ABSTRACT
Probabilistic functional analysis is an important mathematical area of research due to its applications to probabilistic models in applied problems. Random operator theory is needed for the study of various classes of random equations. Indeed,in many cases, the mathematical models or equations used to describe phenomena in the biological, physical, engineering, and systems sciences contain certain parameters or coefficients which have specific interpretations, but whose values are unknown. Therefore, it is more realistic to consider such equations as random operator equations. These equations are much more difficult to handle mathematically than deterministic equations. Important contributions to the study of the mathematical aspects of such random equations have been undertaken in [50, 231,260] among others. The problem of fixed points for random mappings was initiated by the Prague school of probabilities. The first results were studied in 1955-1956 by S˘pac˘ek and Hans˘ in the context of Fredholm integral equations with random kernels. In a separable metric space, random fixed point theorems for contraction mappings were proved by Hans˘ [152, 153], S˘pac˘ek [268], Hans˘ and, S˘pac˘ek [154] and Mukherjee [219,220]. Then random fixed point theorems of Schauder or Krasnosel’skii type were given by Mukherjea (cf. Bharucha-Reid [50], p. 110), Prakasa Rao [246] and Bharucha-Reid [51].
