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Abstract: Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant and for each there are cones with normal constant . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results. [Copyright &y& Elsevier]

Abstract: In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems (Helton et al. (2009) , de Oliviera et al. (2009) ). In the earlier paper (Helton et al. (2009) ) we characterized NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. In this paper we turn to a more general dimension-free ball , called a “pencil ball”, associated with a homogeneous linear pencil For , define and let We study the generalization of NC ball maps to these pencil balls , and call them “pencil ball maps”. We show that every has a minimal dimensional (in a certain sense) defining pencil . Up to normalization, a pencil ball map is the direct sum of with an NC analytic map of the pencil ball into the ball. That is, pencil ball maps are simple, in contrast to the classical result of D''Angelo (1993) showing there is a great variety of such analytic maps from to when . To prove our main theorem, this paper uses the results of our previous paper (Helton et al. (2009) ) plus entirely different techniques, namely, those of completely contractive maps. What we do here is a small piece of the bigger puzzle of understanding how Linear Matrix Inequalities (LMIs) behave with respect to noncommutative change of variables. [Copyright &y& Elsevier]

This paper studies the equivalence of the quasisymmetric mappings on non-connected sets. We introduce a generalized form of weak quasisymmetry and prove that, on a uniformly perfect metric space, a generalized weakly quasisymmetric mapping is quasisymmetric. We further improve this result on Cantor-like sets satisfying the small gap condition. [ABSTRACT FROM AUTHOR]

A well-known result of W. Ray asserts that if C is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping T : C → C that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup S of nonexpansive mappings acting on a closed convex subset C of a Hilbert space, assuming that there is a point c ∈ C with a bounded orbit and assuming that certain subspace of C b ( S ) has a left invariant mean. Left invariant mean (or amenability) is an important notion in harmonic analysis of semigroups and groups introduced by von Neumann in 1929 [28] and formalized by Day in 1957 [5] . In our investigation we use the notion of common attractive points introduced recently by S. Atsushiba and W. Takahashi. [ABSTRACT FROM AUTHOR]

Abstract: A Banach space X is said to satisfy if the set of minimizers of the function is nonempty for each integer , each and each continuous nondecreasing coercive real-valued function f on . We study stability of certain polyhedrality properties under making direct sums, in order to be able to use results from a paper by Fonf, Lindenstrauss and the author to show that if X satisfies and an appropriate polyhedrality property then the function space satisfies for every topological space T. This generalizes the authorʼs result from 1997, proved for finite-dimensional polyhedral spaces X. Moreover, under more restrictive conditions on X and f, the mappings on () are continuous in the Hausdorff metric for each compact K. [Copyright &y& Elsevier]

Abstract: The notion of p-compact sets arises naturally from Grothendieckʼs characterization of compact sets as those contained in the convex hull of a norm null sequence. The definition, due to Sinha and Karn (2002), leads to the concepts of p-approximation property and p-compact operators (which form an ideal with its ideal norm ). This paper examines the interaction between the p-approximation property and certain space of holomorphic functions, the p-compact analytic functions. In order to understand these functions we define a p-compact radius of convergence which allows us to give a characterization of the functions in the class. We show that p-compact holomorphic functions behave more like nuclear than compact maps. We use the ϵ-product of Schwartz, to characterize the p-approximation property of a Banach space in terms of p-compact homogeneous polynomials and in terms of p-compact holomorphic functions with range on the space. Finally, we show that p-compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the -approximation property. Our approach also allows us to solve several questions posed by Aron, Maestre and Rueda (2010). [Copyright &y& Elsevier]

Abstract: Let G be a graph and be a continuous map. Denote by , and the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If is an open arc contained in an edge of G such that for some , then ; (2) Any isolated point of is also an isolated point of ; (3) If for some , then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) and J. Xiong (1983, 1986) on interval maps or tree maps. [Copyright &y& Elsevier]

Abstract: This paper concerns the study of weak and firm local efficiency in constrained mathematical problems governed by set-valued mappings. We derive optimality conditions by means of the Bouligand derivative and by means of the Mordukhovich coderivative as well. [Copyright &y& Elsevier]

