THE EXISTENCE THEOREM OF THE CONE-WEAK SUBDIFFERENTIAL OF SET-VALUED MAPPING

In this paper, the existence theorem of the cone-weak subdiflerential of set-valued mapping in locally convex topological vector space is proved.     

Lipschitz continuity of an approximate solution mapping for parametric set-valued vector equilibrium problems

The purpose of this paper is to generalize and improve some topological properties of solutions set to the set-valued vector equilibrium problems by using the scalar characterization method. Moreover, the Lipschitz continuity of an approximate solution mapping for the parametric set-valued vector equilibrium problems is studied.     

A necessary extremality condition for a set-valued stochastic control problem

For a controlled stochastic dynamic system with set-valued drift coefficient and a terminal cost functional we derive a necessary extremality condition in form of a minimum principle.     

A selection theorem for a new class of set-valued maps

A new class of set-valued maps that includes all upper and lower semicontinuous set-valued maps is introduced. For this class, a selection theorem having applications in the theory of differential inclusions is presented. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 503–507, October, 1999.     

Minimax theorems for set-valued maps without continuity assumptions

We introduce several classes of set-valued maps with new generalized convexity properties. We also obtain minimax theorems for set-valued maps which satisfy these convexity assumptions and which are not continuous. Our method consists of the use of a fixed point theorem for weakly naturally quasiconcave set-valued maps, defined on a simplex in a topological vector space, or of a constant selection of quasiconvex set-valued maps.     

粘结引理对于集值映射的推广

粘结引理是分析中的一个基本定理,在诸多领域都有重要应用.将单值映射情形下的粘结引理,推广至集值映射情形,它在集值分析中具有进一步的应用.     

A degree condition for the existence of k-factors

Let k be an integer such that ≦, and let G be a connected graph of order n with ≦, kn even, and minimum degree at least k. We prove that if G satisfies max(deg(u), deg(v)) ≦ n/2 for each pair of nonadjacent vertices u, v in G, then G has a k-factor.     

The existence of contingent epiderivatives for set-valued maps

This short note deals with the issue of existence of contingent epiderivatives for set-valued maps defined from a real normed space to the real line. A theorem of Jahn-Rauh [1], given for the existence of contingent epiderivatives, is used to obtain more general existence results. The strength and the limitations of the main result are discussed by means of some examples.     

Approximative condition for the continuity of a fractional derivative

For the spaceC of functions having a Marchaud continuous fractional derivative of order > 0 on the closed interval 0, 1] and for the function class it is proved that if and only if .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1719–1722, December, 1992.     

Approximation to a set-valued mapping,I: A proposal

Given a multivalued mappingF, we address the problem of finding another multivalued mappingS that agrees locally withF. We limit ourselves to the case whenF(x) is convex compact for eachx. We propose anS obtained by linearizing the support function ofF(x) and give conditions under which thisS approximatesF in the sense of Hausdorff distance. We show equality betweenS and other proposals obtained from tangent cones to the graph ofF. Finally, we apply these results to the approximate subdifferential of a convex function.     

On the continuity of midpoint hull convex set-valued functions

Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).     

An open mapping principle for set-valued maps

We prove an open mapping principle for set-valued maps on Banach spaces using “kth order variations”.     

A combinatorial condition for the existence of polyhedral 2-manifolds

LetP denote a polyhedral 2-manifold, i.e. a 2-dimensional cell-complex inR ^d (d≧3) having convex facets, such that set (P) is homeomorphic to a closed 2-dimensional manifold. LetE be any subset of odd valent vertices ofP, andc _E its cardinality. Then for the numberc _P(E) of facets containing a vertex ofE the inequality 2c _P(E)≧c_E+1 is proved. This local combinatorial condition shows that several combinatorially possible types of polyhedral 2-manifolds cannot exist.     

Fixed-point theorems for set-valued mappings and existence of best approximants

The paper gives a proof of the following existence result in best approximation. Let M be a compact, convex, and non-empty subset of a normed space E and let g be a continuous almost affine mapping of M onto M. For each continuous mapping f from M into E there exists a point x in M such that g(x) is a best M- approximation to f(x). The proof uses Bohnenblust and Karlin's extension to normed spaces of Kakutani's Fixed Point Theorem for set-valued mappings on compact, convex, and non-empty subsets of Euclidean n-space.