On weakly measurable stochastic processes and absolutely summing operators

A characterization of absolutely summing operators by means of McShane integrable stochastic processes is considered

A characterization of strongly measurable Henstock-Kurzweil integrable functions and weakly continuous operators

We give necessary and sufficient conditions for the Kurzweil–Henstock integrability of functions given by f =n=1 xnχEn , where xn belong to a Banach space and the sets (En)n are measurable and pairwise disjoint. Also weakly completely continuous operators between Banach spaces are characterized by means of scalarly Kurzweil–Henstock integrable functions

MR3088363 Reviewed Robdera, Mangatiana A. More on the equivalence properties of Radon-Nikodým property types and complete continuity types of Banach spaces. Int. J. Funct. Anal. Oper. Theory Appl. 5 (2013), no. 1, 49–61. (Reviewer: Valeria Marraffa) 46B20

Convergence Theorems for Varying Measures Under Convexity Conditions and Applications

AbstractIn this paper, convergence theorems involving convex inequalities of Copson’s type (less restrictive than monotonicity assumptions) are given for varying measures, when imposing convexity conditions on the integrable functions or on the measures. Consequently, a continuous dependence result for a wide class of differential equations with many interesting applications, namely measure differential equations (including Stieltjes differential equations, generalized differential problems, impulsive differential equations with finitely or countably many impulses and also dynamic equations on time scales) is provided.

Operator martingale decomposition and the Radon-Nikodym property in Banach spaces

Abstract We consider submartingales and uniform amarts of maps acting between a Banach lattice and a Banach lattice or a Banach space. In this measure-free setting of martingale theory, it is known that a Banach space Y has the Radon–Nikodým property if and only if every uniformly norm bounded martingale defined on the Chaney–Schaefer l-tensor product E ⊗ ˜ l Y , where E is a suitable Banach lattice, is norm convergent. We present applications of this result. Firstly, an analogues characterization for Banach lattices Y with the Radon–Nikodým property is given in terms of a suitable set of submartingales (supermartingales) on E ⊗ ˜ l Y . Secondly, we derive a Riesz decomposition for uniform …

Non absolutely convergent integrals of functions taking values in a locally convex space

Properties of McShane and Kurzweil-Henstock integrable functions taking values in a locally convex space are considered and the relations with other integrals are studied. A convergence theorem for the Kurzweil-Henstock integral is given

Set-valued Brownian motion

Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach space $X$. The present paper is an application of one the paper of the second author in which an embedding result is obtained which considers also the ordered structure of $ck(X)$ and f-algebras.

Riemann type integrals for functions taking values in a locally convex space

The McShane and Kurzweil-Henstock integrals for functions taking values in a locally convex space are defined and the relations with other integrals are studied. A characterization of locally convex spaces in which Henstock Lemma holds is given.

A Birkhoff type integral and the Bourgain property in a locally convex space

An integral, called the $Bk$-integral, for functions taking values in a locally convex space is defined. Properties of $Bk$-integrable functions are considered and the relations with other integrals are studied. Moreover the $Bk$-integrability of bounded functions is compared with the Bourgain property.

Convergence of Banach lattice valued stochastic processes without the Radon-Nikodyn property

Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity

We are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not assumed to have compact and convex values, nor to be upper semicontinuous concerning the second argument everywhere, as in other related works. A condition involving the contingent derivative relative to the non-decreasing function (recently introduced and applied to initial value problems by R.L. Pouso, I.M. Marquez Albes, and J. Rodriguez-Lopez) is imposed on the set where the upper semicontinuity and the assumption to have compact convex value…

On almost sure convergence of amarts and martingales without the Radon-Nikodym property

It is shown here that for any Banach spaceE-valued amart (X n) of classB, almost sure convergence off(Xn) tof(X) for eachf in a total subset ofE * implies scalar convergence toX.

