Random differential inclusions depending on a parameter
The paper deals with the random multi-valued Cauchy problem \({d\over dt} x(\omega,t)\in F(\omega,t,x(\omega,t),\lambda)\), \(x(\omega,a)=\eta(\omega)\). Here \(F\) is a multifunction from \(\Omega\times[a,b]\times X\times\Lambda\) into \(X\), where \((\Omega,{\mathfrak F},\mu)\) is a complete probability space, \(X\) is a separable Banach space, \(\Lambda\) is a first countable topological space, and \(\eta\) is a measurable function from \(\Omega\) into \(X\). Let \(\Gamma(\eta,\lambda)\) be the set of all random solutions of the problem above. The main result, under suitable hypotheses, insures the lower semicontinuity of the multifunction \(\lambda\to\Gamma(\eta,\lambda)\). It extends to random differential inclusions a previous theorem by \textit{O. Naselli Ricceri} and \textit{B. Ricceri} [Bull. Pol. Acad. Sci., Math. 37, 7-12 (1989)] and improves a result by \textit{A. Nowak} [ibid. 34, 487-494 (1986; Zbl 0617.60059)]. A further result deals with the lipschitzianity of the multifunction \((\eta,\lambda)\to\Gamma(\eta,\lambda)\). It seems that this paper gives a contribution to random differential equations, along directions not previously studied.
scientific article; zbMATH DE number 57291
scientific article