A global existence theorem for autonomous differential equations in a Banach space

Martin, R. H.

doi:10.1090/S0002-9939-1970-0264195-6

A global existence theorem for autonomous differential equations in a Banach space
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by R. H. Martin

Proc. Amer. Math. Soc. 26 (1970), 307-314

DOI: https://doi.org/10.1090/S0002-9939-1970-0264195-6

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Abstract:

Let $E$ be a Banach space and let $A$ be a continuous function from $E$ into $E$. Sufficient conditions are given to insure that the differential equation $u’(t) = Au(t)$ has a unique solution on $[0,\infty )$ for each initial value in $E$. One consequence of this result is that if $-A$ is monotonic, then $-A$ is $m$-monotonic and $A$ is the generator of a nonexpansive semigroup of operators.

References

Felix E. Browder, Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 867–874. MR 232254, DOI 10.1090/S0002-9904-1967-11820-2
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
W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Company, Boston, Mass., 1965. MR 0190463
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

Differential and integral inequalities

R. H. Martin Jr., A theorem on critical points and global asymptotic stability, J. Math. Anal. Appl. 33 (1971), 124–130. MR 269959, DOI 10.1016/0022-247X(71)90186-7
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
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