MR2858094 Musiał, Kazimierz Pettis integrability of multifunctions with values in arbitrary Banach spaces. J. Convex Anal. 18 (2011), no. 3, 769–810. (Reviewer: Luisa Di Piazza) 28B20

Henstock–Kurzweil–Pettis integrability of compact valued multifunctions with values in an arbitrary Banach space

Abstract The aim of this paper is to describe Henstock–Kurzweil–Pettis (HKP) integrable compact valued multifunctions. Such characterizations are known in case of functions (see Di Piazza and Musial (2006)  [16] ). It is also known (see Di Piazza and Musial (2010)  [19] ) that each HKP-integrable compact valued multifunction can be represented as a sum of a Pettis integrable multifunction and of an HKP-integrable function. Invoking to that decomposition, we present a pure topological characterization of integrability. Having applied the above results, we obtain two convergence theorems, that generalize results known for HKP-integrable functions. We emphasize also the special role played in …

Relations among Gauge and Pettis integrals for cwk(X)-valued multifunctions

The aim of this paper is to study relationships among "gauge integrals" (Henstock, Mc Shane, Birkhoff) and Pettis integral of multifunctions whose values are weakly compact and convex subsets of a general Banach space, not necessarily separable. For this purpose we prove the existence of variationally Henstock integrable selections for variationally Henstock integrable multifunctions. Using this and other known results concerning the existence of selections integrable in the same sense as the corresponding multifunctions, we obtain three decomposition theorems. As applications of such decompositions, we deduce characterizations of Henstock and ${\mathcal H}$ integrable multifunctions, toget…

MR2684422 Deville, Robert; Rodríguez, José Integration in Hilbert generated Banach spaces. Israel J. Math. 177 (2010), 285–306. (Reviewer: Luisa Di Piazza)

2010), 285–306, 46Exx (46J10) It is known that each McShane integrable function is also Pettis integrable, while the reverse implication in general is not true. The equivalence of McShane and Pettis integrability depends on the target Banach space X and has been proven: by R. A. Gordon [Illinois J. Math. 34 (1990), no. 3, 557–567, 26A42 (28B15 46G10 49Q15)], and by D. H. Fremlin and J. Mendoza [Illinois J. Math. 38 (1994), no. 1, 127–147, 46G10 (28B05)] if X is separable, by D. Preiss and the reviewer [Illinois J. Math. 47 (2003), no. 4, 1177–1187. 28B05 (26A39 26E25 46G10)] if X=c_0(\Gamma) (for any set \Gamma) or X is super-reflexive, by the second author of the present paper [J. Math. An…

A decomposition of Denjoy-Khintchine-Pettis and Henstock-Kurzweil-Pettis integrable multifunctions

We proved in one of our earlier papers that in case of separable Banach space valued multifunctions each Henstock-Kurzweil-Pettis integrable multifunction can be represented as a sum of one of its Henstock-Kurzweil-Pettis integrable selector and a Pettis integrable multifunction. Now, we prove that the same result can be achieved in case of an arbitrary Banach space. Moreover we show that an analogous result holds true also for the Denjoy-Khintchine-Pettis integrable multifunctions. Applying the representation theorem we describe the multipliers of HKP and DKP integrable functions. Then we use this description to obtain an operator characterization of HKP and DKP integrability.

A variational henstock integral characterization of the radon-nikodým property

A characterization of Banach spaces possessing the Radon-Nikodym property is given in terms of finitely additive interval functions. We prove that a Banach space X has the RNP if and only if each X-valued finitely additive interval function possessing absolutely continuous variational measure is a variational Henstock integral of an X-valued function. Due to that characterization several X-valued set functions that are only finitely additive can be represented as integrals.

ZBL MS 63/6 Satco, Bianca-Renata; Turcu, Corneliu-Octavian Henstock-Kurzweil-Pettis integral and weak topologies in nonlinear integral equations on time scales Mathematica Slovaca, volume 63 (2013) \no 6 pp. 1347-1360

The authors prove an existence result for a nonlinear integral equation on time scales under weak topology assumption in the target Banach space. In the setting of vector valued functions on time scales they consider the Henstock-Kurzweil-Pettis $\Delta$-integral which is a kind of Henstock integral recently introduced by Cichon, M. [Commun. Math. Anal. 11 (2011), no. 1, 94�110]. In this framework they show the existence of weakly continuous solutions for an integral equation x(t)= f(t, x(t))+ (HKP)\int_0^t g(t,s,x(s)) \Delta s governed by the sum of two operators: a continuous operator and an integral one. The main tool to get the solutions is a generalization of Krasnosel'skii fixed point…

Multifunctions determined by integrable functions

Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in the sense of Bochner, McShane or Birkhoff can be transferred to the generated multifunction while Henstock integrability does not guarantee it.

