Selection Theorem Research Articles

On topological existence theorems and applications to optimization-related problems

In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures. Then, we study the solution existence of a number of optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies.

Existence results for random impulsive neutral differential inclusions with lower semicontinuous

This article presents the results on existence of mild solutions for random impulsive neutral functional differential inclusions under sufficient conditions. The results are obtained by using the Krasnoselskii-Schaefer type fixed point theorem combined with the selection theorem of Berssan and Colombo for lower semicontinuous (LSC) maps with decomposable values.

Some New Coincidence Theorems in Product GFC-Spaces with Applications

We first propose a new concept of GFC-subspace. Using this notion, we obtain a new continuous selection theorem. As a consequence, we establish some new collective fixed point theorems and coincidence theorems in product GFC-spaces. Finally, we give some applications of our theorems.

Implicit Vector Integral Equations Associated with Discontinuous Operators

LetI∶=[0,1]. We consider the vector integral equationh(u(t))=ft,∫Ig(t,z),u(z),dzfor a.e.t∈I,wheref:I×J→R, g:I×I→ [0,+∞[,andh:X→Rare given functions andX,Jare suitable subsets ofRn. We prove an existence result for solutionsu∈Ls(I, Rn), where the continuity offwith respect to the second variable is not assumed. More precisely,fis assumed to be a.e. equal (with respect to second variable) to a functionf*:I×J→Rwhich is almost everywhere continuous, where the involved null-measure sets should have a suitable geometry. It is easily seen that such a functionfcan be discontinuous at each pointx∈J. Our result, based on a very recent selection theorem, extends a previous result, valid for scalar casen=1.

Generalized selection theorems for fuzzy mappings and their applications

We develop generalized selection theorems for fuzzy mappings in finite weakly convex spaces (in short, FWC-spaces) without any linear, convex, and topological structure. Applying the generalized selection theorems in noncompact FWC-spaces, we obtain new fuzzy collective coincidence point theorems and fuzzy collectively fixed point theorems. These results generalize many known theorems in the literature.

Existence of solutions for fractional differential inclusions with nonlocal Riemann-Liouville integral boundary conditions

In this paper, we discuss the existence of solutions for a boundary value problem of fractional differential inclusions with nonlocal Riemann-Liouville integral boundary conditions. Our results include the cases when the multivalued map involved in the problem is (i) convex valued, (ii) lower semicontinuous with nonempty closed and decomposable values and (iii) nonconvex valued. In case (i) we apply a nonlinear alternative of Leray-Schauder type, in the second case we combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo, while in the third case we use a fixed point theorem for multivalued contractions due to Covitz and Nadler.

A MISCELLANY OF SELECTION THEOREMS WITHOUT CONVEXITY

Abstract. In this paper, we give sucient conditions for a mapwith nonconvex values to have a continuous selection and the se-lection extension property in LC-metric spaces under the one-pointextension property. And we apply it to weakly lower semicontinu-ous maps and generalize previous results. We also get a continuousselection theorem for almost lower semicontinuous maps with closedsub-admissible values in R-trees. 1. IntroductionIn 1956, Michael [13] stated the important and well-known selectiontheorem;Theorem 1.1. Let Xbe a paracompact space, Ybe a Banach space,and F : X ( Y be a lower semicontinuous map with closed convexvalues. Then Fhas a continuous selection; that is, there is a continuousfunction f: X!Y such that f(x) 2F(x) for all x2X.Michael???s selection theorem has been extended to spaces with a gen-eralized de nition of convexity and maps with a weaker notion of lowersemicontinuity. The following selection theorems are examples of suchresults in LC-metric spaces, hyperconvex spaces and R-trees;Theorem 1.2. (Gutev [5]) Let Xbe a paracompact space, Y be aBanach space, and F: X( Y be a weakly lower semicontinuous mapwith closed convex values. then Fhas a continuous selection.

Lower semi-continuous differential inclusions with p-Laplacian

The existence of solutions to lower semicontinuous, closed-valued differential inclusions with p-Laplacian is investigated under various growth conditions. Proofs exploit the Bressan-Colombo- Fryszkowski Continuous Selection Theorem and fixed point arguments.

Filippov’s selection theorem and the existence of solutions for optimal control problems in time scales

We obtain an extension of Filippov’s Lemma to control systems in time scales, where the multifunction associated with the dynamics is delta measurable on time and Lipschitz on the state variable. Using a modified version of a recent result on compactness of the set of trajectories for inclusions on time scales, in combination with the Filippov’s selection theorem obtained here, we prove the existence of solutions to optimal control problems in time scales. At the outset, we provide some measurability properties for functions and multifunctions defined in time scales.

