Subdifferential characterization of s-lower regular functions

As for Moreau envelopes of primal lower nice as well as prox-regular functions, Moreau s-envelopes of s-lower regular functions have been proved recently to have several remarkable differential properties and to have many important applications. Here, we provide a subdifferential characterization of extended real-valued s-lower regular functions on Banach spaces in terms of a hypomonotonicity-like property of the subdifferential.     

Spaces of lower semicontinuous set-valued maps

We give a negative answer to Problem 2 posed by R. A. McCoy in his paper [McCoy R. A.: Spaces of lower semicontinuous set-valued maps II, Math. Slovaca 60 (2010), 541–570]. Some topological properties of the space L ^?(X) introduced in [McCOY R. A.: Spaces of lower semicontinuous set-valued maps I, Math. Slovaca 60 (2010), 521–540] equipped with the Vietoris topology are also investigated.     

Generalized convexities of lower semicontinuous functions

In this paper a general theorem on the replacement of the condition “for all λ” in the definition of generalized convexity properties of lower semicontinuous functions by the condition “there exists a λ” is shown. This result can be applied to a number of special kinds of convexity and completes, for instance, studies of Behbikgeb concerning (explicitly) quasiconvex functions.     

Topological characterization of the approximate subdifferential in the finite-dimensional case

Topological properties of the approximate Subdifferential introduced by Mordukhovich are studied. Apart from formulating a sufficient condition for connectedness, it is shown that, up to homeomorphy, each compact subset of ^p may occur as the approximate subdifferential of some Lipschitz function. Furthermore, even an exact result is possible when considering the partial approximate Subdifferential, which was introduced as a parametric extension by Jourani and Thibault: Given any compact subset of ^p, there is a locally Lipschitzian function realizing this set as its partial approximate Subdifferential at some predefined point.This research is supported by the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung of the Deutsche Forschungsgemeinschaft. The paper is the written version of a lecture given at theMinisymposium on Stochastic Programming which was held at the Humboldt University of Berlin in January 1994.     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.     

Spaces of lower semicontinuous set-valued maps II

We introduce a lower semicontinuous analog, L ⁻(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ⁻(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ⁻(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ⁻(X) and L ⁻(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ⁻(X) and L ⁻(Y) can be characterized by a unique factorization.     

The structure of approximate solutions of variational problems without convexity

In this work we study the structure of approximate solutions of variational problems with continuous integrands f:[0,∞)×Rn×Rn→R1 which belong to a complete metric space of functions. We do not impose any convexity assumption. The main result in this paper deals with the turnpike property of variational problems. To have this property means that the approximate solutions of the problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.     

A note on the Conley attractors of lower semicontinuous multifunctions

We discuss Conley-type approach to attractive sets for lower semicontinuous multifunctions. Since every iterated function system induces a Barnsley–Hutchinson multifunction which is l.s.c. in such a case it is much more natural to consider a multifunctions of that type then closed relations on compact spaces earlier considered by some authors. We use topological (Kuratowski’s) limit instead of commonly used Hausdorff metric.     

An approximate lower tolerance bound for the three-parameter Weibull applied to lumber property characterization

We investigate a large sample approach for obtaining tolerance bounds where the underlying population is a three-parameter Weibull distribution. Accurate tolerance bounds could play an important role in the development of lumber standards. Properties of the maximum likelihood based approach are compared with those of the standard nonparametric tolerance procedure. The asymptotic normal approximation to the tolerance bound was found to be inadequate for most of the cases considered.     

Full analogy of Sharkovsky's theorem for lower semicontinuous maps

A full analogy of the celebrated Sharkovsky cycle coexistence theorem is established for lower semicontinuous (multivalued) maps on metrizable linear continua. This result is further extended to triangular maps.     

Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959-1000] obtained the strong convergence of uniform bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein-Weierstrass theorem, and obtaining the strong convergence of uniform L^∞ or bounded viscosity solutions for scalar conservation law without convexity.     

Characterization and representation of the lower semicontinuous envelope of the elastica functional

We characterize the lower semicontinuous envelope of the functional , defined on boundaries of sets , where κ_∂E denotes the curvature of ∂E and p>1. Through a desingularization procedure, we find the domain of and its expression, by means of different representation formulas.     

Computation of the lower semicontinuous envelope of integral functionals and non-homogeneous microstructures

A numerical method is established for computing the weakly lower semicontinuous envelope of integral functionals with non-quasiconvex integrands. The convergence of the method is proved and it is shown that the method is capable of capturing curved and non-homogeneous microstructures. Numerical examples are given to show the effectiveness of the method for capturing curved and non-homogeneous laminated microstructures.     

A lower bound for the convexity number of some graphs

Given a connected graphG, we say that a setC ?V(G) is convex inG if, for every pair of verticesx, y ∈ C, the vertex set of everyx-y geodesic inG is contained inC. The convexity number ofG is the cardinality of a maximal proper convex set inG. In this paper, we show that every pairk, n of integers with 2 ≤k ≤ n?1 is realizable as the convexity number and order, respectively, of some connected triangle-free graph, and give a lower bound for the convexity number ofk-regular graphs of ordern withn>k+1.     

下半连续函数的Proximal—次微分与广义中值定理

该文对定义在Hilbert空间E上的一般下半连续函数证明了如［9］中形式的逼近中值定理在Proximal-次微分意义下也成立．若E＝［a，b］R，则得到了不等式形式的中值定理．作为应用给出了函数凸性、Lipschitz性质及常数性质的Proximal-次微分刻划．     

Unbounded critical points for a class of lower semicontinuous functionals

For a general class of lower semicontinuous functionals, we prove existence and multiplicity of critical points, which turn out to be unbounded solutions to the associated Euler equation. We apply a nonsmooth critical point theory developed in [10], [12] and [13] and applied in [8], [9] and [20] to treat the case of continuous functionals.     

Another characterization of convexity for set-valued maps

We give a new necessary and sufficient condition for convexity of a set-valued map F between Banach spaces. It is established for a closed map F having nonconvex values. The main tool in this paper is the coderivative of F which is constructed with the help of an abstract subdifferential notion of Penot . A detailed discussion is devoted to special cases when the contingent, the Fréchet and the Clarke-Rockafellar subdifFerentials Sixe used as this abstract subdifferential.     

Error bounds for systems of lower semicontinuous functions in Asplund spaces

In this paper, using the Fréchet subdifferential, we derive several sufficient conditions ensuring an error bound for inequality systems in Asplund spaces. As an application we obtain in the context of Banach spaces a global error bound for quadratic nonconvex inequalities and we derive necessary optimality conditions for optimization problems.     

Subdifferential set of the supremum of lower semi-continuous convex functions and the conical hull intersection property

In this paper we give some characterizations for the subdifferential set of the supremum of an arbitrary (possibly infinite) family of proper lower semi-continuous convex functions. This is achieved by means of formulae depending exclusively on the (exact) subdifferential sets and the normal cones to the domains of the involved functions. Our approach makes use of the concept of conical hull intersection property (CHIP, for short). It allows us to establish sufficient conditions guarantying explicit representations for this subdifferential set at any point of the effective domain of the supremum function. Research supported by grant SB2003-0344 of SEUI (MEC), Spain.     

Approximate selections of almost lower semicontinuous multimaps in C-spaces

We study the relations of almost lower semicontinuity, lower semicontinuity and other generalized lower semicontinuity; then we establish a new approximate selection theorem for almost lower semicontinuous multimaps with the generalized Zima type condition in C$C$-spaces. Our result unify and extend the approximate selection theorems in many published works.