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1.
Yu. È. Linke 《Siberian Mathematical Journal》2011,52(3):501-511
Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂
c
Q, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: ł2 → C(X) from a separable Hilbert space ł2, where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this
means that the space L
c
(ł2, C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials
of sublinear operators of the class under consideration. 相似文献
2.
We show that Rockafellar's maximal monotonicity and maximal cyclical monotonicity theorems for subdifferentials can be reformulated and proved for the family of -subdifferentials of a proper, lower semicontinuous, convex function defined on a normed space. We also show that the subdifferential map of a lower semicontinuous convex function defined on a Banach space is bothX andX* locally maximal monotone. 相似文献
3.
We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw
*-lower semicontinuous function ϕ defined on aw
*-compact convex setC in a dual Banach spaceX
* and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw
*−H
σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets.
Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second
author. 相似文献
4.
In this paper we initiate a quantitative study of strong proximinality. We define a quantity ϵ(x, t) which we call as modulus of strong proximinality and show that the metric projection onto a strongly proximinal subspace
Y of a Banach space X is continuous at x if and only if ϵ(x, t) is continuous at x whenever t > 0. The best possible estimate of ϵ(x, t) characterizes spaces with
1 \frac121 \frac{1}{2} ball property. Estimates of ϵ(x, t) are obtained for subspaces of uniformly convex spaces and of strongly proximinal subspaces of finite codimension in C(K). 相似文献
5.
Zili Wu 《Journal of Approximation Theory》2002,119(2):181-192
We prove that in a Banach space X with rotund dual X* a Chebyshev set C is convex iff the distance function dC is regular on X\C iff dC admits the strict and Gâteaux derivatives on X\C which are determined by the subdifferential ∂x−
x" height="11" width="10"> for each xX\C and
x" height="11" width="10">PC(x)\{cC:x−c=dC(x)}. If X is a reflexive Banach space with smooth and Kadec norm then C is convex iff it is weakly closed iff PC is continuous. If the norms of X and X* are Fréchet differentiable then C is convex iff dC is Fréchet differentiable on X\C. If also X has a uniformly Gâteaux differentiable norm then C is convex iff the Gâteaux (Fréchet) subdifferential ∂−dC(x) (∂FdC(x)) is nonempty on X\C. 相似文献
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6.
Cyclic Hypomonotonicity,Cyclic Submonotonicity,and Integration 总被引:9,自引:1,他引:8
Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C2 function [respectively, a lower C1 function]. 相似文献
7.
P. B. Zatitskiy 《Journal of Mathematical Sciences》2009,158(6):853-857
Recall the two classical canonical isometric embeddings of a finite metric space X into a Banach space. That is, the Hausdorff–Kuratowsky embedding x → ρ(x, ⋅) into the space of continuous functions on X with the max-norm, and the Kantorovich–Rubinshtein embedding x → δ
x
(where δ
x
, is the δ-measure concentrated at x) with the transportation norm. We prove that these embeddings are not equivalent if |X| > 4. Bibliography: 2 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 153–161. 相似文献
8.
It is shown that if k(x) is an upper semicontinuous and quasi lower semicontinuous function on a Banach space X, then k(x)B
X* is the Clarke subdifferential of some locally Lipschitz function on X. Related results for approximate subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke subdifferential, one can obtain a maximal Clarke subdifferential map via its local Lipschitz-constant function. Finally, some results concerning the characterization and calculus of local Lipschitz-constant functions are developed. 相似文献
9.
V. Yu. Protasov 《Functional Analysis and Its Applications》2011,45(1):46-55
We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x
1 + x
2) ⊂ φ(x
1) + φ(x
2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is
proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces. 相似文献
10.
A variational theory for monotone vector fields 总被引:1,自引:0,他引:1
Nassif Ghoussoub 《Journal of Fixed Point Theory and Applications》2008,4(1):107-135
Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators,
but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from potentials
in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We
describe here how the selfdual variational calculus, developed recently by the author, provides a variational approach to
PDEs and evolution equations driven by maximal monotone operators. To any such vector field T on a reflexive Banach space X, one can associate a convex selfdual Lagrangian L
T
on the phase space X × X
* that can be seen as a “potential” for T, in the sense that the problem of inverting T reduces to minimizing a convex energy functional derived from L
T
. This variational approach to maximal monotone operators allows their theory to be analyzed with the full range of methods—computational
or not—that are available for variational settings. Standard convex analysis (on phase space) can then be used to establish
many old and new results concerned with the identification, superposition, and resolution of such vector fields.
Dedicated to Felix Browder on his 80th birthday 相似文献
11.
We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave
majorant of the modulus of continuity, we obtain the Grüss inequality for the functional L(f) = H(f; x), where H:C[a, b] → C[a, b] is a positive linear operator and x ∈ [a, b] is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the Hermite–Fejér interpolation
operator, and convolution-type operators. Moreover, we deduce Grüss-type inequalities using the Cauchy mean-value theorem,
thus generalizing results of Chebyshev and Ostrowski. The Grüss inequality on a compact metric space for more than two functions
is given, and an analogous Ostrowski-type inequality is obtained. The latter, in turn, leads to one further version of the
Grüss inequality. In the appendix, we prove a new result concerning the absolute first-order moments of the classic Hermite–Fejér
operator. 相似文献
12.
