Hindawi Journal of Function Spaces Volume 2018, Article ID 8169565, 9 pages https://doi.org/10.1155/2018/8169565

Research Article A Riesz Representation Theorem for the Space of Henstock Integrable Vector-Valued Functions

Tomás Pérez Becerra ,1 Juan Alberto Escamilla Reyna,1 Daniela Rodríguez Tzompantzi,1 Jose Jacobo Oliveros Oliveros ,1 and Khaing Khaing Aye2

1 Facultad de Ciencias F´ısico Matematicas,´ Benemerita´ Universidad AutonomadePuebla,Puebla,PUE,Mexico´ 2Department of Engineering Mathematics, Yangon Technological University, Yangon, Myanmar

Correspondence should be addressed to Tomas´ Perez´ Becerra; [email protected]

Received 15 January 2018; Revised 21 March 2018; Accepted 5 April 2018; Published 17 May 2018

Academic Editor: Adrian Petrusel

Copyright © 2018 Tomas´ Perez´ Becerra et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Using a bounded bilinear operator, we defne the Henstock-Stieltjes for vector-valued functions; we prove some theorems for Henstock integral and a Riesz-type theorem which provides an alternative proof of the representation theorem for real functions proved by Alexiewicz.

1. Introduction 2. Preliminaries

Henstock in [1] defnes a Riemann type integral which Troughout this paper �, �,and� will denote three Banach is equivalent to Denjoy integral and more general than spaces, ‖⋅‖�, ‖⋅‖�,and‖⋅‖�, which will denote their respective ∗ the Lebesgue integral, called the Henstock integral. Cao in norms, � the dual of �, �:�×�→�a bounded bilinear [2] extends the Henstock integral for vector-valued func- operator fxed, and [�, �] aclosedfniteintervaloftherealline tions and provides some basic properties such as the Saks- with the usual topology and the Lebesgue , which we Henstock Lemma. denote by �.Forafunction�:[�,�]→R we denote the Schwabik in [3] considers a bilinear form, defnes a Lebesgue integral of � on a measurable �⊂[�,�],whenit Stieltjes type integral, and performs a study about it including exists, by (�) ∫ �. [4]; following his ideas we give integration by parts theorem � involving a bilinear operator and, through it, we prove a Defnition 1. �:�×�→�is a bounded bilinear operator representation theorem for the space of Henstock vector- if � is linear in each variable and there exists �>0such valued functions. that ‖�(�, �)‖� ≤ �‖�‖�‖�‖�;inthiscase,thenormofthe Tis paper is divided into fve sections; in a frst step, in operator � is ‖�‖ = inf{� > 0 : ‖�(�, �)‖� ≤ �‖�‖�‖�‖�}. Section 2 we present some preliminaries and introduce the Henstock-Stieltjes integral via a bilinear We say that �={([��−1,��], ��) : � = 1,...,�}is a tagged and the Bochner integral, together with some basic proper- partition of [�, �] if {[��−1,��]:�=1,...,�}is a fnite collection ties. In Section 3 we provide two useful kinds of integration of nonoverlapping closed intervals whose union is [�, �] such by parts theorems, one of them in terms of the Bochner that �� ∈[��−1,��] for every �.Givenafunction� from [�, �] to integral and the other using Henstock-Stieltjes integral; the (0, ∞),calledgaugeon[�, �], we say that a tagged partition representation theorem is proved in Section 4 which, if {([� ,�], � ):�=1,...,�} � we consider real-valued functions, provides an alternative �−1 � � is -fne if proof of the representation theorem proved by Alexiewicz (Teorem 1 in [5]). [��−1,��] ⊂ (�� −�(��) ,�� +�(��)) for every �. (1) 2 Journal of Function Spaces

Defnition 2. Afunction�:[�,�]→�is Kurzweil integrable Teorem 5 (see [8, Tm. 7.4.5, pp. 217]). Te function �: in [�, �] if there exists �∈�such that for every �>0 [�, �] → � is Henstock integrable on [�, �] with the primitive ∗ there exists a gauge � on [�, �] such that if {([��−1,��], ��):�= � if and only if �:[�,�]→�is continuous and ACG on � 1,...,�}is a �-fne tagged partition of [�, �],then [�, �] such that � (�) = �(�) almost everywhere (a.e.) in [�, �], where the is in the sense of Frechet. � � � � � �∑� (� )[� −� ] −�� <�. �:[�,�]→� ∗ � � � � �−1 � (2) If a function is BV or BV on ,thenitis ��=1 � � �� bounded on �;thatis,�>0exists such that ‖�(�)‖� ≤�,for ∗ every �∈�.AsanAC(�) function is BV on � andanAC (�) � ∗ �=(�)∫ � function is BV on � (immediately from the defnitions), then We write � . ∗ every AC(�) or AC (�) function is also bounded in �.Itis �:[�,�]→� easy to see that if � is AC(�) and �0 ⊂�,then� is AC(�0), Defnition 3. Afunction is Henstock inte- ∗ grable in [�, �] if there exists �:[�,�]→�such that for every similarly if � is AC (�). �>0there exists a gauge � on [�, �] such that if {([��−1,��], ��): Te defnition of a function of strongly bounded variation �=1,...,�}is a �-fne tagged partition of [�, �],then can be extended considering the bilinear operator �:�× �→�. � � � ∑ �� (��)[�� −��−1] − [� (��) −�(��−1)]�� <�. (3) Defnition 6. Let �:[�,�]→�be a function and �= �=1 {�0,�1,...,��} a partition of [�, �]; we defne

