Dimension-like functions of the type dim and universality

In [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Math. Stud., vol. 198, Elsevier Science B.V., Amsterdam, 2005, xvi+559 pp.] base dimension-like functions of the type ind, Ind, and dim were defined. The domains of all these functions are classes of pairs (B,X), where B is a base of the space X. Also, positional base dimension-like functions of the type ind were defined whose domains are triples (Q,B,X), where Q is a subset of a space X and B is a base for Q in X. In [D.N. Georgiou, S.D. Iliadis, A.C. Megaritis, Dimension-like functions and universality, Topology Appl. 155 (2008) 2196-2201] and [D.N. Georgiou, S.D. Iliadis, A.C. Megaritis, The universality property for some dimension-like functions, Questions Answers Gen. Topology, in press] some dimension-like functions and positional dimension-like functions of type Ind, respectively are defined. All the above dimension-like functions are studied with respect to the property of universality. Here, we give positional base dimension-like functions of the type dim and prove the property of universality for these functions.     

On base dimension-like functions of the type Ind

In [5] base dimension-like functions of the type Ind were introduced. These functions were studied only with respect to the property of universality. Here, we first compare these dimensions with the classical large inductive dimension Ind and then study these functions with respect to other standard properties of dimension theory.     

The inductive dimension of a space by its normal base

Properties of the large inductive dimension of a space by its normal base introduced by S. Iliadis are studied. The proposed dimension-like functions generalize both the classic dimensions Ind, Ind₀ and the relative inductive dimensions I. Properties of the normal base characterizing the fulfillment of basic classic theorems of the dimension theory (sum, subset and product theorems) are found.     

Covering dimension d by a normal base

S.D. Iliadis introduced the concept of the dimension-like functions of type dim using the notion of a normal base. Since he considered this notion only from the point of view of the existence of universal spaces in different classes of spaces, in the present paper the basic theorems of dimension theory are obtained for these functions. Their relationship with the relative dimensions of A. Chigogidze and the uniform dimensions of M. Charalambous are shown.     

Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions

The classical existence-and-uniqueness theorem of the solution to a stochastic differential delay equation (SDDE) requires the local Lipschitz condition and the linear growth condition (see e.g. [11], [12] and [20]). The numerical solutions under these conditions have also been discussed intensively (see e.g. [4], [10], [13], [16], [17], [18], [21], [22] and [24]). Recently, Mao and Rassias [14] and [15] established the generalized Khasminskii-type existence-and-uniqueness theorems for SDDEs, where the linear growth condition is no longer imposed. These generalized Khasminskii-type theorems cover a wide class of highly nonlinear SDDEs but these nonlinear SDDEs do not have explicit solutions, whence numerical solutions are required in practice. However, there is so far little numerical theory on SDDEs under these generalized Khasminskii-type conditions. The key aim of this paper is to close this gap.     

Some applications of the condensation of the singularities of families of nonnegative functions

RecentlyPrus-Wi?niowski [14] has proved that the continuous functions of Λ-bounded variation on [0, 1] form a set of the first category in the Banach space C[0, 1] and also in each Banach space CΓBV[0, 1] of continuous functions of Γ-bounded variation on [0, 1] provided that the sequence Γ is adequate. In the present paper these results are extended and completed by using a principle of the condensation of the singularities of a family of nonnegative functions that follows from the theorems given byBreckner [3]. It is shown that Baire category properties similar to those stated in [14] are valid for two very large classes of real-valued functions called functions of bounded λ-variation of orderp and functions of boundedmth variation of orderp, respectively. The benefit of considering these classes is that they comprise several classes of functions of bounded variation type which have occurred so far in Fourier analysis or real analysis; in particular, the functions investigated in [14]. Thus, by specializing the results derived in the present paper, they give at once Baire category information concerning a number of well-known sets of real-valued functions.     

Some new results and generalizations in metric fixed point theory

The main purpose of this paper is the study of the generalization of some results given in [M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl. 326 (2007) 772-782] and references therein. Some generalizations of the Mizoguchi-Takahashi fixed point theorem, Kannan’s fixed point theorems and Chatterjea’s fixed point theorems are established by using our new fixed point theorems.     

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.     

