Set-valued Case Research Articles

Representation theorems, set-valued and fuzzy set-valued Ito integral

In this paper, we shall present representation theorems of set-valued martingales and set-valued processes of finite variation with continuous time. We shall also obtain a representation theorem of a predictable set-valued stochastic process. We shall give a new definition of Ito integral of a set-valued stochastic process with respect to a Brownian motion based on the work [E.J. Jung, J.H. Kim, On set-valued stochastic integrals, Stochastic Anal. Appl. 21(2) (2003) 401–418.]. We shall also discuss some properties of set-valued Ito integral, especially the presentation theorem of set-valued Ito integral. Finally, we extend some of above results to the fuzzy set-valued case.

Hahn-Banach theorems and subgradients of set-valued maps

Some new results which generalize the Hahn-Banach theorem from scalar or vector-valued case to set-valued case are obtained. The existence of the Borwein-strong subgradient and Yang-weak subgradient for set-valued maps are also proven. We present a new Lagrange multiplier theorem and a new Sandwich theorem for set-valued maps.

On least element problems for feasible sets in vector complementarity problems

The purpose of this paper is devoted to the least element problems of feasible sets for vector complementarity problems under certain conditions. We generalize the notion of a Z-map due to Riddell to the set-valued case. Some conditions of the feasible set being a sublattice are presented and the least element problems are discussed under some strict pseudomonotonicity conditions.

Spline subdivision schemes for convex compact sets

The application of spline subdivision schemes to data consisting of convex compact sets, with addition replaced by Minkowski sums of sets, is investigated. These methods generate in the limit set-valued functions, which can be expressed explicitly in terms of linear combinations of integer shifts of B-splines with the initial data as coefficients. The subdivision techniques are used to conclude that these limit set-valued spline functions have shape-preserving properties similar to those of the usual spline functions. This extension of subdivision methods from the scalar setting to the set-valued case has application in the approximate reconstruction of 3-D bodies from finite collections of their parallel cross-sections.

Generalized vector variational inequality

A generalized vector variational inequality (GVVI) is considered. We establish the existence theorem for solutions of (GVVI) with H-convexity assumption. Our existence theorem extends that of Chen [1, Theorem 3.1] to the set-valued case.

Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

A set-valued dynamical systemF on a Borel spaceX induces a set-valued operatorF onM(X) — the set of probability measures onX. We define arepresentation ofF, each of which induces an explicitly defined selection ofF; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method ofS. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation ofT-invariant measures for transformations τ, whereT corresponds to the closure of the graph of τ.

Boundary value problems and periodic solutions for contingent differential equations

It is well known that fixed point theorems play a main role in the proof of existence theorems of differential equations. The papers of Granas (see [5, 61) have extended the notion of topological degree to set-valued mappings and the fixed point theorems of Rothe and Borsuk have also been successfully established for the set-valued case. In this paper we shall prove two existence theorems for contingent differential equations by using the degree theory. The results obtained here are motivated by Theorems 2.1 and 2.2, of [8, p. 4131 and are also generalizations of the results in [4, lo].

Weak Convergence of Set-Valued Functions and Control

Weak convergence on the space of integrably bounded set-valued functions is defined. Generalizations of results of weak convergence of real-valued integrable functions are obtained in the set-valued case. The results are applied to the characterization of the continuous dependence of the attainable set of a linear control system on the restraint set. We show that the weak convergence of the restraint set is a sufficient condition for the uniform convergence of the attainable set, and under the additional condition of uniform integrability the weak convergence is also a necessary condition.