Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Regularizing properties of nonlinear semigroups

Author: Semion Gutman
Journal: Proc. Amer. Math. Soc. 101 (1987), 51-56
MSC: Primary 47H20; Secondary 34G20, 35R20, 47H06
MathSciNet review: 897069
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is known that some classes of $ m$-accretive operators $ A$ generate Lipschitz continuous semigroups of contractions; that is $ \vert\vert S(t + h)x - S(t)x\vert\vert \leqslant L(\vert\vert x\vert\vert)h/t,0 \leqslant t \leqslant t + h \leqslant T,x \in \overline {D(A)} $. If the underlying Banach spaces $ X$ and $ {X^*}$ are uniformly convex and an $ m$-accretive operator $ B$ is bounded, we prove, in particular, that the semigroup generated by $ A + B$ is Hölder continuous. The proof is based on a result on the structure of accretive operators obtained via the Kuratowski-Ryll-Nardzewski Selection Theorem. Also, we consider some applications of these results to the existence of solutions of $ u' + Au + Bu \mathrel\backepsilon Cu,u(0) = {u_0}$.

References [Enhancements On Off] (What's this?)

  • [1] Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
  • [2] Ph. Benilan, Equations d'evolution dans un espace de Banach quelconque et applications, Thése, Orsay, 1972.
  • [3] Haïm Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 101–156. MR 0394323
  • [4] Philippe Bénilan and Michael G. Crandall, Regularizing effects of homogeneous evolution equations, Contributions to analysis and geometry (Baltimore, Md., 1980) Johns Hopkins Univ. Press, Baltimore, Md., 1981, pp. 23–39. MR 648452
  • [5] 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 0287357,
  • [6] Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
  • [7] L. C. Evans, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math. 26 (1977), no. 1, 1–42. MR 0440431,
  • [8] S. Gutman, Evolutions governed by 𝑚-accretive plus compact operators, Nonlinear Anal. 7 (1983), no. 7, 707–715. MR 707079,
  • [9] K. Yosida, Functional analysis, Springer-Verlag, Berlin and New York, 1969.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H20, 34G20, 35R20, 47H06

Retrieve articles in all journals with MSC: 47H20, 34G20, 35R20, 47H06

Additional Information

Keywords: Nonlinear semigroup, regularity, $ m$-accretive, evolution equation, bounded perturbation
Article copyright: © Copyright 1987 American Mathematical Society