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

• Autorzy: Henryk Gacki
• Języki publikacji: EN

Abstrakty

EN

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).

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 1996
• Tom: 16
• Numer: 1
• Strony: 43-51

Twórcy

autor

Henryk Gacki

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

Bibliografia

• [1] J. Banaś, Measure of noncompactness in the space of continuous tempered functions, Demonstratio Math. 14 (1981), 127-133.
• [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.
• [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).
• [4] A. Lasota, M.C. Mackey, Globally asymptotic properties of proliterating cell populations, J. Math. Biology 19 (1984), 43-62.
• [5] A. Lasota, H. Gacki, Markov operators defined by Volterra type integrals with advanced argument, Annales Math. Vol. LI (1990), 155-166.
• [6] C.P. Tsokos, W.J. Padgett, Random integral equations with applications to life sciences and engineering, Academic Press, New York 1974.
• [7] W.J. Padgett, On a stochastic integral equation of Volterra type in tlehone traffic theory, Journal of Applied Probability, 8 (1971), 269-275.
• [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.