Abstract
This paper is concerned with global solutions of the initial value problem (1)du/dt +Au∋0,u(0)=x whereA is a (nonlinear) accretive set in a Banach spaceX. We show that various approximation processes converge to the solution (whenever it exists). In particular we obtain an exponential formula for the solutions of (1).
AssumingX* is uniformly convex, we also prove the existence of a solution under weaker assumptions ofA than those made by previous authors (F. Browder, T. Kato).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Brezis, and A. PazySemigroups of nonlinear contractions on convex sets, J. Functional Analysis, to appear.
F. Browder,Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Nonlinear Functional Anal., Chicago, Amer. Math. Soc., 1968.
P. Chernoff,Note on product formulas for operator semigroups, J. Functional Analysis,2 (1968), 238–242.
M. Crandall,Differential equations on convex sets, J. Math. Soc. Japan, to appear.
T. Kato,Nonlinear semigroups and evolution equations, J. Math. Soc. Japan19 (1967), 508–520.
T. Kato,Accretive operators and nonlinear evolution equations in Banach spaces, Proc. Symp. Nonlinear Functional Anal., Chicago, Amer. Math. Soc., 1968.
Y. Komura,Nonlinear semigroups in Hilbert space, J. Math. Soc. Japan19 (1967) 493–507.
J. Mermin,Aceretive operators and nonlinear semigroups, Thesis, University of California, Berkeley, 1968.
I. Miyadera, and S. Ôharu,Approximation of semigroups of nonlinear operators, to appear.
G. Webb,Nonlinear evolution equations and product integration, to appear.
K. Yosida,Functional Analysis, Springer Verlag, Berlin, 1965.
Author information
Authors and Affiliations
Additional information
Results obtained at the Courant Institute of Mathematical Sciences, New York University, with the National Science Foundation, Grant NSF-GP-11600.
Rights and permissions
About this article
Cite this article
Brezis, H., Pazy, A. Accretive sets and differential equations in Banach spaces. Israel J. Math. 8, 367–383 (1970). https://doi.org/10.1007/BF02798683
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02798683