. Math. Univ. Carol. 27, 455-469 (1986). ISSN 0010-2628

Let X be a Banach space, D a nonempty subset of X, and let B(X) denote the family of all non-empty, bounded and closed subsets of X supplied with the Hausdorff metric H. A mapping $T: D\to B(X)$ is said to be strongly accretive if for some $k<1$ and for each x,y$\in D$, $u\in T(x)$, $v\in T(y):$ $(\lambda -k)\Vert x-y\Vert \le \Vert (\lambda -1)(x- y)+u-v\Vert$for all $\lambda >k$. Suppose $T: D\to B(X)$ is a continuous (with respect to the Hausdorff metric) and strongly accretive mapping. It is shown that if for some $z\in D:$ t(x-z)$\not\in T(x)$ for x in the boundary of D and $t<0$, it is sufficient to guarantee that T has a zero in $\bar D.$ Several implications of this result are considered, particularly on a localized version of it.
[V.Popa]

MSC 2000:: *47H06 Accretive operators, etc. (nonlinear)

Keywords: Hausdorff metric; strongly accretive

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