Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nonlinear evolution equations and product integration in Banach spaces.


Author: G. F. Webb
Journal: Trans. Amer. Math. Soc. 148 (1970), 273-282
MSC: Primary 47.65; Secondary 34.00
MathSciNet review: 0265992
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The method of product integration is used to obtain solutions to the nonlinear evolution equation $ g' = Ag$ where A is a function from a Banach space S to itself and g is a continuously differentiable function from $ [0,\infty )$ to S. The conditions required on A are that A is dissipative on S, the range of $ (e - \varepsilon A) = S$ for all $ \varepsilon \geqq 0$, and A is continuous on S.


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

  • [1] Garrett Birkoff, On product integration, J. Mathematical Phys. 16 (1937), 104-132.
  • [2] H. Brezis and A. Pazy, Semigroups of nonlinear contractions on convex sets, J. Functional Analysis 6 (1970), 237–281. MR 0448185
  • [3] Felix E. Browder, Non-linear equations of evolution, Ann. of Math. (2) 80 (1964), 485–523. MR 0173960
  • [4] 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
  • [5] Michael G. Crandall and Amnon Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis 3 (1969), 376–418. MR 0243383
  • [6] J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
  • [7] J. R. Dorroh, Integral equations in normed abelian groups, Pacific J. Math. 13 (1963), 1143–1158. MR 0158243
  • [8] J. R. Dorroh, Some classes of semi-groups of nonlinear transformations and their generators, J. Math. Soc. Japan 20 (1968), 437–455. MR 0231241
  • [9] 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
  • [10] Tosio Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508–520. MR 0226230
  • [11] 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
  • [12] Yukio Kōmura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan 19 (1967), 493–507. MR 0216342
  • [13] -, Differentiability of nonlinear semi-groups, J. Math. Soc. Japan 21 (1969), 375-402.
  • [14] J. Mermin, Accretive operators and nonlinear semi-groups, Thesis, Univ. of California, Berkeley, 1968.
  • [15] George J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346. MR 0169064
  • [16] J. W. Neuberger, Continuous products and nonlinear integral equations, Pacific J. Math. 8 (1958), 529–549. MR 0102723
  • [17] J. W. Neuberger, An exponential formula for one-parameter semi-groups of nonlinear transformations, J. Math. Soc. Japan 18 (1966), 154–157. MR 0200734
  • [18] Isao Miyadera and Shinnosuke Ôharu, Approximation of semi-groups of nonlinear operators, Tôhoku Math. J. (2) 22 (1970), 24–47. MR 0262874
  • [19] G. F. Webb, Representation of semi-groups of nonlinear nonexpansive transformations in Banach spaces, J. Math. Mech. 19 (1969/1970), 159–170. MR 0247528
  • [20] James A. Yorke, A continuous differential equation in Hilbert space without existence., Funkcial. Ekvac. 13 (1970), 19–21. MR 0264196

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47.65, 34.00

Retrieve articles in all journals with MSC: 47.65, 34.00


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0265992-8
Keywords: Nonlinear evolution equations, product integration, dissipative mapping, semigroup of nonlinear nonexpansive transformations, infinitesimal generator
Article copyright: © Copyright 1970 American Mathematical Society