Ordinarily approximately continuous selections

We give a characterization of set-valued mappings from \(IR^k \) into the class of real intervals admitting an ordinarily approximately continuous selection.     

Continuous selections avoiding a set

Conditions are obtained under which a set-valued function ϕ : X ⇒ 2^Y has a continuous selection which avoids a set E ⊏ Y.     

On the splitting problem for selections

We investigate when does the Repovš-Semenov splitting problem for selections have an affirmative solution for continuous set-valued mappings in finite-dimensional Banach spaces. We prove that this happens when images of set-valued mappings or even their graphs are P-sets (in the sense of Balashov) or strictly convex sets. We also consider an example which shows that there is no affirmative solution of this problem even in the simplest case in R³. We also obtain affirmative solution of the approximate splitting problem for Lipschitz continuous selections in the Hilbert space.     

New Criteria for the Existence of a Continuous <Emphasis Type="Italic">ε</Emphasis>-Selection

We study sets admitting a continuous selection of near-best approximations and characterize those sets in Banach spaces for which there exists a continuous ε-selection for each ε > 0. The characterization is given in terms of P-cell-likeness and similar properties. In particular, we show that a closed uniqueness set in a uniformly convex space admits a continuous ε-selection for each ε > 0 if and only if it is B-approximately trivial. We also obtain a fixed point theorem.     

Characterization and recognition of d.c. functions

A function ${f : \Omega \to \mathbb{R}}$ , where Ω is a convex subset of the linear space X, is said to be d.c. (difference of convex) if f =  g ? h with ${g, h : \Omega \to \mathbb{R}}$ convex functions. While d.c. functions find various applications, especially in optimization, the problem to characterize them is not trivial. There exist a few known characterizations involving cyclically monotone set-valued functions. However, since it is not an easy task to check that a given set-valued function is cyclically monotone, simpler characterizations are desired. The guideline characterization in this paper is relatively simple (Theorem 2.1), but useful in various applications. For example, we use it to prove that piecewise affine functions in an arbitrary linear space are d.c. Additionally, we give new proofs to the known results that C ^1,1 functions and lower-C ² functions are d.c. The main goal remains to generalize to higher dimensions a known characterization of d.c. functions in one dimension: A function ${f : \Omega \to \mathbb{R}, \Omega \subset \mathbb{R}}$ open interval, is d.c. if and only if on each compact interval in Ω the function f is absolutely continuous and has a derivative of bounded variation. We obtain a new necessary condition in this direction (Theorem 3.8). We prove an analogous sufficient condition under stronger hypotheses (Theorem 3.11). The proof is based again on the guideline characterization. Finally, we obtain results concerning the characterization of convex and d.c. functions obeying some kind of symmetry.     

A theorem on martingale selection for relatively open convex set-valued random sequences

For set-valued random sequences (G _n) _n=0 ^N with relatively open convex values G _n(ω), we prove a new test for the existence of a sequence (x _n) _n=0 ^N of selectors adapted to the filtration and admitting an equivalent martingale measure. The statement is formulated in terms of the supports of regular upper conditional distributions of G _n. This is a strengthening of the main result proved in our previous paper [1], where the openness of the set G _n(ω) was assumed and a possible weakening of this condition was discussed.     

LFC bumps on separable Banach spaces

In this note we construct a C^∞-smooth, LFC (Locally depending on Finitely many Coordinates) bump function, in every separable Banach space admitting a continuous, LFC bump function.     

Hadamard Inequality and a Refinement of Jensen Inequality for Set—Valued Functions

We present the following set-valued analogue of the Hadamard inequality: Let Y be a Banach space, I be an open interval and let F: I ? cl(Y) be a continuous and convex set-valued function. Then $${F(a)+F(b)\over 2}\subset{1\over b-a}\int_a^b\ F(x)dx\subset F \bigg({a+b\over 2}\bigg)$$ , for every a, b ∈ I, a < b. Some refinement of Jensen inequality for set-valued functions is also given.     

Linear selections for the metric projection

A characterization is given of those proximinal subspaces of a normed linear space whose (set-valued) metric projections admit linear selections. This characterization is applied in each of the classical Banach spaces C₀(T) and L_p (1 ? p ? ∞), resulting in an intrinsic characterization of those one-dimensional subspaces whose metric projections admit linear selections.     

Smoothing of bump functions

Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C^∞-smooth bump which depends locally on finitely many coordinates.     

Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f:X→Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C¹-smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C¹-smooth mappings together with their first derivatives.     

Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces

In real ordered linear spaces, an equivalent characterization of generalized cone subconvexlikeness of set-valued maps is firstly established. Secondly, under the assumption of generalized cone subconvexlikeness of set-valued maps, a scalarization theorem of set-valued optimization problems in the sense of ?-weak efficiency is obtained. Finally, by a scalarization approach, an existence theorem of ?-global properly efficient element of set-valued optimization problems is obtained. The results in this paper generalize and improve some known results in the literature.     

Variational characterization of the contingent epiderivative

In this paper the existence of the contingent epiderivative of a set-valued map is studied from a variational perspective. We give a variational characterization of the ideal minimal of a weakly compact set. As a consequence we characterize the existence of the contingent epiderivative in terms of an associated family of variational systems. When a set-valued map takes values in Rn we show that these systems can be formulated in terms of the contingent epiderivatives of scalar set-valued maps. By applying these results we extend some existing theorems.     

