A pairing of a class of evolution systems with a class of generators.

Herod, J. V.

doi:10.1090/S0002-9947-1971-0281059-8

A pairing of a class of evolution systems with a class of generators.
HTML articles powered by AMS MathViewer

by J. V. Herod PDF

Trans. Amer. Math. Soc. 157 (1971), 247-260 Request permission

Abstract:

Suppose that $S$ is a Banach space and that $A$ and $M$ are functions such that if $x$ and $y$ are numbers, $x \geqq y$, and $P$ is in $S$ then each of $M(x,y)P$ and $A(y,P)$ is in $S$. This paper studies the relation \[ M(x,y)P = P + \int _x^y {A(t,M(t,y)P)dt.} \] Classes OM and OA will be described and a correspondence will be established which pairs members of the two classes which are connected as $M$ and $A$ are by the relation indicated above.

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
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
Jerome A. Goldstein, Abstract evolution equations, Trans. Amer. Math. Soc. 141 (1969), 159–185. MR 247524, DOI 10.1090/S0002-9947-1969-0247524-5

Multiplicative inverses of solutions for Volterra-Stieltjes integral equations

22

J. V. Herod, Coalescence of solutions for nonlinear Stieltjes equations, J. Reine Angew. Math. 252 (1972), 187–194. MR 291908, DOI 10.1515/crll.1972.252.187
J. V. Herod, A product integral representation for an evolution system, Proc. Amer. Math. Soc. 27 (1971), 549–556. MR 275244, DOI 10.1090/S0002-9939-1971-0275244-4
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Tosio Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508–520. MR 226230, DOI 10.2969/jmsj/01940508
J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148–173. MR 144179
J. S. MacNerney, A nonlinear integral operation, Illinois J. Math. 8 (1964), 621–638. MR 167815
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
J. W. Neuberger, A generator for a set of functions, Illinois J. Math. 9 (1965), 31–39. MR 172143
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
J. W. Neuberger, Product integral formulae for nonlinear expansive semigroups and non-expansive evolution systems, J. Math. Mech. 19 (1969/1970), 403–409. MR 0253086, DOI 10.1512/iumj.1970.19.19037
Dorothy Rutledge, A generator for a semigroup of nonlinear transformations, Proc. Amer. Math. Soc. 20 (1969), 491–498. MR 239464, DOI 10.1090/S0002-9939-1969-0239464-8
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
G. F. Webb, Product integral representation of time dependent nonlinear evolution equations in Banach spaces, Pacific J. Math. 32 (1970), 269–281. MR 257834
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