Vol. 125, No. 2, 1986

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Range of Gateaux differentiable operators and local expansions

Jong Sook Bae and Sangsuk Yie

Vol. 125 (1986), No. 2, 289–300
Abstract

Let X and Y be Banach spaces, and P : X Y a Gateaux differentiable operator having closed graph. Suppose that there is a continuous function c : [0,) (0,) satisfying

dPx(B(0;1)) ⊇ B(0;c(∥x∥)).

Then it is shown that for any K > 0 (possibly K = ), P(B(0;K)) contains B(P(0); 0Kc(s)ds). Similar results are obtained for local expansions and locally strongly ϕ-accretive operators. These results extend a number of known theorems by giving the precise geometric estimations for normal solvability of Px = y.

Mathematical Subject Classification 2000
Primary: 47H06
Milestones
Received: 22 June 1985
Published: 1 December 1986
Authors
Jong Sook Bae
Sangsuk Yie