In this paper we study C0-semigroups on X × Lp( − h, 0; X) associated with linear differential equations with delay, where X is a Banach space. In the case that X is a Banach lattice with order continuous norm, we describe the associated modulus semigroup, under minimal assumptions on
the delay operator. Moreover, we present a new class of delay operators for which the delay equation is well-posed for p in a subinterval of [1,∞).
Dedicated to the memory of H. H. Schaefer 相似文献
13.
J. M. A. M. van Neerven 《Semigroup Forum》1991,43(1):378-394
The adjoint of aC
0-semigroup on a Banach spaceX induces a locally convex σ(X,X
ℴ)-topology inX, which is weaker than the weak topology ofX. In this paper we study the relation between these two topologies. Among other things a class of subsets ofX is given on which they coincide. As an application, an Eberlein-Shmulyan type theorem is proved for the σ(X,X
ℴ)-topology and it is shown that the uniform limit of σ(X,X
ℴ)-compact operators is σ(X,X
ℴ)-compact. Finally our results are applied to the problem when the Favard class of a semigroup equals the domain of the infinitesimal
generator. 相似文献
14.
We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents).
For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini
derivative of f.
Received: April 27, 2001 / Accepted: November 6, 2001?Published online April 12, 2002 相似文献
15.
We study finite set-theoretic solutions (X,r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over ℂ with multipermutation
level two can be put in diagonal form with the associated Yang-Baxter algebra
A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) having a q-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots
of a prescribed form appear as determined by the representation theory of the finite abelian group G\mathcal{G} of left actions on X. We study the structure of
A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) and show that they have a ∙-product form ‘quantizing’ the commutative algebra of polynomials in |X| variables. We obtain the ∙-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite
product for a certain crossed G\mathcal{G}-module (over any field k). We provide first steps in the noncommutative differential geometry of A(k,X,r)\mathcal{A}(k,X,r) arising from these results. As a byproduct of our work we find that every such level 2 solution (X,r) factorises as r = f ∘ τ ∘ f
− 1 where τ is the flip map and (X,f) is another solution coming from X as a crossed G\mathcal{G}-set. 相似文献
16.
LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K. 相似文献
17.
Nicuşor Costea Daniel Alexandru Ion Cezar Lupu 《Journal of Optimization Theory and Applications》2012,155(1):79-99
The aim of this paper is to establish existence results for some variational-like inequality problems involving set-valued maps, in reflexive and nonreflexive Banach spaces. When the set K, in which we seek solutions, is compact and convex, we no dot impose any monotonicity assumptions on the set-valued map A, which appears in the formulation of the inequality problems. In the case when K is only bounded, closed, and convex, certain monotonicity assumptions are needed: We ask A to be relaxed η-α monotone for generalized variational-like inequalities and relaxed η-α quasimonotone for variational-like inequalities. We also provide sufficient conditions for the existence of solutions in the case when K is unbounded, closed, and convex. 相似文献
18.
Zili Wu 《Journal of Mathematical Analysis and Applications》2003,282(2):629-647
For a nonempty closed set C in a normed linear space X with uniformly Gâteaux differentiable norm, it is shown that the distance function dC is strictly differentiable at x∈X?C iff it is regular at x iff its modified upper or lower Dini subdifferential at x is a singleton iff its upper or lower Dini subdifferential at x is nonempty iff its upper or lower Dini derivative at x is subadditive. Moreover if X is a Hilbert space, then dC is Fréchet differentiable at x∈X?C iff its Fréchet subdifferential at x is nonempty. Many characteristics of proximally smooth sets and convex closed sets in a Hilbert space are also given. 相似文献
19.
Among normed linear spacesX of dimension ≧3, finite-dimensional Hilbert spaces are characterized by the condition that for each convex bodyC inX and each ballB of maximum radius contained inC,B’s center is a convex combination of points ofB ∩ (boundary ofC). Among reflexive Banach spaces of dimension ≧3, general Hilbert spaces are characterized by a related but weaker condition
on inscribed balls.
Research of the first author was partially supported by the U.S. National Science Foundation. Research of the second and third
authors was supported by the Consiglio Nazionale delle Ricerche and the Ministero della Pubblica Istruzione of Italy, while
they were visiting the University of Washington, Seattle, USA. 相似文献
20.
Let X be a locally convex Hausdorff space and let C0(S,X) be the space of all continuous functions f:S→X, with compact support on the locally compact space S. In this paper we prove a Riesz representation theorem for a class of bounded operators T:C0(S,X)→X, where the representing integrals are X-valued Pettis integrals with respect to bounded signed measures on S. Under the additional assumption that X is a locally convex space, having the convex compactness property, or either, X is a locally convex space whose dual X′ is a barrelled space for an appropriate topology, we obtain a complete identification between all X-valued Pettis integrals on S and the bounded operators T:C0(S,X)→X they represent. Finally we give two illustrations of the representation theorem proved, in the particular case when X is the topological dual of a locally convex space. 相似文献