Dimitrios Kravvaritis

Nonlinear random equations involving operators of monotone type

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.

Perturbations of Maximal Monotone Random Operators

Abstract Let X be a Banach space, X∗ its dual, and Ω a measurable space. We study the solvability of nonlinear random equations involving operators of the form L + T, where L is a maximal monotone random operator from Ω × X into X∗ and T : Ω × X → X∗ a random operator of monotone type.

Hereditary order convexity inL(X,Y)

LetX be a topological vector space,Y an ordered topological vector space andL(X,Y) the space of all linear and continuous mappings fromX intoY. The hereditary order-convex cover [K]h of a subsetK ofL(X,Y) is defined by [K]h={A∈L(X,Y):Ax∈[Kx] for allx∈X}, where[Kx] is the order-convex ofKx. In this paper we study the hereditary order-convex cover of a subset ofL(X,Y). We show how this cover can be constructed in specific cases and investigate its structural and topological properties. Our results extend to the spaceL(X,Y) some of the known properties of the convex hull of subsets ofX*.

Nonlinear random equations with noncoercive operators in Banach spaces

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