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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a nonlinear stochastic integral equation of the Hammerstein type
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by W. J. Padgett PDF
Proc. Amer. Math. Soc. 38 (1973), 625-631 Request permission

Abstract:

A nonlinear stochastic integral equation of the Hammerstein type in the form \[ x(t;\omega ) = h(t;\omega ) + \int _s {k(t,s;\omega )f(s,x(s;\omega )} )d\mu (s)\] is studied where $t \in S,a$, a $\sigma$-finite measure space with certain properties, $\omega \in \Omega$, the supporting set of a probability measure space $(\Omega ,A,P)$, and the integral is a Bochner integral. A random solution of the equation is defined to be a second order vector-valued stochastic process $x(t;\omega )$ on $S$ which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on $S$, and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 38 (1973), 625-631
  • MSC: Primary 45G99
  • DOI: https://doi.org/10.1090/S0002-9939-1973-0320663-2
  • MathSciNet review: 0320663