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• W2093888005 abstract "If a Banach space E has an equivalent norm such that weak* sequential convergence and norm convergence coincide on the dual unit sphere, then every monotone operator on E is single-valued and norm-norm continuous on a dense Gs subset of E. In particular, this holds for reflexive spaces. Let E be a real Banach space with dual E*. A multivalued mapping T: E -E* is called a monotone operator if 0> whenever x* E Tx and y* E Ty. It is called maximal monotone if, in addition, its graph, {(x, x*): x E E, x* E Tx}, is not properly contained in the graph of any monotone operator on E. We say that a monotone operator T: E -E* is locally bounded at x E E if there is a neighborhood U of x such that T(U) = U{Ty: y E U} is a bounded subset of E*. This does not demnand that x E D(T) = { y E E: Ty # 0}. We say that T is continuous at a point x E D(T) if, whenever n x, x* E Tx,E and x* E Tx, we have x,* x* || 0. If T is continuous at x, then it is necessarily single-valued at x, that is, Tx has exactly one element. We will assume from now on that T is a maximal monotone operator on E, with D(T) = E. This latter hypothesis, while not strictly necessary, simplifies both the statements and the proofs of our results. All the proofs can actually be extended to the case where D(T) # E, provided int conv D(T) # 0. The reason for this stems from the first part of the following result of Rockafellar [9], which will be of further use to us. PROPOSITION A. Let T: E -> E* be a maximal monotone operator with int convD(T) # 0. Then intD(T) is convex, clD(T) = cl intD(T), T is locally bounded at each point of int D(T) and T is not locally bounded at any point of the boundary of D(T). The motivation for studying monotone operators comes from the study of Received by the editors May 11, 1976. AMS (MOS) subject classifications (1970). Primary 47H05. I This work was done under the supervision of Professor R. R. Phelps while the author was a Ph.D. student at the University of Washington. It was supported by a Commonwealth Scientific and Industrial Research Organization Postgraduate Studentship. 2 The author notes that the same result has recently been announced by P. Kenderov and R. Robert, in C. R. Acad. Sci. Paris 282 (1976), No. 16. ? Aiiiericiani Mathemaltical Society 1977" @default.
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• W2093888005 date "1977-01-01" @default.
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• W2093888005 title "Continuity of nonlinear monotone operators" @default.
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• W2093888005 doi "https://doi.org/10.1090/s0002-9939-1977-0425687-6" @default.
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