� � �(�) − �(�) = (�) ∫ � � � � We write � . �� (�, �) = sup {∑ ��(��,�(��)−�(��−1))��}, (4) �=1 Te Henstock integral is also known as Henstock- where the supremum is taken over all possible elections of Lebesgue integral, briefy HL integral ([6]), or variational �� ∈�, �=1,2,...,�,with‖��‖� ≤1. Henstock integral ([7]). � � In[8]wecanfndsomepropertiesofbothintegrals (B) var� (�) = sup {�� (�, �)} , (5) such as the linearity, integrability over subintervals, and the continuity of the function �:[�,�]→�,calledprimitive, where the supremum is taken over all partitions of the � � [�, �] (B) �(�) B � given by �(�) = (�) ∫ � or �(�) = (�) ∫ �, �∈[�,�]. interval and var� is the strong -variation of � � on [�, �]. If we consider � � Defnition 4. Let �:[�,�]→�and let � be a subset of [�, �]. � � � ̂� � � �� (�, �) = sup {�∑� (��,�(��) −�(��−1))� } (6) � � (1) � is said to be of strongly bounded variation (BV) on ��=1 �� � if the number �(�, �) fl sup{∑ ‖�(��)−�(��)‖�} � � � intheequality(4),thenwedefne(B)var̂ (�) = sup{�̂ (�, is fnite, where the supremum is taken over all fnite � � �)} as the B-variation of � on [�, �]. sequences {[��,��]} of nonoverlapping intervals that have endpoints in �. � � If (B)var�(�) < ∞ or (B)var̂�(�) < ∞ we say that � is ∗ (2) � is BV on � if sup{∑� �(�, [��,��])} is fnite, where of strongly bounded B-variation or � is of bounded B-varia- the supremum is taken over all fnite sequences tion, respectively. {[��,��]} of nonoverlapping intervals that have end- It is straightforward that each function of strongly points in �,and�(�; [��,��]) = sup{‖�(�) − �(�)‖ : bounded variation is of strongly bounded B-variation. We �, � ∈ [��,��]} is the oscillation of � on [��,��]. recommend the reader interested in this topic to consult the study exposed in [9]. (3) � is said to be strongly absolutely continuous on � or (�) �>0 �>0 AC if for every there exists such that, 2.1. Stieltjes-Type . As we mentioned in the introduc- for every fnite or infnite sequence of nonoverlapping tion, Schwabik in [3] gives the next defnition and proves {[� ,�]} ∑ (� −�)<� intervals � � ,with � � � ,wehave somebasicpropertiessuchastheUniformConvergence ∑ ‖�(� )−�(�)‖ ≤ � � ,� ∈� � � � � where � � for all . Teorem. � ∗(�) �>0 �> (4) is AC if for every there exists �∈� �: 0 such that, for every fnite or infnite sequence or Defnition 7. is the Kurzweil-Stieltjes integral of [�, �] → � with respect to �:[�,�]→�if for every �>0 nonoverlapping intervals {[��,��]} satisfying ∑ (�� − � there exists a gauge � on [�, �] such that ��)<�,where��,�� ∈�for all �,wehave ∑ �(�; [� ,�]) < �. � � � � � � � � �∑� (� (� ) ,�(� ) −�(� )) −�� <�, ∗ � � � �−1 � (7) (5) � is ACG on � if � is the union of a sequence of ��=1 � ∗ � closed sets {��} such that, on each ��, � is AC (��). for every �-fne tagged partition {([��−1,�1], ��) : � = 1,...,�} of [�, �]. TenextresultgivesusacharacterizationofHenstock � � = (��) ∫ �(�,��) integrability. In this case we write � . Journal of Function Spaces 3