Measure differential inclusions: existence results and minimum problems

AbstractWe focus on a very general problem in the theory of dynamic systems, namely that of studying measure differential inclusions with varying measures. The multifunction on the right hand side has compact non-necessarily convex values in a real Euclidean space and satisfies bounded variation hypotheses with respect to the Pompeiu excess (and not to the Hausdorff-Pompeiu distance, as usually in literature). This is possible due to the use of interesting selection principles for excess bounded variation set-valued mappings. Conditions for the minimization of a generic functional with respect to a family of measures generated by equiregulated left-continuous, nondecreasing functions and to…

MR2911371 Reviewed Raja, Matías; Rodríguez, José Scalar boundedness of vector-valued functions. Glasg. Math. J. 54 (2012), no. 2, 325–333. (Reviewer: Valeria Marraffa) 46G10 (28B05)

Convergence for varying measures in the topological case

In this paper convergence theorems for sequences of scalar, vector and multivalued Pettis integrable functions on a topological measure space are proved for varying measures vaguely convergent.

The McShane, PU and Henstock integrals of Banach valued functions

Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals of vector valued functions are characterized.

On set-valued cone absolutely summing maps

Spaces of cone absolutely summing maps are generalizations of Bochner spaces Lp(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space \( \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] \) of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of \( \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] \) , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We …

Relaxation result for differential inclusions with Stieltjes derivative

The aim of this paper is to provide a Filippov-Wa\.{z}ewski Relaxation Theorem for the very general setting of Stieltjes differential inclusions. New relaxation results can be deduced for generalized differential problems, for impulsive differential inclusions with multivalued impulsive maps and possibly countable impulsive moments and also for dynamic inclusions on time scales.

Strongly measurable Kurzweil-Henstock type integrable functions and series

We give necessary and sufficient conditions for the scalar Kurzweil-Henstock integrability and the Kurzweil-Henstock-Pettis integrability of functions $f:[1, infty) ightarrow X$ defined as $f=sum_{n=1}^infty x_n chi_{[n,n+1)}$. Also the variational Henstock integrability is considered

MR2966106 Reviewed Shahidi, F. A.; Ganiev, I. G. Vector valued martingale-ergodic and ergodic-martingale theorems. Stoch. Anal. Appl. 30 (2012), no. 5, 916–932. (Reviewer: Valeria Marraffa) 60G48

Variational McShane integral in vector spaces

A characterization of absolutely summing operators by means of McShane integrable functions

AbstractAbsolutely summing operators between Banach spaces are characterized by means of McShane integrable functions.

MR3029186 Reviewed Kuo, Wen-Chi; Vardy, Jessica Joy; Watson, Bruce Alastair Mixingales on Riesz spaces. J. Math. Anal. Appl. 402 (2013), no. 2, 731–738. (Reviewer: Valeria Marraffa) 60G48

Closure properties for integral problems driven by regulated functions via convergence results

Abstract In this paper we give necessary and sufficient conditions for the convergence of Kurzweil–Stieltjes integrals with respect to regulated functions, using the notion of asymptotical equiintegrability. One thus generalizes several well-known convergence theorems. As applications, we provide existence and closure results for integral problems driven by regulated functions, both in single- and set-valued cases. In the particular setting of bounded variation functions driving the equations, we get features of the solution set of measure integrals problems.

An equivalent definition of the vector-valued McShane integral by means of partitions of unity

An integral for vector-valued functions on a σ-finite outer regular quasi-radon measure space is defined by means of partitions of unity and it is shown that it is equivalent to the McShane integral. The multipliers for both the McShane and Pettis integrals are characterized

A scalar Volterra derivative for the PoU-integral

A note on the Banach space of preregular maps

The aim of this paper is to give simple proofs for Jeurnink's characterizations of preregular maps in terms of Θ-maps acting between Banach lattices. For Banach lattices E and F, we achieve our goal by considering the space Lβ(E, F) of all those linear maps T: E → F for which there exists a constant K such that {double pipe}Vn i=1 {pipe}Txi{pipe} ≤ K {double pipe}Vn i=1{pipe}xi for all finite sequences x1, ..., xn e{open}E. We show that, if Lβ(E; F), and the spaces L Θ (E; F) of Θ -map and Lpr(E; F) of preregular maps are respectively endowed with their canonical norms, then they are identical Banach spaces

On the integration of Riemann-measurable vector-valued functions

We confine our attention to convergence theorems and descriptive relationships within some subclasses of Riemann-measurable vector-valued functions that are based on the various generalizations of the Riemann definition of an integral.

Variational Henstock integrability of Banach space valued functions

We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum \nolimits _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum \nolimits _{n=1}^{\infty }x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a…

Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces

Abstract In this paper we study the Pettis integral of fuzzy mappings in arbitrary Banach spaces. We present some properties of the Pettis integral of fuzzy mappings and we give conditions under which a scalarly integrable fuzzy mapping is Pettis integrable.

