Discussiones Mathematicae Differential Inclusions 16(1) (1996) 43-51

[BIBTex]

ON THE EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE RANDOM SOLUTIONS OF THE RANDOM INTEGRAL EQUATION WITH ADVANCING ARGUMENT

Henryk Gacki

Institute of Mathematics, Silesian University
40-007 Katowice, Poland

1. Introduction

Random Integral Equations play a significant role in characterizing of many biological and engineering problems [4,5,6,7].

We present here new existence theorems for a class of integral equations with advancing argument. Our method is based on the notion of a measure of noncompactness in Banach spaces and the fixed point theorem of Darbo type. We shall deal with random integral equation with advancing argument

x(t,ω) = h(t,ω) +
t+δ(t)

0
k(t,τ,ω)f(τ,xτ(ω))dτ, (t,ω) ∈ R+×Ω,
(1)

where

(i) (Ω, A,P) is a complete probability space,
(ii) x = x(t,ω) denotes an unknown random function defined for t ∈ R+ and ω ∈ Ω,
(iii) δ is a nonnegative function from R+ into R+,
(iv) xt(ω) denotes the restriction of the function x(t,ω) to the interval [0,t+δ(t)], t>0, with x0(ω) = x(0,ω) ∈ L2(Ω, A,P).

References

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