Vol. 106, No. 1, 1983

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Renorming and the theory of phi-accretive set-valued mappings

David Downing and William O. Ray

Vol. 106 (1983), No. 1, 73–85

Let X and Y be Banach spaces, ϕ : X Y and P : X 2Y ; P is said to be strongly ϕ-accretive if there exists c > 0 so that (w v,ϕ(x y)) cx y2 whenever x,y X and w Px, v Py. Such mappings constitute a simultaneous generalization of monotone mappings (when Y = X) and accretive mappings (when Y = X). By applying a fixed point theorem of J. Caristi, it is shown that if P is strongly ϕ-accretive in a localized sense and if Y can be appropriately renormed, then, under suitable continuity and range restrictions, P is an open mapping. The results generalize a number of known theorems and indicate a firm connection between the theory of ϕ-accretive mappings and the renorming characteristics of the space Y .

Mathematical Subject Classification 2000
Primary: 47H05
Secondary: 47H06
Received: 23 September 1981
Published: 1 May 1983
David Downing
William O. Ray