Pettis Integrable Functions Research Articles

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.

Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals.We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given.

Correction to the paper ‘Copies of $\ell _{\infty }$ in the space of Pettis integrable functions with integrals of finite variation’ (Studia Math. 210 (2012), 93–98)

Correction to the paper ‘Copies of $\ell _{\infty }$ in the space of Pettis integrable functions with integrals of finite variation’ (Studia Math. 210 (2012), 93–98)

Improving integrability via absolute summability: a general version of Diestel’s Theorem

A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the $(p,\sigma)$-absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces ---including $C(K)$, $L^p$ and Hilbert spaces--- and operators ---$p$-summing, $(q,p)$-summing and $p$-approximable operators---.

On coincidence of Pettis and McShane integrability

R.Deville and J.Rodríguez proved that, for every Hilbert generated space X, every Pettis integrable function f: [0, 1] → X is McShane integrable. R.Avilés, G. Plebanek, and J.Rodríguez constructed a weakly compactly generated Banach space X and a scalarly null (hence Pettis integrable) function from [0, 1] into X, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from [0, 1] (mostly) into C(K) spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces K, that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from [0, 1] into C(K) in McShane sense.

Copies of $$c_0$$ c 0 in the space of Pettis integrable functions revisited

If (�, �, μ ) is a finite measure space and X a Banach space whose dual has a countable norming set we provide a proof of the fact that the space of all weakly μ-measurable (classes of scalarly equivalent) Pettis integrable functions f : � → X of finite variation, equipped with the variation norm, contains a copy ofc0 if and only if X does.

Lineability of non-differentiable Pettis primitives

Let \(X\) be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an \(X\)-valued Pettis integrable function on \([0,1]\) whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that \(\mathbf{ND}\), the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to \(\mathbf{ND}\).

Vitali type convergence theorems for Banach space valued integrals

Let (Ω, Σ, µ) be a complete probability space and let X be a Banach space. We introduce the notion of scalar equi-convergence in measure which being applied to sequences of Pettis integrable functions generates a new convergence theorem. We also obtain a Vitali type I-convergence theorem for Pettis integrals where I is an ideal on ℕ.

Copies of l∞in the space of Pettis integrable functions with integrals of finite variation

Let $ ( \varOmega ,\varSigma ,\mu ) $ be a complete finite measure space and $X$ a Banach space. We show that the space of all weakly $\mu $-measurable (classes of scalarly equivalent) $X$-valued Pettis integrable functions with integrals of finite variat

Copies of c 0 in the space of Pettis integrable functions with integrals of finite variation

Let (Ω,Σ,μ) be a complete finite measure space and X a Banach space. If all X-valued Pettis integrals defined on (Ω,Σ,μ) have separable ranges we show that the space of all weakly μ-measurable (classes of scalarly equivalent) X-valued Pettis integrable functions with integrals of finite variation, equipped with the variation norm, contains a copy of c0 if and only if X does.

Moments of vector measures and pettis integrable functions

Conditions, under which the elements of a locally convex vector space are the moments of a regular vector-valued measure and of a Pettis integrable function, both with values in a locally convex vector space, are investigated.

Fractional order boundary value problem with integral boundary conditions involving Pettis integral

In this article, we investigate the existence of Pseudo solutions for some fractional order boundary value problem with integral boundary conditions in the Banach space of continuous function equipped with its weak topology. The class of such problems constitute a very interesting and important class of problems. They include two, three, multi-point and nonlocal boundary-value problems as special cases. In our investigation, the right hand side of the above problem is assumed to be Pettis integrable function. To encompass the full scope of this article, we give an example illustrating the main result.

Integral Representations for a Class of Operators on <em>L</em><sup>1</sup><sub><em>E</em></sub>

Let \((X, \mathcal{A}, \mu)\) be a finite measure space, \(E\) a locally convex Hausdorff space, \(L_{E}^{1}\) the space of functions \(f: X \to E\) which are \(\mu\)-integrable by semi-norms, \(P(\mu, E)\) the space of Pettis integrable functions and \(P_{1}(\mu, E)\) those elements of \(P(\mu, E)\) which are measurable by semi-norms. We prove that a linear continuous mapping \( T: L_{E}^{1} \to E\) is of the form \(T(f)= \int g f d \mu\) (\(g \in L^{\infty}\)) if and only if \( h( T(f))=0\) whenever \( h\circ f=0 \) for any \( f \in L_{E}^{1}, h \in E'\). Similar results are proved for \(P(\mu, E)\) and \(P_{1}(\mu, E)\).

