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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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New results in the perturbation theory of maximal monotone and $M$-accretive operators in Banach spaces
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by Athanassios G. Kartsatos PDF
Trans. Amer. Math. Soc. 348 (1996), 1663-1707 Request permission


Let $X$ be a real Banach space and $G$ a bounded, open and convex subset of $X.$ The solvability of the fixed point problem $(*)~Tx+Cx \owns x$ in $D(T)\cap \overline {G}$ is considered, where $T:X\supset D(T)\to 2^{X}$ is a possibly discontinuous $m$-dissipative operator and $C: \overline {G}\to X$ is completely continuous. It is assumed that $X$ is uniformly convex, $D(T)\cap G \not = \emptyset$ and $(T+C)(D(T)\cap \partial G)\subset \overline {G}.$ A result of Browder, concerning single-valued operators $T$ that are either uniformly continuous or continuous with $X^{*}$ uniformly convex, is extended to the present case. Browder’s method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let $\Gamma = \{\beta :\mathcal {R}_{+}\to \mathcal {R}_{+}~;~\beta (r)\to 0\text { as }r\to \infty \}.$ The effect of a weak boundary condition of the type $\langle u+Cx,x\rangle \ge -\beta (\|x\|)\|x\|^{2}$ on the range of operators $T+C$ is studied for $m$-accretive and maximal monotone operators $T.$ Here, $\beta \in \Gamma ,~x\in D(T)$ with sufficiently large norm and $u\in Tx.$ Various new eigenvalue results are given involving the solvability of $Tx+ \lambda Cx\owns 0$ with respect to $(\lambda ,x)\in (0,\infty )\times D(T).$ Several results do not require the continuity of the operator $C.$ Four open problems are also given, the solution of which would improve upon certain results of the paper.
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Additional Information
  • Athanassios G. Kartsatos
  • Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
  • Email:
  • Received by editor(s): February 7, 1995
  • Additional Notes: The results of this paper were announced in a lecture at the International Conference on Nonlinear Differential Equations, Kiev, Ukraine, August 21-27, 1995.
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 1663-1707
  • MSC (1991): Primary 47H17; Secondary 47B44, 47H09, 47H10
  • DOI:
  • MathSciNet review: 1357397