Now, we introduce the following integral. Teorem 10 (see [8, Cor. 1.4.4, pp. 26]). Astronglymeasurable function �:[�,�]→�is Bochner integrable on [�, �] if there Defnition 8. Afunction�:[�,�]→�is Henstock-Stieltjes exists a function �:[�,�]→R, which is Lebesgue integrable [�, �] �:[�,�]→� integrable in with respect to if there such that ‖�(�)‖� ≤ �(�), �∈[�,�]. exists �:[�,�]→�such that for every �>0there exists a gauge � of [�, �] such that if {([��−1,��], ��) : � = 1,...,�}is a Teorem 11 (see [8, Tm. 7.4.5, pp. 222]). Afunction�: �-fne tagged partition of [�, �],then [�, �] → � is Bochner integrable on [�, �] if and only if there exists a function �:[�,�]→�,whichis�� on [�, �] such � ��(�) = �(�) [�, �] � � that a.e. on . ∑ ��(�(��),�(��)−�(��−1)) −[�(��)−�(��−1)]�� �=1 (8) Given �:[�,�]→�and �:[�,�]→�we defne <�. the function �(�, �) : [�, �] → �,givenby�(�, �)(�) = �(�(�), �(�)); we will use this function from now on. � �(�) − �(�) = (��) ∫ �(�,��) We write � . Lemma 12. If �:[�,�]→�is a continuous function and �:[�,�]→�is Bochner integrable, then the function �(�, �) It is immediate that every Henstock integrable function is Bochner integrable. is Kurzweil integrable and its integrals are the same; we canrepeattheproofofthisfactforthepreviousStieltjes Proof. Since � is continuous, then it is strongly measurable; integrals. Similarly, we can prove the properties of linearity moreover there exists �>0such that ‖�(�)‖� ≤�for all �∈ and integrability over subintervals for the Henstock-Stieltjes [�, �].Because� is strongly measurable and � is continuous, integral directly of the proofs in [8] with slight changes. We �(�, �) is strongly measurable. omit the formulations and the proofs of such results. By Teorem 10 the real function ‖�(�)‖� is Lebesgue integrable. Teorem 9 (see [3, Tm. 11]). Assume that the functions �, Now � : [�, �] → � �:[�,�]→� (B)̂�(�) < � ,and are given. If var� � � � � � ��(�,�)(�)� ≤ ‖�‖‖� (�)‖ �� (�)� ∞ ( )∫ �(� ,��) � �� � � �� , the Kurzweil-Stieltjes integrals KS � � exist and � � (12) the sequence {��} converges on [�, �] uniformly to �, then the ≤�‖�‖ �� (�)� ; � � �� ( )∫ �(�,��) integral KS � exists and hence �(�, �) is Bochner integrable as a consequence of � � Teorem 10. (KS) ∫ �(��,��)= lim (KS) ∫ �(�,��). (9) � �→∞ � It is known that every vector-valued function which is ∗ 2.2. Bochner Integral. Let us recall that a function �:[�,�]→ strongly measurable is weakly measurable; that is, � �: � ∗ ∗ � is called simple if there is a fnite sequence {��}�=1 ⊂ [�, �] [�, �] → R is measurable for each � ∈�; the inverse, in of Lebesgue measurable sets such that �� ∩�� =0for �=� ̸ general, is not true (see [10, Example 5, Chapter II, §1, and pp. � and [�, �] = ⋃�=1 ��,where�(�) = �� ∈�for �∈��, 43]) however under certain conditions is equivalent. � = 1,...,�, and in this case the Bochner integral of � is � Teorem 13 (B)∫ �=∑ � �(� ) (see [10, Tm. 2, Chapter II, §1, pp. 42] (Pettis)). � � � � . �:[�,�]→� �:[�,�]→� Let be a function. Te following conditions are Afunction is strongly measurable if equivalent: there exists a sequence of simple functions that converges � [�, �] pointwise to a.e. on . (i) � is strongly measurable. Afunction�:[�,�]→�isBochnerintegrableifthereis � a sequence of simple functions �� : [�, �] → �, �∈N,such (ii) is weakly measurable and there exists a measurable �⊂[�,�] �([�, �] − �) = 0 �(�) that lim�→∞��(�) = �(�) a.e. in [�, �] and set with such that is separable. � � � lim (�) ∫ ��� −�� =0; (10) Teorem 14. If �:[�,�]→�is continuous a.e., then � is � � strongly measurable. � �:[�,�]→� (B)∫ � the Bochner integral of is denoted by � Proof. As � is continuous a.e., then � is weakly continuous; ∗ and is defned by that is, � �:[�,�]→R is continuous a.e. and, hence, meas- urable. � � We defne � = {� ∈ [�, �] : �(�) is continuous}. � is Leb- (B) ∫ �=lim (B) ∫ ��. (11) � � � esgue measurable with �([�, �] − �) = 0,as[�, �] is separable and then � is separable and, furthermore, � is continuous, We will use the following well-known results of the and then �(�) is separable; hence � is strongly measurable by Bochner integral. the Pettis Teorem. 4 Journal of Function Spaces

Lemma 15 (see [11, Lemma 6]). If �:[�,�]→�is of strongly Proof. By Corollary 16, �(�, �) is Bochner integrable. Let �: bounded variation on [�, �],then� is Bochner integrable on [�, �] → � be a function given by [�, �] . � � (�) =�(� (�) ,�(�)) − (B) ∫ �(�,�). (16) As a consequence of Lemmas 12 and 15, we have the � following result. � is continuous due to the continuity on [�, �] of �, �,and � Corollary 16. Let �:[�,�]→�be Henstock integrable on (B)∫ �(�, �) � . [�, �] and � its primitive, �:[�,�]→�of strongly bounded � ∗ [�, �] �(�, �) [�, �] Teorem 5 implies that is ACG on and Te- variation, then is Bochner integrable on . orem 11 implies that � is AC on [�, �];hencethefunction ∗ ∗ � → �(�(�), �(�)) is ACG on [�, �] by Lemma 18. Finally, ItiseasytoprovethatthesetoffunctionsAC,AC ,ACG, ∗ ∗ Lemma 17 implies that � is ACG on [�, �]. and ACG on �⊂[�,�]form vector spaces with the sum and In order to prove that � is diferentiable on [�, �],using productbyscalars;moreover,thesespacesoffunctionsare the fact that �(�(�)/(� − �), �(�)) = �(�(�), �(�)/(� − �)) for algebras under the bilinear operator �. every �, �∈[�,�],wecalculate Lemma 17. Let �:[�,�]→�and �:[�,�]→�be � ∗ �� (� (�) ,�(�)) −�(� (�) ,�(�)) functions and � asubsetof[�, �].If� is ACG (�) and � is � −�(�(�) ,�(�)) ∗ � �−� ��(�),then�+�is ACG (�). � � � � �� (�) −�(�) � � ∗(�) �=⋃� � −�(�(�) ,�(�))� ≤ ‖�‖ � −�(�)� Proof. Since is ACG then � �,where is � � �−� � ∗(� ) � � (17) AC � . � � �>0 �>0 �� (�) −�(�) � For every there exists such that, for every ⋅ ‖� (�)‖ + ‖�‖‖� (�)‖ � −�(�)� {[� ,�]:�,� ∈�} ∑ (� −�)<� ∑ �(�; � � � � � � � � � ,with � � � ,wehave � � �−� �� [� ,�]) < � ∑ ‖�(� )−�(�)‖ <� � � and � � � � ,and � � + ‖�‖ �� (�)�� ‖� (�) −�(�)‖� , � � ��(�)+�(�)−�(�)−�(�)� � � � � � �� which tends to 0 when �→�;hence�(�, �) is diferentiable � � � � (13) [�, �] (�(�, �))� =�(�,�)+�(�,�) ��(�) = ≤ ��(�)−�(�)� + ��(�)−�(�)� , on and .Ten � � � �� � � � �� (�(�, �))� − �(�, �) = �(�, �) + �(�,�) − �(�, �) = �(�,�). � By Teorem 5, (�) ∫ �(�,�) exists and (15) is fulflled. and then ∑� �(� + �; [��,��]) < 2�. �

Lemma 18. Let �:[�,�]→�and �:[�,�]→�be func- 3.2. Involving Henstock-Stieltjes Integral ∗ tions and � asubsetof[�, �].If� is ACG (�) and � is ��(�), ∗ Teorem 20. Let �:[�,�]→�and �:[�,�]→�be func- then �(�, �) is ACG (�). � (��) ∫ �(�,��) (B) �(�) < tions. If the integral � exists and var� Proof. Te proof is analogous to Lemma 17 changing the ∞,then inequality 2 by � � � � � � � �(��) ∫ �(�,��)� ≤ �� (�)� (B) � (�). � � � � sup � �� var� (18) � � � � � �∈[�,�] ��(�(��),�(��)) − � (� (��),�(��))�� � �� � � � � �>0 � [�, �] ≤ ‖�‖ ��(��)−�(��)�� ��(��)�� (14) Proof. Let . Tere exists a gauge on such that � � � � � � � � + ‖�‖ ��(�)� ��(�)−�(�)� . � � � � � �� � � � �� � � ∑ ��(�(��),�(��)−�(��−1)) − (��) ∫ �(�,��)� � � � �=1 � �−1 �� (19) <�,

3. Integration by Parts Theorem for every �={([��−1,��], ��) : � = 1,...,�},whichis�-fne. Ten, 3.1. Involving Bochner Integral � � � � � � � � � � � Teorem 19. Let �:[�,�]→�be a Henstock integrable �(��) ∫ �(�,��)� ≤ �(��) ∫ �(�,��) � � � � � function with primitive �, �:[�,�]→�of strongly bounded � � � variation, and � theBochnerprimitiveof�.Ten(�) ∫ �(�, � � − ∑�(�(�),�(�)−�(� )) �) exists and � � �−1 �=1 � � � � � � (�) ∫ �(�,�)=�(� (�) ,�(�)) − (B) ∫ �(�,�). (15) + ∑�(�(��),�(��)−�(��−1))� ≤� � � � �=1 �� Journal of Function Spaces 5

� � � � � � � [� (� )−� (� )] ��(�)−� (� )� � �(��) ��(��)� � � � � � � � � �� + ∑ ��( � ,�(�) ≤ ∑ ��( � � , � � � � � ��(�)−� (� )� � ��(��)� �,�(� )=� ̸ (� ) � � � � � �� �=1,‖�(��)‖=0̸ � �� � � � � � � � � � � � � � � � � �(��)−�(��−1))� +2�≤ sup �� (�) −�� (�)�� (B) −�(��−1))� ≤�+ sup �� (�)� (B) var (�). � � � �� � �� �∈[�,�] �� �∈[�,�] � (20) ⋅ var� (�)+ 2� ≤ 3�. (25)