Fat homeomorphisms and unbounded derivate containers

We generalize local and global inverse function theorems to continuous transformations in $\text{R}$ⁿ, replacing nonexistent derivatives by set-valued “unbounded derivate containers.” We also construct and study unbounded and ordinary derivate containers, including extensions of generalized Jacobians.     

Properties of P-sets and trapped compact convex sets

New properties of P-sets, which constitute a large class of convex compact sets in ? ⁿ that contains all convex polyhedra and strictly convex compact sets, are obtained. It is shown that the intersection of a P-set with an affine subspace is continuous in the Hausdorff metric. In this theorem, no assumption of interior nonemptiness is made, unlike in other known intersection continuity theorems for set-valued maps. It is also shown that if the graph of a set-valued map is a P-set, then this map is continuous on its entire effective set rather than only on the interior of this set. Properties of the so-called trapped sets are also studied; well-known Jung’s theorem on the existence of a minimal ball containing a given compact set in ? ⁿ is generalized. As is known, any compact set contains n + 1 (or fewer) points such that any translation by a nonzero vector takes at least one of them outside the minimal ball. This means that any compact set is trapped in the minimal ball. Compact sets trapped in any convex compact sets, rather than only in norm bodies, are considered. It is shown that, for any compact set A trapped in a P-set M ? ? ⁿ , there exists a set A ⁰ ? A trapped in M and containing at most 2n elements. An example of a convex compact set M ? ? ⁿ for which such a finite set A ⁰ ? A does not exist is given.     

A Sard theorem for tame set-valued mappings

If F is a set-valued mapping from Rn into Rm with closed graph, then y∈Rm is a critical value of F if for some x with y∈F(x), F is not metrically regular at (x,y). We prove that the set of critical values of a set-valued mapping whose graph is a definable (tame) set in an o-minimal structure containing additions and multiplications is a set of dimension not greater than m−1 (respectively a σ-porous set). As a corollary of this result we get that the collection of asymptotically critical values of a set-valued mapping with a semialgebraic graph has dimension not greater than m−1. We also give an independent proof of the fact that a definable continuous real-valued function is constant on components of the set of its subdifferentiably critical points.     

Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces

In this paper,continuous homogeneous selections for the set-valued metric generalized inverses T of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces.Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T are given.The results are an answer to the problem posed by Nashed and Votruba.     

Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

We prove that homologically nontrivial generic smooth (2n−1)-parameter families of analytic discs in Cn, n?2, attached by their boundaries to a CR-manifold Ω, test CR-functions in the following sense: if a smooth function on Ω analytically extends into any analytic discs from the family, then the function satisfies tangential CR-equations on Ω. In particular, we give an answer (Theorem 1) to the following long standing open question, so called strip-problem, earlier solved only for special families (mainly for circles): given a smooth one-parameter family of Jordan curves in the plane and a function f admitting holomorphic extension inside each curve, must f be holomorphic on the union of the curves? We prove, for real-analytic functions and arbitrary generic real-analytic families of curves, that the answer is “yes,” if no point is surrounded by all curves from the family. The latter condition is essential. We generalize this result to characterization of complex curves in C² as real 2-manifolds admitting nontrivial families of attached analytic discs (Theorem 4). The main result implies fairly general Morera type characterization of CR-functions on hypersurfaces in C² in terms of holomorphic extensions into three-parameter families of attached analytic discs (Theorem 2). One of the applications is confirming, in real-analytic category, the Globevnik-Stout conjecture (Theorem 3) on boundary values of holomorphic functions. It is proved that a smooth function on the boundary of a smooth strictly convex domain in Cn extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain. The proofs are based on a universal approach, namely, on the reduction to a problem of propagation, from the boundary to the interior, of degeneracy of CR-foliations of solid torus type manifolds (Theorem 2.2).     

The locally 2-arc transitive graphs admitting an almost simple group of Suzuki type

In this paper, seven families of vertex-intransitive locally (G,2)-arc transitive graphs are constructed, where Sz(q)?G?Aut(Sz(q)), q=²2k+1 for some k∈N. It is then shown that for any graph Γ in one of these families, Sz(q)?Aut(Γ)?Aut(Sz(q)) and that the only locally 2-arc transitive graphs admitting an almost simple group of Suzuki type whose vertices all have valency at least three are (i) graphs in these seven families, (ii) (vertex transitive) 2-arc transitive graphs admitting an almost simple group of Suzuki type, or (iii) double covers of the graphs in (ii). Since the graphs in (ii) have been classified by Fang and Praeger (1999) [6], this completes the classification of locally 2-arc transitive graphs admitting a Suzuki simple group     

Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems

This paper is devoted to the Levitin–Polyak well-posedness by perturbations for a class of general systems of set-valued vector quasi-equilibrium problems (SSVQEP) in Hausdorff topological vector spaces. Existence of solution for the system of set-valued vector quasi-equilibrium problem with respect to a parameter (PSSVQEP) and its dual problem are established. Some sufficient and necessary conditions for the Levitin–Polyak well-posedness by perturbations are derived by the method of continuous selection. We also explore the relationships among these Levitin–Polyak well-posedness by perturbations, the existence and uniqueness of solution to (SSVQEP). By virtue of the nonlinear scalarization technique, a parametric gap function g for (PSSVQEP) is introduced, which is distinct from that of Peng (J Glob Optim 52:779–795, 2012). The continuity of the parametric gap function g is proved. Finally, the relations between these Levitin–Polyak well-posedness by perturbations of (SSVQEP) and that of a corresponding minimization problem with functional constraints are also established under quite mild assumptions.