Int. J. Anal. Appl. (2023), 21:49 Sufficient Conditions for Convergence of Sequences of Henstock-Kurzweil Integrable Functions Yassin Alzubaid∗ Department of Mathematical Sciences, Umm Al-Qura University, Makkah Almukarramah, Saudi Arabia ∗Corresponding author: yazubaidi@uqu.edu.sa Abstract. The main aim of this paper is to present our approach of obtaining sufficient conditions for convergence of sequences of Henstock-Kurzweil integrable functions. Our approach involves the use of the concept of multiplier functions, where we define a class Φ of multipliers for the Henstock-Kurzweil integral. We consider a sequence (fn) of Henstock-Kurzweil integrable functions on a non-degenerate interval [a,b] and we assume that (fn) converges point wise to a function f . Then we show that f is Henstock-Kurzweil integrable and its integral is equal to the limit of the sequence ( ∫ b a fn) if there exists φ ∈ Φ such that the defined functionals of the type F (φ,fn) satisfy the imposed conditions. Beside the fact that the results regarding the convergence under the integral sign are always of great importance, the method introduced here can be imitated and used to obtain other results on the related areas. 1. Introduction The Henstock-Kurzweil integral was originally introduced in [5] and [6]. It is a generalization of the Riemann integral. It is a very powerful technique of integration in a way that the space of all Henstock- Kurzweil integrable functions strictly contains the spaces of all Lebesgue and Riemann integrable functions. However, the space of all Henstock- Kurzweil integrable functions, unlike Lebesgue space, lacks the property of completeness. For further reading about the Henstock-Kurzweil integral reader may consult [4], [9], and [11]. In this paper, we investigate the sufficient conditions for the convergence of sequences of Henstock- Kurzweil integrable functions. Let (fn) be a sequence of Henstock-Kurzweil integrable functions that converges point-wise to f on [a,b]. The properties and integrability of the function f was studied by many authors. One of the most well-known results for f was obtained by Received: Apr. 10, 2023. 2020 Mathematics Subject Classification. 26A42. Key words and phrases. Henstock-Kurzweil Integral; equi-Lipschitz; equi-absolutely continuous; multipliers for integral. https://doi.org/10.28924/2291-8639-21-2023-49 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-49 2 Int. J. Anal. Appl. (2023), 21:49 Charles Swartz in [8]. He showed that if the sequence (fn) is uniformly Henstock- Kurzweil integrable then f is as well Henstock- Kurzweil integrable and ∫b a f = limn−→∞ ∫b a fn. In the following section, we obtain a similar result under a different set of conditions. For the convenience of the reader, we state some main definitions and results those will be needed later. Definition 1.1. A sequence of functions (Fn) is said to be equi-Lipschitz on an interval I if there exists C > 0 such that |Fn(x) −Fn(y)| ≤ C|x −y| for all x,y ∈ I and for all