� Now, we shall prove the following Teorem, which is a (��) ∫ �(�,��) Hence, � exists and consequence of Teorem 9. � � Teorem 21 �, � : (uniform convergence theorem). Let � (��) ∫ � (�, ��) = lim (��) ∫ � (��,��) . (26) � � �→∞ � [�, �] → �, � = 1, 2, . . .,and�:[�,�]→�.If(B)var�(�) < � ∞ (��) ∫ �(� ,��) � ,theintegrals � � exist for each ,andthe sequence �� converges uniformly to � in [�, �], then the integral � Teorem 22. If �:[�,�]→�is a step function, then for (��) ∫ �(�,��) exists and � � �:[�,�]→� (��) ∫ �(�,��) every ,theintegral � exists. � � (��) ∫ �(�,��)= lim (��) ∫ �(��,��). (21) � �→∞ � Proof. Analogous to the proof of [12, Lemma 3.2], is enough to prove for functions of the forms �[�,�]�, �[�,�]�, �[�]�,and �>0 � � Proof. Let ;because � converges uniformly to ,there �[�]�,where� ∈ (�, �) and �∈�.Let� ∈ (�, �), �∈�, � >0 �>� �∈[�,�] exists 0 such that for every 0 and and �=�[�,�]�.Given�>0we defne �(�) = � if �=� � � and �(�) = (1/2)|� − �| if �=� ̸ ;thenforany�-fne tagged ��� (�) −�(�)�� <�. (22) partition {([��−1,��], ��) : � = 1,...,�}of [�, �], � is the tag of �, � > � ‖� (�) − � (�)‖ ≤ 2�. � Hence, for every 0, � � � one subinterval, if �� ∈(�,�), �(��)=0,and∫ �(�,��) = 0; �(�) = � Teorem 9 implies the existence of the integral otherwise (��) ∫ �(�,��) for every �⊂[�,�]and lim�→∞��(�) = � � �(�). � ∈ N ‖� (�) − �(�)‖ < � Hence, there exists 1 such that � � ∑ ��(�(�),�(�)−�(� )) � �>� � � � �−1 ,forevery 1. �=1 �> {� ,� } Let max 0 1 be fxed; as the integral � −[�(�,�(�)) − � (�, � (� ))]� � � �−1 �� (27) (��) ∫ �(��,��)fl �� (�) −�� (�) (23) � � � � = ∑ ��(�(��)−�,�(��)−�(��−1))�� =0<�. exists, there exists a gauge � on [�, �] such that, for every �- �=1 {([� ,�], � ):�=1,...,�} [�, �] fnetaggedpartition �−1 � � of , � (��) ∫ �(�,��) = �(�, �(�)) − �(�, �(�)). � � Hence � Te proofs � ∑ ��(� (� ),�(�)−�(� )) of the cases �=�[�,�]�, �=�[�]�,and�=�[�]� are � � � � �−1 �=1 � analogous. (24) � � 1 � Schwabik in [3] introduces the concept of vector-valued − (��) ∫ �(� ,��)� <�. � � ��−1 �� regulated functions; we shall only use the following charac- terization. We have � Teorem 23 �:[�,�]→� � � (see [3, Prop. 2]). is regulated if ∑ ��(�(�),�(�)−�(� )) −[�(�)−�(� )]� � � � �−1 � �−1 �� and only if it is the uniform limit of step functions. �=1

� Teorem 24. �:[�,�]→� �:[�,�]→ � If is regulated and ≤ ∑ ��(�(�),�(�)−�(� )) − � (� (� ),�(�) � � � � �−1 � � � � (B) �(�) < ∞ (��) ∫ �(�, ��) �=1 with var� , then the integral � � exists. � � � −�(� ))� + ∑ ��(� (� ),�(�)−�(� )) − (��) �−1 �� � � � � �−1 � �=1 � Proof. In as much as is regulated, there exists a sequence �� : [�, �] → �, � = 1, 2, . . ., of step functions which � � � � � 1 � � 1 � ⋅ ∫ �(� ,��)� + ∑ �(��) ∫ �(� ,��)−[�(�) converges uniformly to ,byTeorem22;theintegrals � � � � � � ��−1 �� �=1 � ��−1 (��) ∫ �(� ,��) � = 1,2,... � � exist for each .TeUniform � � Convergence Teorem implies the existence of the integral −�(� )]� � �−1 � (��) ∫ �(�, ��) �� � . 6 Journal of Function Spaces

Teorem 25 (integration by parts theorem). If �:[�,�]→� (1) Te essence in the proof of Teorem 19 is the deriva- is Henstock integrable, � its primitive, and �:[�,�]→�with tive of the primitive of the function �→�(�(�), � (B) �(�) < ∞ (�) ∫ �(�,�) �(�)),that is, the Fundamental Teorem of . var� ;then � exists and (2)InTeorem25theFundamentalTeoremofCalculus � does not apply because � is not necessarily diferen- (�) ∫ �(�,�)=�(�(�) ,�(�)) � � tiable, and if it is, the primitive of � , in general, is not (28) � � . − (��) ∫ �(�,��). (3) Te condition of diferentiability on a function �: � [�, �] → � of strongly bounded variation is equiva- Proof. Let �>0.Since� is Henstock integrable, with � its lent to � and has the Radon-Nikodym´ property (see primitive, there exists a gauge �1 such that if �1 = {([��−1, Distel and Uhl [10, Chapter VII §6]). Is it fulflled with B ��], ��); �=1,2,...,�}is a �1-fne tagged partition of [�, �], functions of strongly bounded -variation?

� � � Terefore, the condition of � is one that ensures the Fun- ∑ ��(�)(� −� )−[�(�)−�(� )]� <�. � � � �−1 � �−1 �� (29) damental Teorem of Calculus, so we have the following �=1 theorems. On the other hand, since � is continuous, it is regulated, and � Teorem 26. Let �:[�,�]→�be continuous function and (��) ∫ �(�, ��) � � by Teorem 24, � exists; then there exists a �:[�,�]→�with (B)var (�) < ∞ such that � exists on � � = {([� ,� ], � ); � = 1,2,...,�} � gauge 2 such that if 2 �−1 � � [�, �].Ten is a �2-fne tagged partition of [�, �], � � � � � � (�) ∫ �(�,� )=(��) ∫ �(�,��). (32) � ∑ ��(�(��),�(��)−�(��−1)) � � �=1 � � (30) (��) ∫ �(�, ��) �> � � Proof. exists by Teorem 24; then given � � � � 0 � {([� ,�], � ) : � = 1,...,�} � − (��) ∫ �(�,��)� <�. ,wehaveagauge and �−1 � � a -fne � � � �−1 �� tagged partition of [�, �].Since� exists, for every �∈[�,�] ∗ ∗ ∗ there exists �� >0such that if |� − �| < �� then We defne a gauge � by � (�) = min{�1(�), �2(�)}. Let � = {([� ,� ], � ), � = �,...,�} �∗ � � �−1 � � be a -fne tagged partition �� (�) −�(�) � � [�, �] � −� (�)� <�. (33) of and by the lef-right process (see [13, Section 1, pp. � �−� � 6]) we can assume that the tags are the lef endpoint of each subinterval; then We defne a gauge �1 : [�, �] → (0, ∞) by �1(�) = min{��, �(�)} {([� ,� ], � ) : � = 1,...,�} � � � .Forall �−1 � � which is 1-fne � ∑ ��(�(� ),�(� )) (� −� )−[�(�(� ),�(� )) tagged partition of [�, �] and supposing that each tag is the � � � � �−1 � � �=1 � lef endpoint of its respective subinterval we have