Abstract: 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 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. [Copyright &y& Elsevier]

Abstract: This paper studies topological and tiling properties of a family of self-affine fractal tiles with a real parameter ε. Both interior and boundary structures have been discussed. Moreover, connectedness and tiling properties of the tile are proved to be dependent upon ε. [Copyright &y& Elsevier]

Abstract: In this paper continuity properties of the set-valued map , , are considered where is the closed ball of the space centered at the origin with radius . It is proved that the set-valued map , , is continuous. Applying obtained results, the attainable set of the nonlinear control system with integral constraint on the control is studied. The admissible control functions are chosen from . It is shown that the attainable set of the system is continuous with respect to p. [Copyright &y& Elsevier]

Abstract: In this paper a new notion of differentiability of set-valued maps is introduced. Based on ordered spaces techniques, -lower and -upper contingent derivatives are defined. Conditions for -differentiability are given. Furthermore we show that these concepts are suitable for the formulation of optimality conditions for set-valued optimization problems with set-optimization criteria. [Copyright &y& Elsevier]

Abstract: In this paper we analyze some questions concerning the contingent epiderivative. When the ordering cone is not necessarily pointed we introduce the notion of family of contingent epiderivatives. We also investigate the relationship between the contingent derivative and the generalized epiderivative. Furthermore, we give existence theorems with respect to Daniell cones. As a particular case, we study set-valued maps that take values in . [Copyright &y& Elsevier]

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: In this paper, higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set-valued maps are introduced. Necessary and sufficient conditions for Henig efficient solutions to a constrained set-valued optimization problem are given by employing the higher order generalized epiderivatives. [Copyright &y& Elsevier]

Abstract: In this paper, the existence and uniqueness of positive fixed points for a class of convex operators is obtained by means of the properties of cone, concave operators and the monotonicity of set-valued maps. In the end, we give a simple application to certain integral equations. [Copyright &y& Elsevier]

Abstract: In this paper we present two set-valued variants of the Ekeland variational principle involving the Clarke normal cone and establish sufficient conditions for a set-valued map to have a weak minimizer or a properly positive minimizer when it satisfies Palais–Smale type conditions. [Copyright &y& Elsevier]

Abstract: In this paper we investigate a vector optimization problem where objective and constraints are given by set-valued maps. We show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of as solutions of a scalar optimization problem with single-valued objective and constraint functions. Then applying some classical or recent results in optimization theory to and using estimates of subdifferentials of marginal functions, we obtain optimality conditions for () expressed in terms of Lagrange or sequential Lagrange multipliers associated with various coderivatives of the set-valued data. [Copyright &y& Elsevier]

Abstract: We present in this paper a generalized version of the celebrated Knaster–Kuratowski–Mazurkiewicz–Fan''s principle on the intersection of a family of closed sets subject to a classical geometric condition and a weakened compactness condition. The fixed point formulation of this generalized principle extends the Browder–Fan fixed point theorem to set-valued maps of non-compact convex subsets of topological vector spaces. [Copyright &y& Elsevier]

Abstract: In the present paper, some new almost fixed point theorems and fixed point theorems for lower semicontinuous type multivalued mappings are obtained in metrizable H-spaces. [Copyright &y& Elsevier]

In this paper, the notion of affinelike set-valued maps is introduced and some properties of these maps are presented. Then a new Hahn–Banach extension theorem with a K-convex set-valued map dominated by an affinelike set-valued map is obtained. [Copyright &y& Elsevier]

The aim of this paper is to show that under a mild semicontinuity assumption (the so-called segmentary epi-closedness), the cone-convex (respectively, cone-quasiconvex) set-valued maps can be characterized in terms of weak cone-convexity (respectively, weak cone-quasiconvexity), i.e., the notions obtained by replacing in the classical definitions the conditions of type “for all x,y in the domain and for all t in ]0,1[ …” by the corresponding conditions of type “for all x,y in the domain there exists t in ]0,1[ … .” [Copyright &y& Elsevier]