The McShane integral in weakly compactly generated spaces

Di Piazza and Preiss asked whether every Pettis integrable function defined on [ 0 , 1 ] and taking values in a weakly compactly generated Banach space is McShane integrable. In this paper we answer this question in the negative. Moreover, we give a counterexample where the target Banach space is reflexive.

Evaluating norms of Pettis integrable functions

Assuming that (Ω, Σ, μ) is a complete probability space and that X is a Banach space, we evaluate both the semivariation and the variation norm of a wide class of Pettis integrable functions f : Ω → X.

Convergence theorems for Pettis integrable functions and regular methods of summability

We characterize the relatively sequentially compact subsets of P 1 ( μ , X ) , the space of all X -valued Pettis integrable functions, where X is a separable Banach space, for the weak topology of P 1 ( μ , X ) by using the regular methods of summability. These characterizations are alternative descriptions of the results already done by Amrani and Castaing in [A. Amrani, C. Castaing, Weak compactness in Pettis integration, Bull. Pol. Acad. Sci. Math. 45 (2) (1997) 139–150]. We also study the theorem of Komlós in P 1 ( μ , X ) , which is a generalization of a result of E.J. Balder in [E.J. Balder, Infinite-dimensional extension of a theorem of Komlós, Probab. Theory Related Fields 81 (1989) 185–188, Theorem B]. We also prove some convergence theorems by applying the theorem. We also prove convergence theorems in P 1 ( μ , X ) analogous to the results of A. Amrani [A. Amrani, Lemme de Fatou pour l'intégrale de Pettis, Publ. Math. 42 (1998) 67–79] and H. Ziat [H. Ziat, Convergence theorems for Pettis integrable multifunctions, Bull. Pol. Acad. Sci. Math. 45 (2) (1997) 123–137]. Finally, we prove some convergence theorems in P 1 ( μ , X ) which are generalizations of some results of N.C. Yannelis [N.C. Yannelis, Weak sequential convergence in L p ( μ , X ) , J. Math. Anal. Appl. 141 (1989) 72–83] and A. Ülger [A. Ülger, Weak compactness in L 1 ( μ , X ) , Proc. Amer. Math. Soc. 113 (1991) 143–149].

APPROXIMATION OF BANACH SPACE VALUED NON-ABSOLUTELY INTEGRABLE FUNCTIONS BY STEP FUNCTIONS

AbstractThe approximation of Banach space valued non-absolutely 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 approximated 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.

On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies

This paper is devoted to investigate the existence of Pseudo solutions for the nonlinear m -point boundary value problem of fractional type D α x ( t ) + q ( t ) f ( t , x ( t ) ) = 0 , a.e. on [ 0 , 1 ] , α ∈ ( n − 1 , n ] , n ≥ 2 , x ( 0 ) = x ′ ( 0 ) = x ″ ( 0 ) = ⋯ = x ( n − 2 ) ( 0 ) = 0 , x ( 1 ) = ∑ i = 1 m − 2 ζ i x ( η i ) , where 0 < η 1 < ⋯ < η m − 2 < 1 , ζ i > 0 with ∑ i = 1 m − 2 ζ i η i α − 1 < 1 . It is assumed that q is a real-valued continuous function and f is a nonlinear Pettis integrable function.

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions f : [ 0 , 1 ] → L 1 ( μ ) , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function h : [ 0 , 1 ] → X and an absolutely summing operator u from X to another Banach space Y such that the composition u ○ h : [ 0 , 1 ] → Y is not Bochner integrable; in particular, h is not McShane integrable.

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions f : [ 0 , 1 ] → L 1 ( μ ) , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function h : [ 0 , 1 ] → X and an absolutely summing operator u from X to another Banach space Y such that the composition u ○ h : [ 0 , 1 ] → Y is not Bochner integrable; in particular, h is not McShane integrable.