� � � � � � � − � (� (� ),�(� )) − (��) ∫ �(�,��)]� � � �−1 �−1 � ∑ ��(�(��),� (��))(�� −��−1)−(��) � � � �−1 �� �=1 �

� � � � � � ≤ ∑ ��(�(��)(� � −��−1) ⋅ ∫ �(�,��)� � �=1 (31) ��−1 �� � � −[�(� )−�(� )],�(�))� � � �(�)−�(� ) � �−1 � �� � � � �−1 ≤ ∑ ��(�(��),� (��)− ) (34) � � �� −��−1 � � �=1 � + ∑ ��(�(��),�(��)−�(��−1)) − (��) � � � � � � �=1 � � ⋅(�� −��−1)� + ∑ ��(�(��),�(��)−�(��−1)) � � � � � � �� �=1 � ⋅ ∫ �(�,��)� ≤�(B) � (�)+ �. � var� � � � � �−1 �� � � − (��) ∫ �(�,��)� <�(� ‖�‖ (�−�) +1) , � ��−1 ��

Aswecansee,wehavetwotypesofintegrationbyparts where �>0is the bound of � due to it is continuous. theorems, one is of the Stieltjes type and the other is non- Stieltjes; it is possible to ask for the conditions so that the Obviously,wecanchangetheconditionover� in integral of the Stieltjes type becomes a non-Stieltjes; for that, Teorem 26 if we ask for strongly bounded variation and we must do the following analysis: impose the Radon-Nikodym´ property on �; either with these Journal of Function Spaces 7 conditions or with those of Teorem 26 we can write equality By the integration by parts theorem we have (28) as � � � � � � � � � � � ��(�)� = �(�) ∫ �(�,�)� = ��(�(�) ,�(�)) � ∫ � (�, �) =�(� � ,� � ) − � ∫ � (�, ��) . � �� � � � ( ) ( ) ( ) ( ) (35) � � �� � � � � � � � �:[�,�]→R � � Afunction satisfes the Lipschitz condi- − (��) ∫ �(�,��)� ≤ ��(�(�) ,�(�)) �>0 |�(�) − �(�)| < �|� − �| � � � tion if there exists such that , , � �� � �∈[�,�],and� is of bounded slope variation (BSV) if � � � � � �−2 ��(� )−�(� ) �(� )−�(�)� − ∑�(�(�),�(�)−�(� ))� ∑ � �+2 �+1 − �+1 � � � � �−1 � � � (36) �=1 �� �=0 � ��+2 −��+1 ��+1 −�� � � � � is bounded for all divisions �=�0 <�1 <⋅⋅⋅<�� =�. + �∑�(�(�),�(�)−�(� )) � � � �−1 Leein[14,p.75]provesthatafunction�:[�,�]→R ��=1 is the primitive (in the sense of Henstock) of a function of � � � (42) strongly bounded variation on [�, �] if and only if � satisfes � � � − (��) ∫ �(�,��)� ≤ ��(�(�) ,�(�))�� the Lipschitz condition and is of bounded slope variation � � �� on [�, �]; this same characterization can be extended for our case (see the proof of [15, Tm. 10]). So we can see that the � � � + ∑ ��(�(�),�(�)−�(� ))� +� Fundamental Teorem of Calculus applies and we can restate � � � �−1 �� �=1 the above integration by parts formula as follows. � � � � ≤ ��(�(�) ,�(�))� + ��� Corollary 27. Let �:[�,�]→�be a function and � its � � �:[�,�]→� � � � primitive. If satisfes the Lipschitz condition and � �(��) � � ⋅ ∑ ��( ,�(�)−�(� ))� (�) ∫ �(�,�) � � � � �−1 � is of bounded slope variation, then � exists and � ��(��)� � �=1,‖�(��)‖�=0̸ � �� � � � � � � � � � �� � +�≤��� ‖�‖ �� (�)� + ��� (B) � (�)+ �. (�) ∫ �(�,�)= �(�,�)�� − (�) ∫ �(�,� ). (37) � �� � �� � �� var� � � It follows that 4. Representation Theorem � � ��(�)� � � � � ≤ ‖�‖ �� (�)� + (B) � (�). Now, we will establish an important connection between the � � � �� var� (43) ��� space of Henstock integrable functions and its : a � Riesz representation theorem. So � is bounded and continuous. �([�, �], �) Defnition 28. Let be the space of all Henstock �(�, �) denotes the space of continuous linear operators integrable functions from [�, �] to �. We defne a norm on � � �:�×�(�,�)→� �([�, �], �) from to and is the bilinear bounded , called Alexiewicz norm (see e.g., [5] or [16]), by operator given by �(�, �) = �(�). � � � � � � � ��� = {� � ∫ �� :�≤�≤�}. � �� sup �( ) � (38) Teorem 30. Let � : �([�, �], �) → � be a linear continuous � � � � operator defned on the space of the Henstock integrable � �:[�,�]→�(�,�) Teorem 29. Let �:[�,�]→�be a function. If (B)var�(�) functions. Tere exists a function with � <∞,then (B)var�(�) < ∞ such that � � �(�)= (�) ∫ �(�,�) (39) � � (�) = (�) ∫ � (�, �) , (44) � defnes a continuous linear operator on the space of Henstock � ∈ �([�, �], �) integrable functions into � with for every . � � � � ∈ [�, �] �∈� � �∈ ‖�‖ ≤ ‖�‖ �� (�)�� + (B) var� (�). (40) Proof. For each and we have [�,�] �([�, �], �). We defne a function �� :�→�by Proof. Let �:[�,�]→�the primitive of � and �>0; � {([� ,�], by Teorem 24 there exists a gauge such that if �−1 � �� (�) =�(�[�,�]�). (45) ��); �=1,...,�}is �-fne, � � � � � � � In this form, � is linear and it is also continuous due to � � ∑ ��(�(��),�(��)−�(��−1)) − (��) ∫ �(�,��)� � � � � � � � �� (�)� = ��(� �)� ≤�‖�‖ , �=1 � �� (41) � � �� � [�,�] �� � (46) <�. where � = ‖�‖(� − �). 8 Journal of Function Spaces

