The principle of symmetric criticality for non-differentiable mappings

Let X be a Banach space on which a symmetry group G linearly acts and let J be a G-invariant functional defined on X. In 1979, R. Palais (Comm. Math. Phys. 69 (1979) 19) gave some sufficient conditions to guarantee the so-called “Principle of Symmetric Criticality”: every critical point of J restricted on the subspace of G-symmetric points becomes also a critical point of J on the whole space X. This principle is generalized to the case where J is not differentiable within the setting which does not require the full variational structure under the hypothesis that the action of G is isometry or G is compact.     

Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities

In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11], but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11] and [12].     

Openness properties for parametric set-valued mappings and implicit multifunctions

The aim of this paper is to obtain some openness results in terms of normal coderivative for parametric set-valued mappings acting between infinite dimensional spaces. Then, implicit multifunction results are obtained by simply specializing the openness results. Moreover, we study a kind of metric regularity of the implicit multifunction. The results of the paper generalize several recent results in literature.     

Locally bounded set-valued mappings and monotone countable paracompactness

We prove that the following statements are equivalent for a space X: (1) X is monotonically countably paracompact; (2) for every metric space Y there exists an operator Φ assigning to each locally bounded mapping , a locally bounded l.s.c. mapping with ?⊂Φ(?) such that Φ(?)⊂Φ(?^′) whenever ?⊂?^′, where B(Y) is the set of all non-empty closed bounded sets of Y; (3) for every metric space Y, there exist operators Φ and Ψ assigning to each u.s.c. mapping , an l.s.c. mapping and a u.s.c. mapping with ?⊂Φ(?)⊂Ψ(?) such that Φ(?)⊂Φ(?^′) and Ψ(?)⊂Ψ(?^′) whenever ?⊂?^′.     

Continuous selections for set-valued mappings with finite-dimensional convex values

We show selection theorems for set-valued mappings with finite-dimensional convex values, one of which is a partial extension of a selection theorem due to S. Barov (2008) [1].     

Locally compact set-valued solutions for the general linear equation

Summary. We give a representation for the set-valued solutions, with closed convex and locally compact values, of a general linear equation as a sum of an additive mapping and a constant set.     

Linearly perturbed polyhedral normal cone mappings and applications

Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed polyhedra in reflexive Banach spaces. Our focus point is a positive linear independence condition, which is a relaxed form of the linear independence condition employed recently by Henrion et al. (2010) [1], and Nam (2010) [3]. The formulae obtained allow us to get new results on solution stability of affine variational inequalities under linear perturbations. Thus, our paper develops some aspects of the work of Henrion et al. (2010) [1] Nam (2010) [3] Qui (in press) [12] and Yao and Yen (2009) [6] and [7].     

Miscellaneous incidences of convergence theories in optimization and nonlinear analysis I: Behavior of solutions

We examine some connections between convergence theories and optimization. In particular we study the Lipschitzian character of the infimal value function with respect to variations of the objective function. We also study the approximate solution multifunction for convex and nonconvex objective functions.Dedicated to the many real and putative authors of La Marseillaise, on its bicentennial.     

Convergence of epigraphs and of sublevel sets

Let LSC(X) be the set of the proper lower semicontinuous extended real-valued functions defined on a metric spaceX. Given a sequence f _n in LSC(X) and a functionf LSC(X), we show that convergence of f _n tof in several variational convergence modes implies that for each , the sublevel set at height off is the limit, in the same variational sense, of an appropriately chosen sequence of sublevel sets of thef _n, at height _n approaching . The converse holds true whenever a form of stability of the sublevel sets of the limit function is verified. The results are obtained by regarding a hyperspace topology as the weakest topology for which each member of an appropriate family of excess functionals is upper semicontinuous, and each member of an appropriate family of gap functionals is lower semicontinuous. General facts about the representation of hyperspace topologies in this manner are given.     

On Hypertopologies

In this article a unified approach is presented to hypertopologies on collections of nonempty closed subsets of a Hausdorff uniform space generated by a saturated and separating family of pseudo-metrics. One identifies here a suitable topology on the family of proper, convex and lower semicontinuous functions defined on a Hausdorff locally convex space for which the Young Fenchel transform is bicontinuous. This improves a well known result due to Mosco, Joly and Beer.     

