In this paper we shall present some of the most general results which have been obtained to date concerning random integral equations of the Volterra type. Some results of Tsokos [4] are given for the random integral equation; ∫ ( ) t x t w h t w k t w f x w d 0 ( ; ) = ( )+ ( ,τ ; ) τ , (τ ; ) τ (1.1) where t ≥ 0 and 1) w is appoint of Ω ; 2) h(t;w) is the stochastic free term or free random variable defined for 0 ≤ t and w ∈Ω ; 3) x(t;w) is the unknown random variable for each t ≥ 0 4) the stochastic kernel k(t,τ ;w) is defined for 0 ≤τ ≤ t < ∞ and w ∈Ω .
References
[1]
Tsokos, C.P. (1969) On a Monlinear Differential System with a Random Parameter. International Conference on System Sciences, IEEE Proceedings, Honolulu, Hawaii.
[2]
Tsokos, C.P. (1969) On Some Nonlinear Differential System with Random Parameters. IEEE Proceedings Ann. Princeton Conf. on Information Sci. and Systems, 3rd, 228-234.
[3]
Tsokos, C.P. (1969) On the Classical Stability Theorem of Poincare-Lyapunov. Proceedings of the Japan Academy, 45, 780-785. http://dx.doi.org/10.3792/pja/1195520593
[4]
Tsokos, C.P. (1969) On the Stochastic Integral Equation of the Volterra Type. Mathematical Systems Theory, 3, 222-231. http://dx.doi.org/10.1007/BF01703921
[5]
Fortet, R. (1956) Random Distributions with Application to Telephone Engineering. Proc. Berkeley Symp. Math. Statist and Probability, 3rd, 11, 81-88.
[6]
Bharucha-Reid, A.T. (1972) Random Integral Equations. Academic Press, New York.
[7]
Corduneanu, C. (1968) Admissibility with Respect to an Integral Operator and Applications. Math. Tech. Rep., Univ. of Rhode Island.
[8]
Tserpes, N.A. and Mukherjea, A. (1971) Invariant Measures on Semigroups with Closed Translations. Z. Wahr schein- lichk eits-theorie verw. Geb., 17, 33-38. http://dx.doi.org/10.1007/BF00538472
[9]
Yosida, K. (1965) Functional Analysis. Springer-Verlag, Berlin and New York.
[10]
Adomian, G. (1970) Random Operator Equations in Mathematical Physics 1. Journal of Mathematical Physics, 11.
[11]
Anderson, M.W. (1970) A Stochastic Integral Equation. SIAM Journal on Applied Mathematics, 18, 526-532.
http://dx.doi.org/10.1137/0118045