∗ We defne a function �:[�,�]→�(�,�)by Consider � a and �=� in the defnition of strongly bounded (B)-variation function; then this is � (�) =��. (47) equivalent to the defnition of strongly bounded variation � Now, (B)var�(�) < ∞, indeed, for every arbitrary parti- function,sowehavethenextresult. tion {[��−1,��]:�=1,...,�}and for every ��, �=1,...,�,with Corollary 32. � �([�, �], ‖��‖� ≤1, is a linear continuous functional on �) �:[�,�]→�∗ � if and only if there exists a function and � � �(�, [�, �]) < ∞ such that ∑ ��(��,�(��)−�(��−1))�� �=1 � �(�)= (�) ∫ � (�) ∘�(�) , � (53) � � � = ∑ ��� (��)−�� (��)� � � �−1 �� � ∈ �([�, �], �) �=1 for every .

� � � (48) Hence, the dual space of �([�, �], �) is isometrically = ∑ ��(�[�,� ]��)−�(�[�,� ]��)� � � �−1 �� isomorphic to the space of functions of strongly bounded �=1 variation. � � � � � � Since the space of Henstock integrable real-valued func- = ∑ ��(� � )� ≤ ‖�‖ ∑ �� � (� −� ) � [��−1,��] � �� � ��� � �−1 tions coincides with the Kurzweil integrable functions we will �=1 �=1 name the integral as the integral of Henstock-Kurzweil. ≤ ‖�‖ (�−�) . As a particular case of Corollary 31 (with �=�(�,�)= �=R) we have obtained the following representation theo- Suppose that �:[�,�]→�is a step function; by Teorem rem for the space of Henstock-Kurzweil integrable functions. � 22 we have that the integral (��) ∫ �(�, ��) exists. � Corollary 33. � : ��([�, �], R)→R Given � ∈ �([�, �], �), we take its primitive �: is a linear continuous functional if and only if exists a function �:[�,�]→R, [�, �] → �.Since� is continuous, there exists (��) sequence �(�[�, �], ) < ∞,suchthat of linear piecewise functions such that �� →�uniformly � [�, �] (��) ∫ �(� ,��) � � on .Teintegral � � exists for every by � (�) = (��) ∫ �� , (54) Teorem 24. � By the Uniform Convergence Teorem, the integral � � for every Henstock-Kurzweil integrable function. (��) ∫ �(�, ��) � exists; hence � Remarks 34. Te result above was proved in [5] by Alex- iewicz; later other proofs of the theorem arose, for example, � (�) = lim �(��)= lim (��) ∫ �(��,��) �→∞ �→∞ � thoseprovidedbySargentandLee(Teorem4in[17] (49) � and Teorem 12.7 in [14, pp. 76], resp.), who use diferent = (��) ∫ �(�,��). techniques from those used in this work; for example, Hahn- � BanachTeoremisnotnecessaryfortheproofofTeorem30. Let (��) be the sequence of of ��; �� is simple for each �,since�� →�uniformly Te Integration by parts Corollary 27 yields a new � � � � representation theorem without using Stieltjes integral, which �� −�� = �� −�� �→ 0 ; � � �� � � �∞ (50) we shall establish next. Te proof of the frst result below is by the continuity of � and the Integration by parts Teo- analogous to Teorem 29; using equality (35), we will only rem25wehavethat sketch the proof of the second result. � �(�)= �(�)= (�) ∫ �(�,�) Teorem 35. If �:[�,�]→�is of strongly bounded B- �→∞lim � �→∞lim � � � variation and diferentiable and � is bounded, then �(�) = (51) � � (�) ∫ �(�,�) � is a continuous linear operator on the space = (�) ∫ �(�,�). �([�, �], �) � � into .

