Nonlinear random equations involving operators of monotone type

Abstract

The systematic study of random equations has been initiated by SpaEek [19] and Hans [9]. In recent years Kannan and Salehi [13] and Itoh [ 111 have treated nonlinear random equations with monotone operators. In [ 14, 151 the author studied nonlinear random equations and inequalities with singlevalued or multivalued operators of monotone type. It is the purpose of this paper to give some new existence theorems for nonlinear random equations with operators of monotone type. More precisely, in Section 3, we present an existence and perturbation theory for solutions of nonlinear random equations involving multivalued maximal monotone operators. In Section 4 we study a random Hammerstein equation in a Hilbert space involving a closed linear maximal monotone random operator and a random operator of type (M). In Section 5 we consider random equations with noncoercive pseudomonotone operators. Instead of coercivity a Leray-Schauder-type boundary condition is assumed.