Decompositions of Weakly Compact Valued Integrable Multifunctions

We give a short overview on the decomposition property for integrable multifunctions, i.e., when an &ldquo

MR2886259 Naralenkov, Kirill Several comments on the Henstock-Kurzweil and McShane integrals of vector-valued functions. Czechoslovak Math. J. 61(136) (2011), no. 4, 1091–1106. (Reviewer: Luisa Di Piazza) 26A39 (28B05)

In this paper the author essentially discusses the difference between the Henstock-Kurzweil and McShane integrals of vector-valued functions from the descriptive point of view. He first considers three notions of absolute continuity for vector-valued functions AC, AC*, AC_{\delta}) and studies the relationships between the corresponding classes of functions. Then he uses such notions to give descriptive characterizations of the Henstock-Kurzweil and McShane integrable functions.

Integration of multifunctions with closed convex values in arbitrary Banach spaces

Integral properties of multifunctions with closed convex values are studied. In this more general framework not all the tools and the technique used for weakly compact convex valued multifunctions work. We pay particular attention to the "positive multifunctions". Among them an investigation of multifunctions determined by vector-valued functions is presented. Finally, decomposition results are obtained for scalarly and gauge-defined integrals of multifunctions and a full description of McShane integrability in terms of Henstock and Pettis integrability is given.

Kurzweil-Henstock type integration on Banach spaces

In this paper properties of Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrals for vector-valued functions are studied. In particular, the absolute integrability for Kurzweil-Henstock integrable functions is characterized and a Kurzweil-Henstock version of the Vitali Theorem for Pettis integrable functions is given.

Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values

Fremlin (Ill J Math 38:471–479, 1994) proved that a Banach space valued function is McShane integrable if and only if it is Henstock and Pettis integrable. In this paper we prove that the result remains valid also in case of multifunctions with compact convex values being subsets of an arbitrary Banach space (see Theorem 3.4). Di Piazza and Musial (Monatsh Math 148:119–126, 2006) proved that if \(X\) is a separable Banach space, then each Henstock integrable multifunction which takes as its values convex compact subsets of \(X\) is a sum of a McShane integrable multifunction and a Henstock integrable function. Here we show that such a decomposition is true also in case of an arbitrary Banac…

Approximation by step functions of Banach space valued nonabsolute integrals.

The approximation of Banach space valued nonabsolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock-Kurzweil-Pettis and a Denjoy-Khintchine-Pettis integrable function can be only scalarly approximate in the Alexiewicz norm by a sequence of step functions. In case of Henstock-Kurzweil-Pettis and Denjoy-Khintchine-Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact. It is also proved that if the target Banach space X does not contain any isomorphic copy of c_0, then the range of t…

A Decomposition of Henstock-Kurzweil-Pettis Integrable Multifunctions

We proved in our earlier paper [9] that in case of separable Banach space-valued multifunctions each Henstock-Kurzweil-Pettis integrable multifunction can be represented as a sum of one of its Henstock-Kurzweil-Pettis integrable selectors and a Pettis integrable multifunction. Now, we prove that the same result can be achieved in case of an arbitrary Banach space. Applying the representation theorem we describe the multipliers of the Henstock-Kurzweil-Pettis integrable multifunctions. Then we use this description to obtain a characterization of the Henstock-Kurzweil-Pettis integrability in terms of subadditive operators.

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…

MR3093276 Reviewed Naralenkov, K. M. On continuity and compactness of some vector-valued integrals. Rocky Mountain J. Math. 43 (2013), no. 3, 1015–1022. (Reviewer: Luisa Di Piazza)

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.

Radon-Nikodym derivatives of finitely additive interval measures taking values in a Banach space with basis

Let X be a Banach space with a Schauder basis {en}, and let Φ(I)= ∑n en ∫I fn(t)dt be a finitely additive interval measure on the unit interval [0, 1], where the integrals are taken in the sense of Henstock–Kurzweil. Necessary and sufficient conditions are given for Φ to be the indefinite integral of a Henstock–Kurzweil–Pettis (or Henstock, or variational Henstock) integrable function f:[0, 1] → X.

