On the Existence and Uniqueness of a Random Solution to a Perturbed Random Integral Equation of the Fredholm Type

Abstract

A study of a random or stochastic integral equation of the Fredholm type of the form\[ x( {t;\,\omega } ) = h( t,x( t;\omega ) ) + \int_0^\infty {k_0 ( t,\tau ;\omega )e( \tau ,x( \tau ;\omega ) )d\tau },\qquad t\geqq 0, \]is presented, where $\omega \in \Omega $, the supporting set of the probability measure space $( \Omega ,A,\mu )$. The existence and uniqueness of a random solution of the equation is considered by first investigating a stochastic integral equation of the mixed Volterra–Fredholm type of the form\[ \begin{gathered} \hfill x( t;\omega ) = h( t,x( t;\omega ) ) + \int_0^t k ( t,\tau ;\omega )f( {\tau ,x( \tau ;\omega ) )d\tau } \\ \hfill + \int_0^\infty {k_0 ( t,\tau ;\omega )e( \tau ,x( \tau ;\omega ) )d\tau },\qquad t\geqq 0. \\ \end{gathered} \]A random solution $x( t;\omega )$ of an equation such as that above is defined to be a random function which satisfies the equation $\mu $-a.e. Several theorems and special cases are presented which give conditions such that a random solution exists for each type of equation.