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Proceedings of the American Mathematical Society

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

 

 

A random Fredholm integral equation


Authors: W. J. Padgett and Chris P. Tsokos
Journal: Proc. Amer. Math. Soc. 33 (1972), 534-542
MSC: Primary 60H20
DOI: https://doi.org/10.1090/S0002-9939-1972-0292197-4
MathSciNet review: 0292197
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Abstract: The aim of this paper is the study of a random or stochastic integral equation of the Fredholm type given by $ x(t;\omega ) = h(t;\omega ) + \smallint_0^\infty {{k_0}(t,} \tau ;\omega )e(\tau ,x(\tau ;\omega ))\;d\tau, t \geqq 0$, where $ \omega \in \Omega $, the supporting set of the probability measure space $ (\Omega ,A,P)$. The existence and uniqueness of a random solution to the above stochastic integral equation is considered. A random solution, $ x(t;\omega )$, of such a random equation is defined to be a random function which satisfies the equation almost surely. Several theorems and useful special cases are presented which give conditions such that a random solution exists.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0292197-4
Keywords: Stochastic integral equation of Fredholm type, random Fredholm integral equation, random mixed Volterra-Fredholm integral equation, random solutions
Article copyright: © Copyright 1972 American Mathematical Society