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Random Integral Equation of the Volterra Type with Applications

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ABSTRACT
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 .
 
Cite this paper
Alafif, H. (2014) Random Integral Equation of the Volterra Type with Applications. Journal of Applied Mathematics and Physics, 2, 138-149. doi: 10.4236/jamp.2014.25018.
 
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                                                                                           eww150120lx
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