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1.
General Existence Theorem of Zero Points   总被引:2,自引:0,他引:2  
Let X be a nonempty, compact, convex set in and let be an upper semicontinuous mapping from X to the collection of nonempty, compact, convex subsets of . It is well known that such a mapping has a stationary point on X; i.e., there exists a point X such that its image under has a nonempty intersection with the normal cone of X at the point. In the case where, for every point in X, it holds that the intersection of the image under with the normal cone of X at the point is either empty or contains the origin 0 n , then must have a zero point on X; i.e., there exists a point in X such that 0 n lies in the image of the point. Another well-known condition for the existence of a zero point follows from the Ky Fan coincidence theorem, which says that, if for every point the intersection of the image with the tangent cone of X at the point is nonempty, the mapping must have a zero point. In this paper, we extend all these existence results by giving a general zero-point existence theorem, of which the previous two results are obtained as special cases. We discuss also what kind of solutions may exist when no further conditions are stated on the mapping . Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.  相似文献   

2.
A family of convex sets is said to be in convex position, if none of its members is contained in the convex hull of the others. It is proved that there is a function N(n) with the following property. If is a family of at least N(n) plane convex sets with nonempty interiors, such that any two members of have at most two boundary points in common and any three are in convex position, then has n members in convex position. This result generalizes a theorem of T. Bisztriczky and G. Fejes Tóth. The statement does not remain true, if two members of may share four boundary points. This follows from the fact that there exist infinitely many straight-line segments such that any three are in convex position, but no four are. However, there is a function M(n) such that every family of at least M(n) segments, any four of which are in convex position, has n members in convex position.  相似文献   

3.
A code is called (t, 2)-identifying if for all the words x, y(x y) and the sets (B t (x) B t (y)) C and are nonempty and different. Constructions of such codes and a lower bound on the cardinality of these codes are given. The lower bound is shown to be sharp in some cases. We also discuss a more general notion of -identifying codes and introduce weakly identifying codes.  相似文献   

4.
In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set XE, two multifunctions :X2X and , find such that We extend to the above problem a result established by Ricceri for the case (x)X, where in particular the multifunction is required only to satisfy the following very general assumption: each set (x) is nonempty, convex, and weakly-star compact, and for each yX–:X the set is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.  相似文献   

5.
This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equation of nth order with complex coefficients M[y] – wy = wf(t, y [0],... , y [n–1]), t [a, b) provided that all rth quasi-derivatives of solutions of M[y] – wy = 0 and all solutions of its normal adjoint are in and under suitable conditions on the function f.  相似文献   

6.
In vector optimization, topological properties of the set of efficient and weakly efficient points are of interest. In this paper, we study the connectedness of the setE w of all weakly efficient points of a subsetZ of a locally convex spaceX with respect to a continuous mappingp:X Y,Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions thatZ is convex and closed and thatp is a pointwise quasiconvex mapping (i.e., a generalized quasiconvex concept), the setE w is connected, if the lower level sets ofp are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of Refs. 1–3. Moreover, some examples in vector approximation are given.The author is grateful to Dr. D. T. Luc and to a referee for pointing out an error in an earlier version of this paper.  相似文献   

7.
Multivalued differential equations in separable Banach spaces   总被引:3,自引:0,他引:3  
This paper is concerned with multivalued differential equations of the form F(t,x), whereF is a multivalued mapping taking as its values nonempty compact, but not necessarily convex, subsets in a separable Banach space. The main result is connected with the existence of solutions of these equations.  相似文献   

8.
We prove that the efficient point set Max(Q|K) of a compact convex set QX in a Hausdorff topological vector space X ordered by a closed convex pointed cone KX with nonempty K +i:={lK\{0}:l(x)>0} is arcwise connected.  相似文献   

9.
Kashin  B. S. 《Mathematical Notes》2002,72(3-4):473-478
In this paper, we establish lower bounds for n-term approximations in the metric of L 2(I 2 ) of characteristic functions of plane convex subsets of the square I 2 with respect to arbitrary orthogonal systems. It is shown that, as n, these bounds cannot decrease more rapidly than .  相似文献   

