Renorming and the theory of phi-accretive set-valued mappings
David J. Downing, William D. Ray · Pacific Journal of Mathematics · 1983
Let X and Y be Banach spaces, φ: X -> Y* and P: X-*2 Y \P is said to be strongly φ-accretive if there exists c > 0 so that (w -v,φ(x -y)) > c\\x -y II 2 whenever x 9 y E X and w E Px, v E 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 7.