# 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

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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

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