66
Views
3
CrossRef citations to date
0
Altmetric
Original Articles

The Existence, Uniqueness, and Measurability of a Stopped Semilinear Integral Equation

&
Pages 493-518 | Received 15 Oct 2002, Accepted 15 May 2006, Published online: 07 May 2007

REFERENCES

  • Ahmed , N.U. 1979 . Nonlinear integral equations on reflexive Banach spaces with applications to stochastic integral equations and abstract evolution equations . J. Integral Equations 1 : 1 – 15 .
  • Ahmed , N.U. 1985 . Abstract stochastic evolution equations on Banach spaces . J. Stochastic Analysis Applications 3 ( 4 ): 397 – 432 .
  • Banks , H. , and Burns , T. 1978 . Hereditary control problems: Numerical methods based on averaging approximations . SIAM J. Control Optimization 16 ( 2 ): 169 – 208 .
  • Brezis , H. , and Browder , F.E. 1975 . Existence theorems for nonlinear integral equations Hammerstein type . Bull. Am. Math. Soc. 81 : 73 – 79 .
  • Brezis , H. 1987 . Analyse Fonctionnelle, Théorie et Applications . MASSON , Paris .
  • Broweder , F.E. 1964 . Nonlinear equations of evolution . Ann. Math. 80 : 485 – 523 .
  • Chernoff , P.R. 1976 . Two counterexamples in semigroup theory on Hilbert space . Proc. Amer. Math. Soc. 56 : 253 – 255 .
  • Curtain , R.F. , and Pritchard , A.J. 1978 . Infinite dimensional linear system theory . In Control and Information Sciences . Vol. 8 . Springer-Verlag , Berlin-Heidelberg-New York .
  • Da Prato , G. , and Zabczyk , J. 1992 . Stochastic equations in infinite dimensions . In Encyclopedia of Mathematics and Its Applications . Rota , G.C. (ed.), Vol. 45 . Cambridge University Press .
  • Da Prato , G. , and Zabczyk , J. 1996 . Ergodicity for Infinite Dimensional Systems . Cambridge University Press .
  • Dunford , N. , and Schwarz , J.T. 1958 . Linear Operators, I: General Theory , Pure and Appl. Math. Vol. 7 . Interscience , New York .
  • Gelfand , I.M. , and Vilenkin , N.Y. 1964 . Generalized Functions, Applications of Harmonic Anaylsis . Vol. 4 . New York .
  • Hamedani , H.D. , and Zangeneh , B.Z. 2001 . Stopped Doob inequality for pth moment, > 0, stochastic convolution integrals . J. Stochastic Analysis and Applications 19 ( 5 ).
  • Higham , D.J. , Mao , X. , and Stuart , A.M. 2002 . Strong convergence of Euler-type methods for nonlinear stochastic differential equations . SIAM J. Numer. Anal. 40 ( 3 ): 1041 – 1063 .
  • Kato , T. 1953 . Integration of the equation of evolution in a Banach space . J. Math. Soc. Japan 5 : 208 – 234 .
  • Kato , T. 1964 . Nonlinear evolution equations in Banach spaces . Proc. Symp. Appl. Math. 17 : 50 – 67 .
  • Kato , T. 1973 . Linear evolution equations of “hyperbolic” type, II . J. Math. Soc. Japan 25 ( 4 ).
  • Léon , J.A. 1989 . Stochastic evolution equations with respect to semimartingales in Hilbert space . Stochastics and Stochastics Reports 27 : 1 – 21 .
  • Metivier , M. , and Pellaumail , J. 1980 . Stochastic Integration . Academic Press , New York .
  • Metivier , M. ( 1982 ) Semimartingales . Walter de Gruyter , Berlin, New York .
  • Pardoux , E. 1975. Equations aux derivees Partielles Stochastiques non linéaires monotones, Etude de solutions fortes de type ito. These doct. Sci. Math. Univ. Paris Sud.
  • Pazy , A. 1983 . Semigroups of Linear Operators and Applications to Partial Differential Equations . Applied Mathematical Sciences, 44 . Springer Verlag , Berlin .
  • Reed , M. , and Simon , B. 1972 . Methods of Modern Mathematical Physics, I: Functional analysis . Academic Press , New York London .
  • Tanabe , H. 1979 . Equations of Evolution . Pitman , London .
  • Vainberg , M.M. 1973 . Variational Method and Method of Monotone Operator in the Theory of Nonlinear Equations . John Wiley & Sons, LTD .
  • Walsh , J.B. 1986 . An introduction to stochastic partial differential equations . LN in Math. 1180 : 266 – 439 .
  • Zangeneh , B.Z. 1991 . Measurability of the Solution of a Semilinear Evolution Equation. Progress in Probability . Vol. 24 . Birkhauser Boston , MA , pp. 335 – 351 .
  • Zangeneh , B.Z. 1995 . Semilinear stochastic evolution equations with monotone nonlinearities . Stochastics and Stochastics Reports 53 ( 1–2 ): 129 – 174 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.