On Spaces of Bochner and Pettis Integrable Functions and Their Set-Valued Counterparts

The aim of this paper is to give a brief summary of the Pettis and Bochner integrals, how they are related, how they are generalized to the set-valued setting and the canonical Banach spaces of bounded maps between Banach spaces that they generate. The main tool that we use to relate the Banach space-valued case to the set-valued case, is the R ̊adstr ̈om embedding theorem.

MR2640176 Reviewed Liu, PeiDe; Hou, YouLiang; Wang, MaoFa Weak Orlicz space and its applications to the martingale theory. Sci. China Math. 53 (2010), no. 4, 905–916. (Reviewer: Valeria Marraffa) 60G46

The Fubini and Tonelli Theorems for Product Local Systems

The notion of product local system and of the Kurzweil-Henstock type integral related to a product local system is introduced. The main result is a version of the Fubini and Tonelli theorems for product local systems.

Stochastic processes of vector valued Pettis and McShane integrable functions

Set valued integrability in non separable Fréchet spaces and applications

AbstractWe focus on measurability and integrability for set valued functions in non-necessarily separable Fréchet spaces. We prove some properties concerning the equivalence between different classes of measurable multifunctions. We also provide useful characterizations of Pettis set-valued integrability in the announced framework. Finally, we indicate applications to Volterra integral inclusions.

Approximating the solutions of differential inclusions driven by measures

The matter of approximating the solutions of a differential problem driven by a rough measure by solutions of similar problems driven by “smoother” measures is considered under very general assumptions on the multifunction on the right-hand side. The key tool in our investigation is the notion of uniformly bounded $$\varepsilon $$-variations, which mixes the supremum norm with the uniformly bounded variation condition. Several examples to motivate the generality of our outcomes are included.

On Regulated Solutions of Impulsive Differential Equations with Variable Times

In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined.

The Variational Mcshane Integral in Locally Convex Spaces

The variational McShane integral for functions taking values in a locally convex space is defined, and it is characterized by means of the p-variation of the indefinite Pettis integral

On a step method and a propagation of discontinuity

In this paper we analyze how to compute discontinuous solutions for functional differential equations, looking at an approach which allows to study simultaneously continuous and discontinuous solutions. We focus our attention on the integral representation of solutions and we justify the applicability of such an approach. In particular, we improve the step method in such a way to solve a problem of vanishing discontinuity points. Our solutions are considered as regulated functions.

Quadratic variation of martingales in Riesz spaces

We derive quadratic variation inequalities for discrete-time martingales, sub- and supermartingales in the measure-free setting of Riesz spaces. Our main result is a Riesz space analogue of Austinʼs sample function theorem, on convergence of the quadratic variation processes of martingales http://www.journals.elsevier.com/journal-of-mathematical-analysis-and-applications/ http://dx.doi.org/10.1016/j.jmaa.2013.08.037 National Research Foundation of South Africa (Grant specific unique reference number (UID) 85672) and by GNAMPA of Italy (U 2012/000574 20/07/2012 and U 2012/000388 09/05/2012)

MR2376860 (2009e:28048) Khurana, Surjit Singh Weak compactness of vector measures on topological spaces. Publ. Math. Debrecen 72 (2008), no. 1-2, 69--79. (Reviewer: Valeria Marraffa)

Weak compactness of vector measures on topological spaces.

MR2261655 (2007j:28019) Rodríguez, José On integration of vector functions with respect to vector measures. Czechoslovak Math. J. 56(131) (2006), no. 3, 805--825. (Reviewer: Valeria Marraffa) 28B05 (46G10)

On strongly measurable Kurzweil-Henstock type integrable functions

We consider the integrability, with respect to the scalar Kurzweil-Henstock integral, the Kurzweil-Henstock-Pettis integral and the variational Henstock integral, of strongly measurable functions de ned as f = P1 n=1 xn [n;n+1),where (xn) belongs to a Banach space. Examples which indicate the difference between the scalar Henstock-Kurzweil integral and the Henstock- Kurzweil-Pettis integral and between the variational Henstock integral and the Henstock-Kurzweil-Pettis integral are given.

A negative answer to a problem of Fremlin and Mendoza

A negative answer to a problem of Fremlin and Mendoza