Summary
Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ω) = h(t; ω) + ∫ t o k(t, τ; ω)f(τ, x(τ; ω)) dτ, whereω ∈ Ω, the supporting set of a probability measure space (Ω,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of Banach's contraction-mapping principle, it is shown here that a unique random solution of (*) exists whenf is (∈, λ)-uniformly locally Lipschitz in the same subset of the Banach space considered in [12].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. W. Anderson,Stochastic Integral Equations. Ph.D. Dissertation, University of Tennessee, 1966.
A. T. Bharucha-Reid, On random solutions of Fredholm integral equations,Bull. Amer. Math. Soc. 66, (1960), 104–109.
A. T. Bharucha-Reid, On the theory of random equations, inProceedings of Symposia in Applied Mathematics, XVI, pp. 40–69. American Mathematical Society, Providence, Rhode Island, 1964.
A. T. Bharucha-Reid,Random Integral Equations, Academic Press, New York (to appear, 1972).
M. Edelstein, An extension of Banach's contraction principle,Proc. Amer. Math. Soc. 12 (1961), 7–10.
O. Hans, Random operator equations,Proc. Fourth Berkeley Sympos. Math. Statist. and Prob., II, pp. 185–202. University of California Press, Berkeley, 1961.
T. Morozan, Stability of linear systems with random parameters,J. Differential Equations 3 (1967), 170–178.
W. J. Padgett andC. P. Tsokos, On a semi-stochastic model arising in a biological system,Math. Biosciences 9 (1970), 105–117.
W. J. Padgett andC. P. Tsokos, Existence of a solution of a stochastic integral equation in turbulence theory,J. Mathematical Phys. 12 (1971), 210–212.
W. J. Padgett andC. P. Tsokos, On a stochastic integral equation of the Volterra type in telephone traffic theory,J. Appl. Probability 8 (1971), 269–275.
N. U. Prabhu,Stochastic Processes, Macmillan, New York, 1965.
C. P. Tsokos, On a stochastic integral equation of the Volterra type,Math. Systems Theory 3 (1969), 222–231.
C. P. Tsokos, On some nonlinear differential systems with random parameters,IEEE Proc., Third Annual Princeton Conference on Information Sciences and Systems, 1969, 228–234.
C. P. Tsokos, On the classical stability theorem of Poincaré-Lyapunov with a random parameter,Proc. Japan Acad. 45 (1969), 780–785.
C. P. Tsokos, The method of V. M. Popov for differential systems with random parameters,J. Appl. Probability 8 (1971), 298–310.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Padgett, W.J. On a random Volterra integral equation. Math. Systems Theory 7, 164–169 (1973). https://doi.org/10.1007/BF01762234
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01762234