Application of random fixed point theorems in solving nonlinear stochastic integral equation of the Hammerstein type

Debashis Deya,* and Mantu Sahab

Application of random fixed point theorems in solving nonlinear stochastic integral equation of the Hammerstein type

Authors :

Debashis Dey^a,* and Mantu Saha^b

Author Address :

^aKoshigram Union Institution, Koshigram-713150, Burdwan , West Bengal, India.
^bDepartment of Mathematics, The University of Burdwan, Burdwan-713104, West Bengal, India.

*Corresponding author.

Abstract :

In the present paper, we apply random analogue Kannan fixed point theorem [10] to analyze the existence of a solution of a nonlinear stochastic integral equation of the Hammerstein type of the form

\begin{eqnarray}x(t;\omega)=h(t;\omega)+\int_{S}k(t,s;\omega)f(s,x(s;\omega))d\mu(s)\nonumber\end{eqnarray}where $t\in S$,

a $\sigma$-finite measure space with certain properties, $\omega\in\Omega$, the supporting set of a probability measure space $\left(\Omega,\beta,\mu\right)$ and the integral is a Bochner integral.