The Pettis and McShane integrals for vector valued functions

VARIATIONAL MEASURES GENERATED BY FUNCTIONS AND ASSOCIATED WITH LOCAL SYSTEMS OF SETS

MR3266136 Porcello, G., Decomposability in the space of HKP-integrable functions. Math. Nachr. 287 (2014), no. 14-15, 17331744. 26A39 (26E25 28B20 54C60)

The notion of decomposability for families of Banach space valued functions is a certain kind of generalization of convexity. Decomposability is usually de- ned (in a space, or some subspaces, of measurable functions as the space of Bochner integrable or Pettis integrable functions) with respect to a -algebra of sets. In the paper under review the author introduces the notion of decom- posability for vector-valued functions integrable in Henstock sense. Since the Henstock-type integrals act only on intervals, the author modi es in a slight but essential way the classical"de nition of decomposability: instead of a - algebra of sets, one has to work with the ring A generated by the subinterva…

MR3058477 Reviewed Ereú, Thomás; Sánchez, José L.; Merentes, Nelson; Wróbel, Małgorzata Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Schramm. Pr. Nauk. Akad. Jana Długosza Częst. Mat. 16 (2011), 23–32. ISBN: 978-83-7455-209-7

In this paper it is established a property of a composition operator between spaces of functions of bounded variation in the sense of Schramm. Let X and Y be two real normed spaces, C a convex cone in X and I a closed bounded interval of the real line. Moreover let cc(Y) be the family of all non-empty closed convex and compact subsets of Y. The authors study the Nemytskij (composition) operator (HF)(t)=h(t,F(t)), where F: I \rightarrow C and h: I\times C \rightarrow cc(Y) is a given set-valued function. They show that if the Nemytskij operator $H$ is uniformly continuous and maps the space \Phi BV (I;C) of functions (from I to C) of bounded \Phi-variation in the sense of Schramm into the sp…

MR3191427 Naralenkov, Kirill M., A Lusin type measurability property for vector- valued functions. J. Math. Anal. Appl. 417 (2014), no. 1, 293307. 28A20

In the paper under review the author introduces the notion of Riemann measurability for vector-valued functions, generalizing the classical Lusin condition, which is equivalent to the Lebesgue measurability for real valued functions. Let X be a Banach space, let f : [a; b] ! X and let E be a measurable subset of [a; b]. The function f is said to be Riemann measurable on E if for each " > 0 there exist a closed set F E with (E n F) < 0 (where is the Lebesgue measure) and a positive number such that k XK k=1 ff(tk) ?? f(t0 k)g (Ik)k < " whenever fIkgKk =1 is a nite collection of pairwise non-overlapping intervals with max1 k K (Ik) < and tk; t0 k 2 Ik T F. The Riemann measurabilit…

Differentiation of an additive interval measure with values in a conjugate Banach space

We present a complete characterization of finitely additive interval measures with values in conjugate Banach spaces which can be represented as Henstock-Kurzweil-Gelfand integrals. If the range space has the weak Radon-Nikodým property (WRNP), then we precisely describe when these integrals are in fact Henstock-Kurzweil-Pettis integrals.

Multi-integrals of finite variation

The aim of this paper is to investigate different types of multi-integrals of finite variation and to obtain decomposition results.

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.

Kurzweil--Henstock and Kurzweil--Henstock--Pettis integrability of strongly measurable functions

We study the integrability of Banach valued strongly measurable functions defined on $[0,1]$. In case of functions $f$ given by $\sum _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ belong to a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for the Bochner and for the Pettis integrability of $f$ (cf Musial (1991)). In this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions.

Variational measures related to local systems and the Ward propery of P-adic path bases

Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a P-adic path system that defines a differentiation basis which does not possess Ward property.

Radon–Nikodým Theorems for Finitely Additive Multimeasures

In this paper we deal with interval multimeasures. We show some Radon–Nikodým theorems for such multimeasures using multivalued Henstock or Henstock–Kurzweil–Pettis derivatives. We do not use the separability assumption in the results.

Some new results on integration for multifunction

It has been proven in previous papers that each Henstock-Kurzweil-Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable.

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…

A new full descriptive characterization of Denjoy-Perron integral

It is proved that the absolute continuity of the variational measure generated by an additive interval function \(F\) implies the differentiability almost everywhere of the function \(F\) and gives a full descriptive characterization of the Denjoy-Perron integral.

