Nonlinear random equations with noncoercive operators in Banach spaces

Abstract

admits random solutions. In [ 121 Itoh proved the existence of solutions of nonlinear random equations with monotone operators. His method of proving the measurability of solutions was based on the selection theorem of Kuratowski and Ryll-Nardzewski [16]. In [13, 141 Itoh’s method was extended to obtain existence theorems for nonlinear random equations and inequalities with single-valued or multivalued operators of monotone type. The above results were derived under the basic assumption that the random operators are coercive. It is the purpose of this paper to give some new existence results concerning the solvability of nonlinear random equations with noncoercive operators. As in the coercive case, the measurability of solutions depends mainly on the selection theorem proved in [ 161. In Section 3, we prove the existence of solutions of nonlinear random equations with multivalued operators satisfying a Leray-Schauder-type boundary condition, generalizing thus some results in [ 151. In Section 4, we study the solvability of the random equation