On summability, multipliability, product integrability and parallel translation
OOn summability, multipliability, productintegrability, and parallel translation
Seppo Heikkil¨a ∗ and Anton´ın Slav´ık ∗∗ ∗ Department of Mathematical Sciences, University of OuluBOX 3000, FIN-90014, Oulu, FinlandE-mail: [email protected] ∗∗ Charles University in Prague, Faculty of Mathematics and PhysicsSokolovsk´a 83, 186 75 Praha 8, Czech RepublicE-mail: [email protected]ff.cuni.czDedicated to Professor Heikki Haahti on the occasion of his 85th birthday
Abstract
In this paper we provide necessary and sufficient conditions for the existence of the Kurzweil, McShaneand Riemann product integrals of step mappings with well-ordered steps, and for right regulated map-pings with values in Banach algebras. Our basic tools are the concepts of summability and multipliabilityof families in normed algebras indexed by well-ordered subsets of the real line. These concepts also leadto the generalization of some results from the usual theory of infinite series and products. Finally, wepresent two applications of product integrals: First, we describe the relation between Stieltjes-type prod-uct integrals, Haahti products, and parallel translation operators. Second, we provide a link betweenthe theory of strong Kurzweil product integrals and strong solutions of linear generalized differentialequations.
Keywords: product integral, well-ordered set, summability, multipliability, parallel translation,generalized ordinary differential equation
MSC classification:
The aim of this paper is to apply the concepts of summability and multipliability in order to generalize someresults in the theory of infinite series and products, and also to derive criteria for product integrability ofmappings which take values in Banach algebras. Product integrals and Haahti products are then used todefine parallel translation operators and to study their properties. The last part of the paper is devoted tothe relation between product integrals and linear generalized differential equations.The paper is organized as follows.In Section 2, we begin by recalling the definition of summability introduced by S. Heikkil¨a in [9]. We considersums of the form Σ α ∈ Λ x α , where the index set Λ is a well-ordered subset of R ∪ {∞} , and x α are elements ofa normed vector space; sums of this type were used in [9] as a tool in the study of integrability and impulsivedifferential equations. Then we proceed to the related novel concept of multipliability and consider productsof the form Π α ∈ Λ x α , where x α are elements of a normed algebra. In the case when Λ = N , our definitions andresults correspond to the usual theory of infinite series and products in normed spaces and algebras.In Section 3, we recall the general definition of the Kurzweil and McShane product integrals of the form (cid:81) ba V ( t, d t ), which were studied in [11, 17, 19, 21, 23], and which include the product integrals (cid:81) ba ( I + A ( t ) d t )1 a r X i v : . [ m a t h . F A ] A ug nd (cid:81) ba ( I +d A ( t )) considered in the next sections. In infinite-dimensional Banach algebras, the Kurzweil andMcShane product integrals lose some of their pleasant properties. To overcome this difficulty, we follow theideas from A. Slav´ık’s paper [21], introduce the strong Kurzweil and McShane product integrals (cid:81) ba V ( t, d t ),and establish some of their basic properties.In Sections 4 and 5, we focus on the product integrals (cid:81) ba ( I + A ( t ) d t ) in the sense of Kurzweil, McShane andRiemann. We apply the results from Sections 2, 3 and from the papers [9, 21] to derive new sufficient andnecessary conditions for product integrability of right-continuous step mappings having well-ordered steps,and then for right regulated mappings.Section 6 is devoted to the Riemann-Stieltjes and Kurzweil-Stieltjes product integrals (cid:81) ba ( I + d A ( t )). Themain result here is concerned with Kurzweil-Stieltjes product integrability of right-continuous step mappingswith well-ordered steps. The results from Sections 4, 5, 6 are illustrated on a number of examples.In Section 7, we present an application of Stieltjes-type product integrals to differential geometry. In [8],H. Haahti and S. Heikkil¨a studied operators corresponding to parallel translation of vectors along paths onmanifolds, and used product and Riemann-Stieltjes product integration techniques to establish the existenceof these operators; their results are recalled and generalized in Section 7.In Section 8, we provide a link between the theory of Kurzweil product integrals (cid:81) ba V ( t, d t ) and generalizeddifferential equations. We show that under fairly general assumptions, strong Kurzweil product integrabilityis equivalent to the existence of a strong Kurzweil-Henstock solution of a certain linear generalized differentialequation. In this section, we generalize some results of the theory of infinite series and products. A nonempty subsetΛ of R ∪ {∞} , ordered by the natural ordering < of R together with the relation t < ∞ for every t ∈ R , iswell-ordered if every nonempty subset of Λ has the smallest element. In particular, to every number β of Λ,different from its possible maximum, there corresponds the smallest element in Λ that is greater than β . It iscalled the successor of β and is denoted by S ( β ). There are no numbers of Λ in the open interval ( β, S ( β )).If an element γ of Λ is not a successor or the minimum of Λ, it is called a limit element. For every γ ∈ R ,we denote Λ <γ = { α ∈ Λ; α < γ } , Λ ≤ γ = { α ∈ Λ; α ≤ γ } . One of our basic tools in this paper is the following principle of transfinite induction: If Λ is well-ordered and P is a property such that P ( γ ) is true whenever P ( β ) is true for all β ∈ Λ <γ , then P ( γ ) is true of all γ ∈ Λ . The following definition of summability is adopted from [9].
Definition 2.1.
Let E be a normed space, and let Λ be a well-ordered subset of R ∪{∞} . Denote a = min Λ,and b = sup Λ. The family ( x α ) α ∈ Λ with elements x α ∈ E is called summable if for every γ ∈ Λ ∪ { b } , thereis an element Σ α ∈ Λ <γ x α of E , called the sum of the family ( x α ) α ∈ Λ <γ , satisfying the following conditions:(i) Σ α ∈ Λ β ε ∈ Λ <γ such that (cid:13)(cid:13)(cid:13)(cid:13) Σ α ∈ Λ <β x α − Σ α ∈ Λ <γ x α (cid:13)(cid:13)(cid:13)(cid:13) < ε, β ∈ Λ ∩ [ β ε , γ ) .
2e define the sum Σ α ∈ Λ x α of a summable family ( x α ) α ∈ Λ as Σ α ∈ Λ