10.
We prove the maximum principle for an optimal control problem governed by the system
with state constraint , under three different hypotheses: (H1) C is a convex set with nonempty interior; (H2) a convex set with nonempty interior in H and the evolution system satisfying compactness hypotheses; (H3) the periodic case , with the evolution system satisfying compactness hypotheses. We do not assume the controls to be bounded. We give some examples for distributed control problems.  相似文献   

11.
Let Hp be the Hardy space in the polydisk. Denote by the set of all holomorphic polynomials. A vector f ∈ Hp is called weakly cyclic if the product f is weakly dense in Hp, 0 < p < 1. We construct weakly cyclic vectors with a prescribed lower semi-continuous modulus of the boundary values. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 111–118.  相似文献   

12.
Let be a cardinal number with the usual order topology. We prove that all subspaces of 2 are weakly sequentially complete and, as a corollary, all subspaces of are sequentially complete. Moreover we show that a subspace of (1 + 1)2 need not be sequentially complete, but note that X = A × B is sequentially complete whenever A and B are subspaces of .  相似文献   

13.
In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems where –p is the p-Laplace operator, p > 1 and is a C 1,-domain in . We prove an analogue of [7, 16] for the eigenvalue problem with and obtain a non-existence result of positive solutions for the general systems.  相似文献   

14.
The purpose of this paper is to study the existence of fixed points for nonexpansive multivalued mappings in a particular class of Banach spaces. Furthermore, we demonstrate a relationship between the weakly convergent sequence coefficient WCS(X) and the Jordan–von Neumann constant CNJ(X) of a Banach space X. Using this fact, we prove that if CNJ(X) is less than an appropriate positive number, then every multivalued nonexpansive mapping has a fixed point where E is a nonempty weakly compact convex subset of a Banach space X, and KC(E) is the class of all nonempty compact convex subsets of E.  相似文献   

15.
Let (Y,,,T) be an ergodic dynamical system. LetA be an nonempty subset ofL 2() such that , whereA=sup{||sȒt||2 ,s, tA} andN(A, u) is the smallest number ofL 2()-open balls of radiusu, centered inA, enough to coverA. Let . We prove as a consequence of a more general result, thatC(A) is aGB subset ofL 2().  相似文献   

16.
Let P be a probability distribution on d and let be the family of the uniform probabilities defined on compact convex sets of d with interior non-empty. We prove that there exists a best approximation to P in , based on the L 2-Wasserstein distance. The approximation can be considered as the best representation of P by a convex set in the minimum squares setting, improving on other existent representations for the shape of a distribution. As a by-product we obtain properties related to the limit behavior and marginals of uniform distributions on convex sets which can be of independent interest.  相似文献   

17.
The measure (A) of nonconvexity for a bounded subset A of a normed linear space L is the Hausdorff distance between A and the convex hull coA. It is proved that if L is an L p-space, then , where d(A) is the diameter of A and . Furthermore, this estimate is sharp. Bibliography: 6 titles.  相似文献   

18.
A formation is called radical (weakly n-radical) if it contains every group G which can be represented in the formG=M 1M2 ...M n,M iG, where the subgroups Mi belong to (belong to and have pairwise prime indices). It is proved that a local formation is radical (weakly n-radical,n 2) if and only if its complete inner local screen f has the following property: f(p) is a radical (a weakly n-radical,n 2) formation for every prime number p.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 861–864, June, 1977.  相似文献   

19.
For the lattice point enumerator of a lattice and a convex body K we give bounds in terms of the intrinsic volumes of K and of minimal determinants of . The intrinsic volumes are the normalized Minkowski quermassintegrals and the minimal determinants are analogous functionals of .  相似文献   

20.
For a nonempty setE of nonnegative integers letH E p, q, a andH E p be the closed linear span of
  相似文献   

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