If �:[�,�]→�is of bounded slope variation and satis- Finally, we have the following representation theorem. fes the Lipschitz condition then the previous theorem is also true; we shall prove the second part in the new representation Corollary 31. � : �([�, �], �) → � is a linear continuous theorem. operator if and only if there exists a function �:[�,�]→�(�, � Teorem 36. Let � : �([�, �], �) → � be a linear contin- �) with (B)var�(�) < ∞ such that uous operator defned on the space of the Henstock integrable � �:[�,�]→�(�,�) �(�)= (�) ∫ �(�,�), (52) functions. Tere exists a function � of bounded slope variation and Lipschitz such that �(�) = � � ∈ �([�, �], �) (�) ∫ �(�,�) � ∈ �([�, �], �) for every . � ,forevery . Journal of Function Spaces 9

Proof. Let �∈�and � be the primitive of � ∈ �([�, �], �). [11] Y. Guoju, W. Congxin, and L. P. Yee, “Integration by Parts for We defne the function ��(�) = �(�[�,�]�);thisfunction the Denjoy-Bochner Integral,” Southeast Asian Bulletin of Math- is in �(�, �), and we defne �:[�,�]→�(�,�)by ematics,vol.26,no.4,pp.693–700,2003. �(�) = ��; � is of bounded slope variation and satisfes the [12] G. A. Monteiro and M. Tvrdy,´ “On Kurzweil-Stieltjes integral Lipschitz condition (see the proof of [15, Tm. 10]); hence in a Banach space,” Mathematica Bohemica,vol.137,no.4,pp. � it is diferentiable and � is of strongly bounded variation. 365–381, 2012. Following the proof of Teorem 30 using � which is also [13] R.G.Bartle,AModernTeoryofIntegration,vol.32ofGraduate of strongly bounded variation, by Teorem 26 the integral Studies in Mathematics, American Mathematical Society, 2001. � � (��) ∫ �(�, ��) (�) ∫ �(�, ��) [14] P. Y. Lee, “Lanzhou Lectures on Henstock Integration,” in Real � is equal to � and by integration � Analysis, vol. 2, World Scientifc, Singapore, 1989. �(�) = (�) ∫ �(�,�). by parts Corollary 27, we have � [15]R.M.ReyandP.Y.Lee,“ARepresentationTeoremforthe Space of Henstock-Bochner integrable functions,” Southeast �([�, �], �) Te last representation theorem identifes AsianBulletinofMathematics, pp. 129–136, 1993. with the space of primitives of the functions of strongly [16] S. Saks, Teory of the Integral, PWN, Monografe Matematyczne, bounded variation, unlike the Corollary 31 which identifes it Warsaw, Poland, 2nd edition, 1937. B with the space of functions of strongly bounded -variation. [17] W. L. C. Sargent, “On linear functionals in spaces of condition- ally integrable functions,” Quarterly Journal of Mathematics,vol. Conflicts of Interest 1, no. 1, pp. 288–298, 1950. Te authors declare that they have no conficts of interest. Acknowledgments Tis research has been able to see the light thanks to the help, comments, suggestions, and unconditional support of Professor Lee Peng Yee and the work team and the staf of MME/NIE of Nanyang Technological University in Singa- pore. Te authors will be eternally grateful. Tis research has been supported by Conacyt, VIEP-BUAP, DGRIIA-BUAP, and the Academic Group of Mathematical Modeling and Diferential Equations-FCFM-BUAP. References

[1]R.Henstock,“ANewDescriptiveDefnitionoftheWardInte- gral,” Journal of the London Mathematical Society,vol.s1-35,no. 1,pp.43–48,1960. [2]S.S.Cao,“TeHenstockintegralforBanach-valuedfunctions,” Southeast Asian Bulletin of Mathematics,vol.16,pp.35–40,1992. [3] S. Schwabik, “Abstract Perron-Stieltjes integral,” Mathematica Bohemica,vol.121,no.4,pp.425–447,1996. [4] S. Schwabik, “Anote on integration by parts for abstract Perron- Stieltjes integral,” Math. Bohem,vol.2001,no.3,pp.613–629, 2001. [5] A. Alexiewicz, “Linear functionals on Denjoy-integrable func- tions,” Colloquium Mathematicum,vol.1,no.4,pp.289–293, 1947. [6] B. Satco, “Solution tube method for impulsive periodic dif- ferential inclusions of frst order,” Nonlinear Analysis: Teory, Methods & Applications,vol.75,no.1,pp.260–269,2012. [7] V. Marrafa, “A descriptive characterization of the variational Henstock integral,” in Proceedings of the International Mathe- matics Conference,vol.22,pp.73–84,Manila,Philippines,1998. [8] S. Schwabik and G. Ye, “Topics in Banach Space Integration, Real Analysis,” in Real Analysis, vol. 10, World Scientifc, Singapore, 2005. [9] G. A. Monteiro, “On Functions of Bounded Semivariation,” Real Analysis Exchange,vol.40,no.2,pp.233–276,2015. [10] J. Diestel and J. J. Uhl, VectorMeasures, Mathematical surveys and monographs, American Mathematical Society, Mathemat- ical surveys and monographs, 1977. Advances in Advances in Journal of The Scientifc Journal of Operations Research Decision Sciences Applied Mathematics World Journal Probability and Statistics Hindawi Hindawi Hindawi Hindawi Publishing Corporation Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018

International Journal of Mathematics and Mathematical Sciences

Journal of Optimization Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Submit your manuscripts at www.hindawi.com

International Journal of Engineering International Journal of Mathematics Analysis Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Journal of Advances in Mathematical Problems International Journal of Discrete Dynamics in Complex Analysis Numerical Analysis in Engineering Dierential Equations Nature and Society Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

International Journal of Journal of Journal of Abstract and Advances in Stochastic Analysis Mathematics Function Spaces Applied Analysis Mathematical Physics Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018