A theorem and a counterexample in the theory of semigroups of nonlinear transformations
HTML articles powered by AMS MathViewer
- by Michael G. Crandall and Thomas M. Liggett PDF
- Trans. Amer. Math. Soc. 160 (1971), 263-278 Request permission
Abstract:
This paper studies the basic method in current use for constructively obtaining a generator from a given semigroup of nonlinear transformations on a Banach space. The method is shown to succeed in real two-dimensional Banach spaces and to fail in a particular three-dimensional example. Other results of independent interest are obtained. For example, it is shown that the concepts of “maximal accretive” and “hyperaccretive” (equivalently, $m$-accretive or hypermaximal accretive) coincide in ${R^n}$ with the maximum norm.References
- H. Brezis and A. Pazy, Accretive sets and differential equations in Banach spaces, Israel J. Math. 8 (1970), 367–383. MR 275243, DOI 10.1007/BF02798683
- Felix E. Browder, Nonlinear maximal monotone operators in Banach space, Math. Ann. 175 (1968), 89–113. MR 223942, DOI 10.1007/BF01418765
- M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298. MR 287357, DOI 10.2307/2373376
- J. R. Dorroh, Some classes of semi-groups of nonlinear transformations and their generators, J. Math. Soc. Japan 20 (1968), 437–455. MR 231241, DOI 10.2969/jmsj/02030437
- Samuel Eilenberg and Deane Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214–222. MR 16676, DOI 10.2307/2371832
- B. Grünbaum, On a theorem of Kirszbraun, Bull. Res. Council Israel Sect. F 7 (1957/58), 129–132. MR 106423
- Tosio Kato, Note on the differentiability of nonlinear semigroups, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 91–94. MR 0270208
- Tosio Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508–520. MR 226230, DOI 10.2969/jmsj/01940508
- Yukio K\B{o}mura, Differentiability of nonlinear semigroups, J. Math. Soc. Japan 21 (1969), 375–402. MR 250118, DOI 10.2969/jmsj/02130375
- George J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346. MR 169064
- J. W. Neuberger, An exponential formula for one-parameter semi-groups of nonlinear transformations, J. Math. Soc. Japan 18 (1966), 154–157. MR 200734, DOI 10.2969/jmsj/01820154
- Shinnosuke Ôharu, Note on the representation of semi-groups of non-linear operators, Proc. Japan Acad. 42 (1966), 1149–1154. MR 220100
- R. S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 193–254. MR 104919, DOI 10.1090/S0002-9947-1959-0104919-1
- Sten Olof Schönbeck, On the extension of Lipschitz maps, Ark. Mat. 7 (1967), 201–209 (1967). MR 241962, DOI 10.1007/BF02591628
- G. F. Webb, Representation of semi-groups of nonlinear nonexpansive transformations in Banach spaces, J. Math. Mech. 19 (1969/1970), 159–170. MR 0247528, DOI 10.1512/iumj.1970.19.19016
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 160 (1971), 263-278
- MSC: Primary 47H99; Secondary 47D05
- DOI: https://doi.org/10.1090/S0002-9947-1971-0301592-X
- MathSciNet review: 0301592