n. Definition 1.2. A sequence of functions (Fn) where Fn(x) = ∫ x a fn is said to be equi-absolutely continuous on an interval I if for all � > 0 there exists δ > 0 such that n∑ k=1 | ∫ xk xk−1 fn| < � (1.1) for all n whenever {[xk−1,xk] : k = 1, ...,n} is a collection of disjoint subintervals of I satisfying n∑ k=1 |xk −xk−1| < δ. (1.2) Definition 1.3. A sequence of functions (Fn) is said to be uniformly Cauchy on an interval I if for all � > 0 there exists N� such that |Fn(x) −Fm(x)| < � for all x ∈ I whenever n,m ≥ N� . A connection between a sequence of Lebesgue integrable functions and the sequence of their prim- itives (indefinite integrals) is given via the following theorem (see [4] and [7]). Theorem 1.1. Let fn be a sequence of Lebesgue integrable functions that converges point-wise to a function f on [a,b]. If the sequence Fn(x) = ∫ x a fn is equi-absolutely continuous on [a,b], then f is Lebesgue integrable and lim n−→∞ ∫ b a fn = ∫ b a f . (1.3) Another result that will be needed in the coming sections is the so-called Hake’s theorem (see [2]), which can be stated as follows. Theorem 1.2. Let f : [a,b] −→R be a function, then f is Henstock- Kurzweil integrable on [a,b] if and only if f is Henstock- Kurzweil integrable on [a,c] for all c ∈ [a,b) and lim c→b− ∫ c a f dt exists. (1.4) In this case ∫ b a f = lim c→b− ∫ c a f . (1.5) Int. J. Anal. Appl. (2023), 21:49 3 2. The Main Results Definition 2.1. Φ([a,b]) =: {φ ∈ C1([a,b]) such that φ is monotonic and φ(b) = 0 6= φ′(b) } . In the above definition C1([a,b]) denotes the class of all continuously differentiable functions on [a,b]. We note that Φ([a,b]) ⊆ BV ([a,b]) since Φ consists of monotonic functions. Therefore, the function φf is Henstock-Kurzwel integrable for all φ ∈ Φ([a,b]) and f ∈ HK([a,b]), (see [11]). Lemma 2.1. Let fn be a sequence of HK-integrable functions that converges point-wise to a function f on [a,b]. If there exists φ ∈ Φ such that ∫ x a φfn is equi-Lipschitz continuous on [a,b], then i. φf ∈ L1([a,b]), and ∫b a φfn −→ ∫b a φf . ii. f ∈ L1([a,r]) and ∫ r a fn −→ ∫ r a f for all r ∈ [a,b). Proof. It is clear that φfn converges point-wise to φf and the sequence ∫ r a φfn is equi-absolutely continuous on [a,b] since it is equi-Lipschitz on [a,b] where m∑ k=1 | ∫ xk xk−1 φfn| = m∑ k=1 | ∫ xk a φfn − ∫ xk−1 a φfn| ≤ C m∑ k=1 |xk −xk−1|. Therefore, (i) follows immediately from Theorem 1.1. Now since φ is continuous on [a,b] it attains its extreme values on any closed subinterval of [a,b]. Thus, Mr = Min{|φ(t)| : t ∈ [a,r]} exists for all r ∈ [a,b]. Moreover, since φ is a monotonic function and φ(b) = 0, then |φ(x)| is a decreasing function. Also, since φ′(b) 6= 0, then φ is a non-constant function in [r,b] for all r ∈ [a,b). Thus, there is s ∈ [r,b] such that φ(s) 6= φ(b) = 0 and hence, Mr > 0 for all r ∈ [a,b) since |φ(t)| ≥ |φ(r)| ≥ |φ(s)| > 0 for all t ∈ [a,r]. Using this result, we get∫ r a |f | = ∫ r a | φf φ | ≤ 1 Mr ∫ r a |φf | for all r ∈ [a,b), and (2.1) | ∫ r a fn − f | = | ∫ r a