# On the existence and asymptotic behavior of the random solutions of the random integral equation with advancing argument

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1996)

- Volume: 16, Issue: 1, page 43-51
- ISSN: 1509-9407

## Access Full Article

top## Abstract

top## How to cite

topHenryk Gacki. "On the existence and asymptotic behavior of the random solutions of the random integral equation with advancing argument." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 16.1 (1996): 43-51. <http://eudml.org/doc/275950>.

@article{HenrykGacki1996,

abstract = {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
$x(t,ω) = h(t,ω) + ∫^\{t+δ(t)\}₀ k(t,τ,ω)f(τ,x_τ(ω))dτ$, (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).},

author = {Henryk Gacki},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {random integral equations with advancing argument; fixed point theorem of Darbo type},

language = {eng},

number = {1},

pages = {43-51},

title = {On the existence and asymptotic behavior of the random solutions of the random integral equation with advancing argument},

url = {http://eudml.org/doc/275950},

volume = {16},

year = {1996},

}

TY - JOUR

AU - Henryk Gacki

TI - On the existence and asymptotic behavior of the random solutions of the random integral equation with advancing argument

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 1996

VL - 16

IS - 1

SP - 43

EP - 51

AB - 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
$x(t,ω) = h(t,ω) + ∫^{t+δ(t)}₀ k(t,τ,ω)f(τ,x_τ(ω))dτ$, (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).

LA - eng

KW - random integral equations with advancing argument; fixed point theorem of Darbo type

UR - http://eudml.org/doc/275950

ER -

## References

top- [1] J. Banaś, Measure of noncompactness in the space of continuous tempered functions, Demonstratio Math. 14 (1981), 127-133. Zbl0462.47035
- [2] J. Banaś, K. Goebel, Measure of noncompactness in Banach space, Notes in Pure and Applied Mathematics 60 Marcel Dekker, New York and Basel 1980. Zbl0441.47056
- [3] H. Gacki, On the existence and uniqueness of a solution of the random integral equation with advancing argument, Demonstratio Math. Vol. XIV (4) (1981). Zbl0501.60067
- [4] A. Lasota, M.C. Mackey, Globally asymptotic properties of proliterating cell populations, J. Math. Biology 19 (1984), 43-62. Zbl0529.92011
- [5] A. Lasota, H. Gacki, Markov operators defined by Volterra type integrals with advanced argument, Annales Math. Vol. LI (1990), 155-166. Zbl0721.34094
- [6] C.P. Tsokos, W.J. Padgett, Random integral equations with applications to life sciences and engineering, Academic Press, New York 1974. Zbl0287.60065
- [7] W.J. Padgett, On a stochastic integral equation of Volterra type in tlehone traffic theory, Journal of Applied Probability, 8 (1971), 269-275. Zbl0221.60072
- [8] D. Szynal, St.Wędrychowicz, On existence and an asymptotic behavior of random solutions of a class of stochastic functional-integral equations, Colloquium Math. Vol. LI (1987) 349-364. Zbl0625.60071