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

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

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## Abstract

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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 $x\left(t,\omega \right)=h\left(t,\omega \right)+{\int }^{t+\delta \left(t\right)}₀k\left(t,\tau ,\omega \right)f\left(\tau ,{x}_{\tau }\left(\omega \right)\right)d\tau$, (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).

## How to cite

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Henryk 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

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1. [1] J. Banaś, Measure of noncompactness in the space of continuous tempered functions, Demonstratio Math. 14 (1981), 127-133. Zbl0462.47035
2. [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. [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. [4] A. Lasota, M.C. Mackey, Globally asymptotic properties of proliterating cell populations, J. Math. Biology 19 (1984), 43-62. Zbl0529.92011
5. [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. [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. [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. [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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