φ(fn − f ) φ | ≤ 1 Mr | ∫ r a φ(fn − f )| for all r ∈ [a,b). (2.2) Therefore, we can obtain (ii) directly form (i), (2.1) and (2.2). � Remark 2.1. If φf is Lebesgue integrable on [a,b] for some f ∈ HK and φ ∈ Φ, then f can only attain a point of Lebesgue singularity at b (f oscillates very rapidly as approaching b). In fact, we would concentrate on the local case when f is Lebesgue integrable on any subinterval [c,d] ⊂ [a,b] with d 6= b (see [4]). For the special case φ = b−x, the reader my refer to the results in [1]. In the coming sections we set F (r) = ∫ r a f and Fn(r) = ∫ r a fn. Lemma 2.2. Let fn be a sequence of HK-integrable functions that converges point-wise to a function f on [a,b]. If there exists φ ∈ Φ such that the sequence 1 φ(r) ∫b r Fndφ(t) is uniformly Cauchy on [a,b), then there are C > 0 and s ∈ [a,b) such that | 1 φ(r) ∫ b r Fndφ(t)| ≤ C for all r ∈ (s,b) and for all n. (2.3) 4 Int. J. Anal. Appl. (2023), 21:49 Proof. Since 1 φ(r) ∫b r Fndφ(t) is uniformly Cauchy on [a,b), then there is N1 such that∣∣∣∣ 1φ(r) ∫ b r (Fn −Fm)dφ(t) ∣∣∣∣ < 1 for all n,m ≥ N1. (2.4) Therefore, ∣∣∣∣ ∫ b r (Fn −Fm)dφ(t) ∣∣∣∣ < |φ(r)| for all n,m ≥ N1. (2.5) Also, since φ′ and Fn are continuous we have that the function ∫ r a Fndφ(t) is Lipschitz . Thus, for all n there is Kn > 0 such that ∣∣∣∣ ∫ b r Fndφ(t) ∣∣∣∣ ≤ Kn(b− r). (2.6) Choosing K = Max{K1,K2, ...,KN1}, we get∣∣∣∣ ∫ b r Fndφ(t) ∣∣∣∣ ≤ K(b− r) for all n ∈{1, 2, ...,N1}. (2.7) For the case n > N1, we have ∣∣∣∣ ∫ b r Fndφ(t) ∣∣∣∣ ≤ ∣∣∣∣ ∫ b r (Fn −FN1dφ(t) ∣∣∣∣ + ∣∣∣∣ ∫ b r FN1dφ(t) ∣∣∣∣ ≤ |φ(r)| + K(b− r) for all n ≥ N1 Combining the above results, we get∣∣∣∣ ∫ b r Fndφ(t) ∣∣∣∣ ≤ |φ(r)| + K(b− r) for all n. (2.8) Now, using the fact that φ(b) = 0 6= φ′(b), we get lim r→b− b− r −φ(r) = 1 φ′(b) . (2.9) Therefore, there exists s ∈ (a,b) such that (b− r) |φ(r)| ≤ 2 |φ′(b)| for all r ∈ (s,b). (2.10) Letting Ms = Min{|φ(t)| : t ∈ [a,s]} be as defined in Lemma 2.2, we obtain (b− r) |φ(r)| ≤ 2 |φ′(b)| + |b−a| Ms for all r ∈ [a,b), (2.11) and hence, (b− r) ≤ ( 2 |φ′(b)| + |b−a| Ms ) |φ(r)| for all r ∈ [a,b) (2.12) Choosing C = 1 + K ( 2 |φ′(b)| + |b−a| Ms ) and and using it with (2.8 ) and (2.12), we get the desired result. � Theorem 2.1. Let fn be a sequence of HK-integrable functions that converges point-wise to a function f on [a,b]. If there exists φ ∈ Φ such that: Int. J. Anal. Appl. (2023), 21:49 5 • the sequence 1 φ(r) ∫b r Fndφ(t) is uniformly Cauchy on [a,b), and • the sequence ∫b r φfndt is equi-Lipschitz on [a,b], then f ∈ HK([a,b]) and lim n→∞ ∫ b a fndt = ∫ b a f dt. (2.13) Proof. Similar to the argument above and since ∫b r φfndt is equi-Lipschitz, there exists M0 > 0 such that ∣∣∣∣ 1φ(r) ∫ b r φfndt ∣∣∣∣ ≤ M0 |b− r||φ(r)| for all r ∈ (a,b) and all n. (2.14) Therefore, choosing s as in Lemma 2.2 and C2 = 2M0 φ′(b) , we get∣∣∣∣ 1φ(r) ∫ b r φfndt ∣∣∣∣ ≤ C2 for all r ∈ (s,b) and all n. (2.15) Now integrating by parts and using the fact Fn(a) = 0, we get φ(r)Fn(r) = ∫ r a φfndt + ∫ r a Fndφ(t). (2.16) Also, since