Fixed Point Theory, Volume 5, No. 1, 2004, 37-52 http://www.math.ubbcluj.ro/ ∼ nodeacj/sfptcj.htm AN APPLICATION OF SCHAUDER’S FIXED POINT THEOREM IN STOCHASTIC MCSHANE’S MODELLING GH. CONSTANTIN AND R. NEGREA Faculty of Mathematics and Informatics West University of Timi¸ soara Bvd. V. Parvan, 300223 Timi¸ soara, Romania e-mail: gconst@math.uvt.ro, negrea@math.uvt.ro Abstract. The aim of this work is to establish an existence theorem for solutions of Mc- Shane’s stochastic systems, applying Schauder’s fixed point theorem. Also, we give new conditions relative to the coefficients for the continuity of solution with respect to the initial condition and respectively, the problem of parametric dependence of the solution process on the coefficients in McShane stochastic integral equations, generalizing the results of [1], [3], [4]. A short comment on continuous dependence of the solution on the disturbance and on modelling problems is given. Key Words and Phrases: Stochastic integral equation of McShane’s type, McShanes’s stochastic belated integral, Schauder’s fixed point theorem. 2000 Mathematics Subject Classification: 60H10, 47H10. 1. The mathematical modelling of several real-world problems leads to differential systems that involve radomness due to ignorance or uncertainties. In the formulation of a mathematical model for a physical, biological or economical problems, we make errors in constructing the coefficients and errors in the initial conditions. For theoretical purposes it is sufficient to know that the change in the solution can be made arbitrary small by making the change in the coefficients and the initial values sufficiently small. We consider families of stochastic integral equation systems of McShane type X i λ (t, ω)= α i λ (t, ω)+ r  j =1  t 0 g i λ,j (s, X λ (s, ω))dz j (s, ω)+ 37