Tangency conditions for multivalued mappings

We prove that interiority conditions imply tangency conditions for two multivalued mappings from a topological space into a normed vector space. As a consequence, we obtain the lower semicontinuity of the intersection of two multivalued mappings. An application to the epi-upper semicontinuity of the sum of convex vector-valued mappings is given.     

Generic continuity of restricted weak upper semi-continuous set-valued mappings

Set-valued mappings from a topological space into subsets of a Banach space which satisfy a restricted form of weak upper semi-continuity, have particularly noteworthy properties. We establish a selection theorem for certain set-valued mappings from a (-) unfavourable topological space into subsets of a Banach space and as a consequence derive the property that restricted weak upper semi-continuous set-valued mappings which satisfy a minimality condition, from a (-) unfavourable topological space into subsets of a Banach space are single-valued and norm upper semi-continuous at the points of a residual subset of their domain.     

Ordinals and set-valued zero-selections for hyperspaces

A continuous zero-selection f for the Vietoris hyperspace F(X) of the nonempty closed subsets of a space X is a Vietoris continuous map f:F(X)→X which assigns to every nonempty closed subset an isolated point of it. It is well known that a compact space X has a continuous zero-selection if and only if it is an ordinal space, or, equivalently, if X can be mapped onto an ordinal space by a continuous one-to-one surjection. In this paper, we prove that a compact space X has an upper semi-continuous set-valued zero-selection for its Vietoris hyperspace F(X) if and only if X can be mapped onto an ordinal space by a continuous finite-to-one surjection.     

A note on a set-valued extension property

Principal result: Suppose Y is metrizable. Then: (a) if X is metrizable and A⊂X is closed, then every continuous g:A→Y extends to an l.s.c. ψ:X→K(Y); (b) Y satisfies (a) for all paracompact X if and only if Y is completely metrizable.     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

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T contains all weakly Lindelöf Banach spaces;

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l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

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T is stable under weak homeomorphisms;

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E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

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E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

A set-valued type inequality system

A set-valued type inequality system is introduced along with two solvability questions composed of existence and perturbation, and main theorems are obtained, which include three necessary and sufficient conditions concerning the existence and a continuity property concerning the perturbation. As applications, two existence criteria with respect to a single-valued type inequality system have also been obtained.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces

In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the S-iteration process recently introduced by Sahu in [14] converges strongly to a unique fixed point of a mapping T, where T is κ-strongly pseudocontractive mapping from a nonempty, closed and convex subset C of a smooth Banach space into itself. It is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of λ_i-strictly pseudocontractive mappings from C into itself. Our results extend and improve some very recent theorems in fixed point theory and variational inequality problems. Particularly, the results presented here extend some theorems of Reich (1980) [1] and Yamada (2001) [15] to a general class of λ-strictly pseudocontractive mappings in uniformly smooth Banach spaces.     

Some representations of midconvex set-valued functions

Summary In this note we establish conditions under which every midconvex set-valued function can be represented as sum of an additive function and a convex set-valued function. These results improve some theorems obtained in [8], [10] and [3]. Some results on local Jensen selections of midconvex set-valued functions are also given.     

Proto-derivative formulas for basic subgradient mappings in mathematical programming

Subgradient mappings associated with various convex and nonconvex functions are a vehicle for stating optimality conditions, and their proto-differentiability plays a role therefore in the sensitivity analysis of solutions to problems of optimization. Examples of special interest are the subgradients of the max of finitely manyC ² functions, and the subgradients of the indicator of a set defined by finitely manyC ² constraints satisfying a basic constraint qualification. In both cases the function has a property called full amenability, so the general theory of existence and calculus of proto-derivatives of subgradient mappings associated with fully amenable functions is applicable. This paper works out the details for such examples. A formula of Auslender and Cominetti in the case of a max function is improved in particular.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983 for the first author and by the National Science Foundation under grant DMS-9200303 for the second author.