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Let K be a nonempty compact Δ-convex subset of a topological semilattice with path-connected intervals. In this paper, under new assumptions, we establish some existence theorems of x ∈ K such that τ(A) ∩ V EP(ƒ) ≠ ∅, where τ(A) is the set of all fixed points of the multifunction A : K → 2K and V EP(ƒ) is the set of all solutions for the vector equilibrium problems of the multifunction ƒ from K × K to a topological vector space Y. These results generalize and improve the recent ones in the literature. Some examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

We present some significant open problems in Euclidean spaces Rn, n ∊ N* which can be extended, mutatis mutandis, to RI for every non-empty set I. [ABSTRACT FROM AUTHOR]

In 1970, Schirmer provided some results about topological properties of the fixed points set of multifunctions. Later, some authors continued this review by providing different conditions. In 2008, 2009 Sintamarian proved some results on absolute retractivity of the common fixed points set of two multivalued operators. In this paper we shall present some new results on absolute retractivity of tripled fixed points set for multifunctions of the form F : X x X x X → Pb,cl(X). [ABSTRACT FROM AUTHOR]

We introduce the notion of upper/lower (τ, m)-continuous multifunctions and obtain many characterizations of such multifunctions. The notion of upper/lower (τ, m)-continuous multifunctions is a generalization of (τ, m)-continuous functions [34] and upper/lower ultra continuous multifunctions due to Navalagi et al. [26]. [ABSTRACT FROM AUTHOR]

In this paper, we study the behaviors of fixed points sets of non necessarily pseudo-contractive multifunctions. Rather than comparing the images of the involved multifunctions, we make use of some conditions on the fixed points sets to establish general results on their stability and continuous dependence. We illustrate our results by applications to differential inclusions and give stability results of fixed points sets of non necessarily pseudo-contractive multifunctions with respect to the bounded proximal convergence. [ABSTRACT FROM AUTHOR]

We introduce the notion of mIT-structures determined by operators mInt and mCl on an m-space (X;mX). By using mIT- structures, we introduce and investigate new classes of multifunctions F : (X;mX) → (Y; σ) called upper/lower mIT-continuous. As special cases of upper/lower mIT-continuity, we obtain upper/lower -M continuity [34] and upper/lower δ-M-precontinuity [35]. [ABSTRACT FROM AUTHOR]

Abstract: The class of nonatomic finite measure spaces with the saturation property, as developed in Maharam (1942) and Hoover–Keisler (1984), is characterized by the Fatou (and Lebesgue) property of a well-dominated sequence of multifunctions taking values in a Banach space. With multifunctions reduced to functions, this Fatou characterization also extends to a variant of the closure property found in optimal control theory. The results are developed through a considered overview of the relevant literature on the exact and approximate Fatou lemma phrased in terms of Bochner integration. [Copyright &y& Elsevier]

The connection between transitivity and existence of a dense orbit for multifunctions $\phi \: X\multimap X$ in generalized topological spaces is studied. Moreover strongly transitive multifunctions and functions in generalized topological spaces are investigated. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of viable or monotone solutions of a first order functional differential inclusion in a separable Banach space. We consider two cases with moving constraints. [ABSTRACT FROM AUTHOR]

Ekeland's variational principle (EVP) has many equivalent formulations and generalizations. In this article, we present new minimal point theorems in product spaces and the corresponding vector variational principles for set-valued functions. As special cases we derive many of the existing variational principles of Ekeland's type. Moreover, we use our new approach to get extensions of EVPs of Isac–Tammer and Ha types, as well as extensions of EVPs for bi-functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce two kinds of generalized continuity for multifunctions. Basic properties and characterizations of such multifunctions are established. These two generalized continuities include many of the variations of multifunction continuity already in the literature as special cases. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce two new classes of multifunctions called almost δ-semicontinuous multifunctions and weakly δ-semicontinuous multifunctions which are generalization of δ-semicontinuous multifunctions. In this paper we obtain several properties of such multifunctions. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a metric space with metric d, denote the family of all nonempty compact subsets of X and, given , let be the Hausdorff excess of F over G. The excess variation of a multifunction , which generalizes the ordinary variation V of single-valued functions, is defined by where the supremum is taken over all partitions of the interval . The main result of the paper is the following selection theorem: If , , and , then there exists a single-valued function of bounded variation such that for all , , and . We exhibit examples showing that the conclusions in this theorem are sharp, and that it produces new selections of bounded variation as compared with [V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (1) (2004) 1–82]. In contrast to this, a multifunction F satisfying for some constant and all with (Lipschitz continuity with respect to ) admits a Lipschitz selection with a Lipschitz constant not exceeding C if and may have only discontinuous selections of bounded variation if . The same situation holds for continuous selections of when it is excess continuous in the sense that as for all and as for all simultaneously. [Copyright &y& Elsevier]

We define a multifunction F: X ⇝ Y to be upper (lower) D*-supercontinuous if F +( V) ( F −( V)) is d*-open in X for every open set V of Y. We obtain some characterizations and several properties concerning upper (lower) D*-supercontinuous multifunctions. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the sensitivity analysis in optimization problems and variational inequalities. The concept of proto-differentiability of set-valued maps (see [R.T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 449–482]) plays the key role in our investigation. It is proved that, under some suitable qualification conditions, the generalized perturbation maps (that is, the solution set map to a parameterized constraint system, to a parameterized variational inequality, or to a parameterized optimization problem) are proto-differentiable. [Copyright &y& Elsevier]

Abstract: The purpose of this note is to present some fixed point and data dependence theorems in complete gauge spaces and in hyperconvex metric spaces for the so-called Meir–Keeler multivalued operators and admissible multivalued -contractions. Our results extend and generalize several theorems of Espínola and Kirk [R. Espínola, W.A. Kirk, Set-valued contractions and fixed points, Nonlinear Anal. 54 (2003) 485–494] and Rus, Petruşel, and Sîntămărian [I.A. Rus, A. Petruşel, A. Sîntămărian, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal. 52 (2003) 1947–1959]. [Copyright &y& Elsevier]

A continuous selection theorem for multifunctions, with nonempty Kuratowski–Painleve´ lower limit, whose values are C-convex and L-convex subsets of C(S,Rn) and L∞(T,Rn) is presented. [Copyright &y& Elsevier]