Vol. 87, No. 1, 1980

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The approximation of upper semicontinuous multifunctions by step multifunctions

Gerald A. Beer

Vol. 87 (1980), No. 1, 11–19

Let P be a right rectangular parallelepiped in Rm and let Y be a metric space. If Γ : P Y is an upper semicontinuous multifunction such that for each x in P the set Γ(x) is nonempty and closed, then there exists a sequence {Γk} of upper semicontinuous closed valued step multifunctions convergent in terms of Hausdorff distance to Γ from above. If Γ is compact valued and increasing and P is a closed interval, then the convergence can be made uniform. As a consequence of a Dini-type theorem for mutifunctions, the convergence can also be made uniform if Γ is compact valued and continuous.

Mathematical Subject Classification 2000
Primary: 54C60
Secondary: 90C30, 90C48
Received: 14 July 1978
Published: 1 March 1980
Gerald A. Beer