A random variational inequality and its applications to random best approximation and fixed point theorems. (English) Zbl 0841.47039
Summary: A random variational inequality is established which, in turn, is used to give a random best approximation theorem for a continuous set-valued mapping defined on a compact convex subset of a normed space. As an application, random fixed points are then derived. Finally, non-compact versions are also given. Our results include the corresponding results of N. M. Sehgal and S. P. Singh [Proc. Am. Math. Soc. 95, 91-94 (1985; Zbl 0607.47057)] as special cases and our approach is different from those in the literature.
MSC:
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 47H10 | Fixed-point theorems |
| 49J40 | Variational inequalities |
| 54C60 | Set-valued maps in general topology |
| 46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |
| 42A50 | Conjugate functions, conjugate series, singular integrals |
| 47H40 | Random nonlinear operators |
| 47S50 | Operator theory in probabilistic metric linear spaces |
| 41A50 | Best approximation, Chebyshev systems |