CONNEXION PROPERTIES AND FACTORISATION THEOREMS

Abstract

With the introduction of several new factorisation theorems, this paper is intended to show that previous efforts of the authors [3] [5] and of Strecker [15] to describe the factorisations involving connectedness are incomplete. In Section 1 we give a purely topological construction of such a factorisation, in which the right factor is the class of spreads and the left factor has a certain property hereditarily: crucially, not all members of the left factor need be quotients. Section 2 shows that, given a left factor consisting of onto maps in the category T of topological spaces, then the class of mappings with the relevant properties hereditarily is also a left factor, and the result of section 1 is a particular case of this. Section 3 combines the material in [3] on intrinsic connexion properties with ideas of Preuss (see [1]) on disconnectednesses to yield another range of factorisations, for example, involving the maps with strongly connected fibres; and Section 4 notes some outstánding problems which our work has provoked.     

On an affirmative answer to S. Lin?s problem

In this paper, we give an affirmative answer to the problem posed by S. Lin (2002, 2007) in [7] and [8], and give another answer to the question posed by Y. Ikeda, C. Liu and Y. Tanaka (2002) in [5].     

Monotonic positive solution of a nonlinear quadratic functional integral equation

The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equation have been studied in [4], [5], [6], [7] and [8]. Here we are concerning with a nonlinear quadratic integral equation of Volterra type and we shall prove the existence of at least one L₁-positive monotonic solution for that equation under Carathèodory condition.     

Doubly stochastic measures with hairpin support

Summary Seethoff and Shiflett [5] proved nice uniqueness theorems concerning doubly stochastic measures supported on the union of the graphs of two functions, but the existence theorems were more elusive. In this present paper, using a functional equations approach, not only uniqueness results but also existence theorems are obtained for doubly stochastic measures with support sets of the form gg^-1 where g is an increasing homeomorphism of [0,1] onto itself such that g(x) whenever 0<x<1.     

Coeuklidische Hjelmslevgruppen

Over a three-dimensional metric vector space V over a field K of characteristic 2 can be defined geometrical structures in two ways. You can define reflections on lines either as reflections of V on one-dimensional non-isotropic subspaces of V or as involutory elements of the orthogonal group G of V with negative norm (in respect to a weak order of K). In this paper we characterize the geometry structures defined by the second method in the case that Ind V=0, dim rad V=1 via their reflection groups (G,S) — S the set of all reflections on lines. We show that these are certain Hjelmslevgroups. For an analogues characterization of the other seven classes of geometries see [1],[5], [7],[9].     

Towards Lim

The paper contains an elegant extension of the Nadler fixed point theorem for multivalued contractions (see Theorem 21). It is based on a new idea of the α-step mappings (see Definition 17) being more efficient than α-contractions. In the present paper this theorem is a tool in proving some fixed point theorems for “nonexpansive” mappings in the bead spaces (metric spaces that, roughly speaking, are modelled after convex sets in uniformly convex spaces). More precisely the mappings are nonexpansive on a set with respect to only one point - the centre of this set (see condition (4)). The results are pretty general. At first we assume that the value of the mapping under consideration at this central point looks “sharp” (see Definition 6). This idea leads to a group of theorems (based on Theorem 7). Their proofs are compact and the theorems, in particular, are natural extensions of the classical results for (usual) nonexpansive mappings. In the second part we apply the idea of Lim to investigate the regular sequences and here the proofs are based on our extension of Nadler's Theorem. In consequence we obtain some fixed point theorems that generalise the classical Lim Theorem for multivalued nonexpansive mappings (see e.g. Theorem 26).     

On continuous extensions

We consider various possiblities concerning the continuous extension of continuous functions taking values in an ultrametric space. In Section 1 we consider Tietze-type exetension theorems concerning continuous extendibility of continuous functions from compact and closed subsets to the whole space. In Sections 2 and 3 we consider extending separated continous functions in such a way that, certain continuous extensions remain separated. Functgions taking values in a complete ultravalued field are dealt with in Section 2, and the real and complex cases in Section 3.     

On the topological entropy of continuous and almost continuous functions

We prove some results concerning the entropy of continuous and almost continuous functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. Based on these facts we give theorems about approximation of functions defined and assuming their values on compact manifold by functions having strong entropy points.     

Continuity of sublinear operators on F-spaces

In this note, some of the fundamental theorems concerning the automatic continuity of positive linear functionals and positive linear operators (see [9,14,21]) as well as certain uniform boundedness type theorems (see [5,13,15,19]) will be extended to derive various continuity properties of (even discontinuous) sublinear operators from some metrizable topological vector space to some ordered topological vector space.     

HC-continuous functions

Long and Herrington [3] have investigated some properties of the strongly -continuous functions. In this paper we strengthen some of these properties of Long and Herrington [3] by studying a new class of functions, called HC-continuous functions.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

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.