Differential equations on closed subsets of a Banach space

Martin, R. H.

doi:10.1090/S0002-9947-1973-0318991-4

Differential equations on closed subsets of a Banach space
HTML articles powered by AMS MathViewer

by R. H. Martin

Trans. Amer. Math. Soc. 179 (1973), 399-414

DOI: https://doi.org/10.1090/S0002-9947-1973-0318991-4

PDF | Request permission

Abstract:

In this paper the problem of existence of solutions to the initial value problem $u’(t) = A(t,u(t)),u(a) = z$, is considered where $A:[a,b) \times D \to E$ is continuous, D is a closed subset of a Banach space E, and $z \in D$. With a dissipative type condition on A, we establish sufficient conditions for this initial value problem to have a solution. Using these results, we are able to characterize all continuous functions which are generators of nonlinear semigroups on D.

References

Jean-Michel Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 277–304 xii (French, with English summary). MR 262881, DOI 10.5802/aif.319
Haïm Brezis, On a characterization of flow-invariant sets, Comm. Pure Appl. Math. 23 (1970), 261–263. MR 257511, DOI 10.1002/cpa.3160230211
Felix E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875–882. MR 232255, DOI 10.1090/S0002-9904-1967-11823-8
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Michael G. Crandall, Differential equations on convex sets, J. Math. Soc. Japan 22 (1970), 443–455. MR 268491, DOI 10.2969/jmsj/02240443

A generalization of Peano’s theorem and flow invariance

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
Philip Hartman, On invariant sets and on a theorem of Ważewski, Proc. Amer. Math. Soc. 32 (1972), 511–520. MR 298091, DOI 10.1090/S0002-9939-1972-0298091-7
J. V. Herod, A pairing of a class of evolution systems with a class of generators, Trans. Amer. Math. Soc. 157 (1971), 247–260. MR 281059, DOI 10.1090/S0002-9947-1971-0281059-8
Tosio Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508–520. MR 226230, DOI 10.2969/jmsj/01940508
Tosio Kato, Accretive operators and nonlinear evolution equations in Banach spaces. , Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 138–161. MR 0271782
David Lowell Lovelady and Robert H. Martin Jr., A global existence theorem for a nonautonomous differential equation in a Banach space, Proc. Amer. Math. Soc. 35 (1972), 445–449. MR 303035, DOI 10.1090/S0002-9939-1972-0303035-5
Robert H. Martin Jr., The logarithmic derivative and equations of evolution in a Banach space, J. Math. Soc. Japan 22 (1970), 411–429. MR 298467, DOI 10.2969/jmsj/02230411
R. H. Martin Jr., A global existence theorem for autonomous differential equations in a Banach space, Proc. Amer. Math. Soc. 26 (1970), 307–314. MR 264195, DOI 10.1090/S0002-9939-1970-0264195-6
R. M. Redheffer, The theorems of Bony and Brezis on flow-invariant sets, Amer. Math. Monthly 79 (1972), 740–747. MR 303024, DOI 10.2307/2316263
G. F. Webb, Nonlinear evolution equations and product integration in Banach spaces, Trans. Amer. Math. Soc. 148 (1970), 273–282. MR 265992, DOI 10.1090/S0002-9947-1970-0265992-8
G. F. Webb, Nonlinear evolution equations and product stable operators on Banach spaces, Trans. Amer. Math. Soc. 155 (1971), 409–426. MR 276842, DOI 10.1090/S0002-9947-1971-0276842-9