φ(b) = 0, we have ∫ b a φfndt + ∫ b a Fndφ(t) = 0, (2.17) which implies φ(r)Fn(r) = − ∫ b r φfndt − ∫ b r Fndφ(t) (2.18) and hence, Fn(r) = − 1 φ(r) ∫ b r φfndt − 1 φ(r) ∫ b r Fndφ(t). (2.19) Applying Lemma 2.2 and (2.15), we get |Fn(r)| ≤ C (2.20) for all n and all r ∈ (s,b). Thus, passing the limit as n −→∞, we get∣∣∣∣ limn→∞ ∫ r a fndt ∣∣∣∣ ≤ C. (2.21) Applying lemma 2.1, we get ∣∣∣∣ ∫ r a f dt ∣∣∣∣ ≤ C for all r ∈ (s,b). (2.22) Therefore, lim r→b− |φ(r)F (r)| ≤ lim r→b− C|φ(r)| = 0. (2.23) using this result with the equation φ(r)F (r) = ∫ r a φf dt + ∫ r a Fdφ(t), (2.24) we get 6 Int. J. Anal. Appl. (2023), 21:49 0 = ∫ b a φf dt + ∫ b a Fdφ(t) (2.25) Thus, in asimilar way to that for (2.19) we get F (r) = − 1 φ(r) ∫ b r φf dt − 1 φ(r) ∫ b r Fdφ(t). (2.26) Now, we use (2.19), (2.26) and Lemma 2.2 to get lim n→∞ 1 φ(r) ∫ b r Fndφ(t) = lim n→∞ − 1 φ(r) ∫ b r φfndt −Fn(r) = − 1 φ(r) ∫ b r φf dt −F (r) = 1 φ(r) ∫ b r Fdφ(t) Combining the above point-wise convergence with the given that ( 1 φ(r) ∫b r Fndφ(t) ) is uniformly Cauchy, we obtain that lim n→∞ 1 φ(r) ∫ b r Fndφ(t) = 1 φ(r) ∫ b r Fdφ(t) uniformly (2.27) Now, since φ′Fn is continuous then by the Fundamental Theorem of Calculus ∫ r a Fndφ(t) = ∫ r a φ′Fndt is differentiable and d dr ( ∫ r a Fndφ(t)) = φ ′Fn By L’Hopital rule, we have lim r→b 1 φ(r) ∫ b r Fndφ(t) = − lim r→b φ′(r) ·Fn(r) φ′(r) = − lim r→b Fn(r) (2.28) Similarly, lim r→b 1 φ(r) ∫ b r Fdφ(t) = − lim r→b F (r) (2.29) On the other hand, passing the limit in (2.20) as r −→ b and using Theorem 1.2, we get |Fn(b)| < C. (2.30) Therefore, by the completeness of R lim nk→∞ ∫ b a fnkdt = A for some A ∈R (2.31) Also, in view of (2.27) we have (see [10] Theorem 7.11) lim r→b lim nk→∞ 1 φ(r) ∫ b r Fnkdφ(t) = lim nk→∞ lim r→b 1 φ(r) ∫ b r Fnkdφ(t). (2.32) Int. J. Anal. Appl. (2023), 21:49 7 Now, using (2.29), (2.27), (2.32), (2.28), Theorem 1.2 and (2.31) respectively, we get lim r→b ∫ r a f dt = − lim r→b 1 φ(r) ∫ b r Fdφ(t) = − lim r→b lim nk→∞ 1 φ(r) ∫ b r Fnkdφ(t) = − lim nk→∞ lim r→b 1 φ(r) ∫ b r Fnkdφ(t) = − lim nk→∞ lim r→b ∫ r a fnk = lim nk→∞ ∫ b a fnk = A. Thus, by Theorem 1.2 f is Henstock-Kurzwel integrable on [a,b] and∫ b a f dt = lim r→b− ∫ r a f dt = A (2.33) Reversing the steps above and using the the result that f is Henstock-Kurzwel integrable, we obtain the convergence for the whole sequences as follows lim n→∞ ∫ b a fndt = − lim n→∞ lim r→b 1 φ(r) ∫ b r Fndφ(t) = − lim r→b lim n→∞ 1 φ(r) ∫ b r Fndφ(t) = − lim r→b 1 φ(r) ∫ b r Fdφ(t) = ∫ b a f dt This completes the proof. � Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] Y. Alzubaidi, A Complete Normed Space of a Class of Guage Integrable Functions, J. Math. 2022 (2022), 2022/2354758. https://doi.org/10.1155/2022/2354758. [2] R.G. Bartle, Return to the Riemann Integral, Amer. Math. Mon. 103 (1996), 625–632. https://doi.org/10. 1080/00029890.1996.12004798. [3] J. Depree, C. Swartz, Introduction to Real Analysis, Wily, New York, 1987. [4] R.A. 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