MR2817222 Ursescu, Corneliu, A mean value inequality for multifunctions. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54(102) (2011), no. 2, 193–200

The paper is devoted to extend some mean value inequalities from the function setting to the multifunction one. Let (M,d) be a metric space, let F be a multifunctions defined on D \subset R and taking values in the family of nonempty subsets of M, and let g: D\rightarrow R be a strictly increasing function. The author proves the following inequality: \frac{\delta(F(b),F(a))}{g(b)-g(a)} \leq \sup_{s\in [a,b)\cap D} \sup_{S\in F(s)} \sup_{t\in (s,b)\cap D} \frac{\delta(F(t),S)}{g(t)-g(s)}, where a and b are two points of D with a<b and, if Q and P are nonempty subsets of M, then \delta(Q,P)=\sup_{p\in P} \inf_{q\in Q}d(q,p). An application of the previous inequality to the Dini derivatives of…

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.

MR2569913: Rodríguez, José. Some examples in vector integration. Bull. Aust. Math. Soc. 80 (2009), no. 3, 384–392. (Reviewer: Luisa Di Piazza),

The paper deals with some classical examples in vector integration due to Phillips, Hagler and Talagrand, revisited from the point of view of the Birkhoff and McShane integrals. More precisely, the author considers: - Phillips' example of a Pettis integrable function f which is not Birkhoff integrable [R. S. Phillips, Trans. Amer. Math. Soc. 47 (1940), 114--145; MR0002707 (2,103c)]. It is proved here that f is universally McShane integrable. - Hagler's example of a scalarly measurable l∞-valued function g which is not strongly measurable. The function g is proved to be universally Birkhoff integrable. - Talagrand's example of a bounded Pettis integrable function φ having no conditional expe…

MR2481817 (2010e:46040): Haluška, Ján; Hutník, Ondrej On vector integral inequalities. Mediterr. J. Math. 6 (2009), no. 1, 105–124. (Reviewer: Luisa Di Piazza),

I. Dobrakov in his papers [Czechoslovak Math. J. 40(115) (1990), no. 1, 8--24; MR1032359 (90k:46097); Czechoslovak Math. J. 40(115) (1990), no. 3, 424--440; MR1065022 (91g:46052)] developed a theory for integrating vector-valued functions with respect to operator-valued measures: Let X and Y be two Banach spaces, Δ be a δ-ring of subsets of a nonempty set T, L(X,Y) be the space of all continuous operators L:X→Y, and m:Δ→L(X,Y) be an operator-valued measure σ-additive in the strong operator topology of L(X,Y). A measurable function f:T→X is said to be integrable in the sense of Dobrakov if there exists a sequence of simple functions fn:T→X, n∈N, converging m-a.e. to f and the integrals ∫.fnd…

MR2524292 (2010f:26007): Kolyada, V. I.; Lind, M. On functions of bounded p-variation. J. Math. Anal. Appl. 356 (2009), no. 2, 582–604. (Reviewer: Luisa Di Piazza),

For p∈(1,+∞), let f∈Lp be a 1-periodic function on the real line, with the norm of f given by ∥f∥p=(∫10|f(x)|pdx)1/p. The Lp-modulus of continuity of f is defined by ω(f,δ)p=sup0≤h≤δ(∫10|f(x+h)−f(x)|pdx)1/p, 0≤δ≤1. A partition of period 1 (or simply a partition) is a set Π={x0,x1,…,xn} of points such that x0<x1<…<xn=x0+1. For a given partition Π={x0,x1,…,xn} let vp(f;Π)=(∑k=0n−1|f(xk+1)−f(xk)|p)1/p. The modulus of p-continuity of f is defined by ω1−1/p(f,δ)=sup∥Π∥≤δvp(f;Π), where the supremum is taken over all partitions Π such that ∥Π∥=maxk(xk+1−xk)≤δ. In this paper, improving a previous estimate given by A. P. Terehin [Mat. Zametki 2 (1967), 289--300; MR0223512 (36 #6560)], it is shown th…

Variational measures in the theory of integration

{Variational measures in the theory of integration} {Luisa Di Piazza} {Palermo , Italy} We will present here some results concerning the variational measures associated to a real valued function, or, in a more general setting, to a vector valued function. Roughly speaking, given a function $\Phi$ defined on an interval $[a,b]$ of the real line it is possible to construct, using suitable families of intervals, a measure $\mu_{\Phi}$ which carries information about $\Phi$. If $\Phi$ is a real valued function, then the $\sigma$-finiteness of the measure $\mu_{\Phi}$ implies the a.e. differentiability of $\Phi$, while the absolute continuity of the measure $\mu_{\Phi}$ characterizes the functio…