Some existence results for differential inclusions of fractional order with nonlocal strip conditions

In this paper, the existence of solutions for differential inclusions of fractional order with nonlocal strip conditions is investigated. Our study includes two cases: (i) the multivalued map involved in the problem is not necessarily convex valued, (ii) the multivalued map consists of non-convex values. We combine the nonlinear alternative of Leray-Schauder type coupled with the selection theorem of Bressan and Colombo to establish the first result, while the second result relies on Wegrzyk’s fixed point theorem for generalized contractions. MSC:34A08, 34B10, 34B15.

Approximate selection theorems with n-connectedness

We establish new approximate selection theorems for almost lower semicontinuous multimaps with n-connectedness. Our results unify and extend the approximate selection theorems in many published works and are applied to topological semilattices with path-connected intervals. MSC:54C60, 54C65, 55M10.

Functional inequalities motivated by the Lax–Milgram theorem

By adopting some ideas of the theory of functional inequalities we obtain a modification of the Lax–Milgram theorem. We drop linearity assumptions by assuming that respective inequalities hold true and we impose only a mild regularity assumption. Moreover, we do not assume the coercivity, however, as a consequence, our assertions are adequately weaker. One of our tools is a selection theorem for multifunctions.

An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities

In this paper, we introduce and study the system of generalized vector quasi-variational-like inequalities in Hausdorff topological vector spaces, which include the system of vector quasi-variational-like inequalities, the system of vector variational-like inequalities, the system of vector quasi-variational inequalities, and several other systems as special cases. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the g-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for the system of generalized vector quasi-variational-like inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.

Set valued F-variational inequalities and set valued vector F-complementarity problems

In this work, we study a new type of set-valued vector F-variational inequalities and a new kind of set valued vector F-complementarity problem in Hausdorff topological vector spaces. We establish the equivalence between the set valued vector F -variational inequalities and set valued vector F -complementarity problems under certain conditions. By considering the existence of solutions for the vector F -variational inequalities and using the continuous selection theorem, we obtain some new existence theorems of solutions for the set valued vector F-variational inequalities and set valued vector F -complementarity problems, respectively.

On a simultaneous selection theorem

On a simultaneous selection theorem

Anℓ1-oracle inequality for the Lasso in finite mixture Gaussian regression models

We consider a finite mixture of Gaussian regression models for high-dimensional heterogeneous data where the number of covariates may be much larger than the sample size. We propose to estimate the unknown conditional mixture density by an l 1 -penalized maximum likelihood estimator. We shall provide an l 1 -oracle inequality satisfied by this Lasso estimator with the Kullback–Leibler loss. In particular, we give a condition on the regularization parameter of the Lasso to obtain such an oracle inequality. Our aim is twofold: to extend the l 1 -oracle inequality established by Massart and Meynet [12] in the homogeneous Gaussian linear regression case, and to present a complementary result to Stadler et al. [18], by studying the Lasso for its l 1 -regularization properties rather than considering it as a variable selection procedure. Our oracle inequality shall be deduced from a finite mixture Gaussian regression model selection theorem for l 1 -penalized maximum likelihood conditional density estimation, which is inspired from Vapnik’s method of structural risk minimization [23] and from the theory on model selection for maximum likelihood estimators developed by Massart in [11].

Equilibrium existence under generalized convexity

Abstract We introduce, in the first part, the notion of weakly convex pair of correspondences, we give its economic interpretation, we state a fixed point and a selection theorem. Then, by using a tehnique based on a continuous selection, we prove existence theorems of quilibrium for an abstract economy. In the second part, we define the weakly biconvex correspondences, we prove a selection theorem and we also demonstrate the existence of equilibrium for a generalized quasi-game (2003 Kim's model). In the last part of the paper, we give other applications in the game theory, finding equilibrium for abstract economies having correspondences with weakly convex graph. We show that the equilibrium exists without continuity assumptions.

ON A THEOREM OF ARVANITAKIS

Arvanitakis established recently a theorem which is a common generalization of Michael's convex selection theorem and Dugundji's extension theorem. In this note we provide a short proof of a more general version of Arvanitakis' result.

Countable Random Sets: Uniqueness in Law and Constructiveness

The first part of this article deals with theorems on uniqueness in law for \sigma-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two approaches on uniqueness theorems: First, the study of generators for \sigma-fields used in this context and, secondly, the analysis of hitting functions. The last section of this paper deals with the notion of constructiveness. We will prove a measurable selection theorem and a decomposition theorem for constructive countable random sets, and study constructive countable random sets with independent increments.

A nonlocal three-point inclusion problem of Langevin equation with two different fractional orders

This article studies the existence of solutions for a three-point inclusion problem of Langevin equation with two fractional orders. Our main tools of study include a nonlinear alternative of Leray-Schauder type, selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps, and a fixed point theorem for multivalued map due to Covitz and Nadler. An illustrative example is also presented. Mathematical Subject Classification 2000: 26A33; 34A12; 34A40.