Czechoslovak Mathematical Journal, first online, pp. 1-22
On almost periodicity defined via non-absolutely convergent integrals
Dariusz Bugajewski, Adam Nawrocki
Received January 9, 2023. Published online April 2, 2024. OPEN ACCESS
Abstract: We investigate some properties of the normed space of almost periodic functions which are defined via the Denjoy-Perron (or equivalently, Henstock-Kurzweil) integral. In particular, we prove that this space is barrelled while it is not complete. We also prove that a linear differential equation with the non-homogenous term being an almost periodic function of such type, possesses a solution in the class under consideration.
Keywords: almost periodic function in view of the Lebesgue measure; barrelled space; Bohr almost periodic function; Denjoy-Bochner almost periodic function; Denjoy-Perron integral; Henstock-Kurzweil integral; linear differential equation
References: [1] A. Alexiewicz: Linear functionals on Denjoy-integrable functions. Colloq. Math. 1 (1948), 289-293. DOI 10.4064/cm-1-4-289-293 | MR 0030120 | Zbl 0037.32302
[2] J. Andres, A. M. Bersani, R. F. Grande: Hierarchy of almost-periodic function spaces. Rend. Mat. Appl., VII. Ser. 26 (2006), 121-188. MR 2275292 | Zbl 1133.42002
[3] M. Borkowski, D. Bugajewska: Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations. Math. Slovaca 68 (2018), 77-88. DOI 10.1515/ms-2017-0082 | MR 3764318 | Zbl 1473.45008
[4] M. Borkowski, D. Bugajewska, P. Kasprzak: Selected Topics in Nonlinear Analysis. Lecture Notes in Nonlinear Analysis 19. Nicolaus Copernicus University, Juliusz Schauder Center for Nonlinear Studies, Toruń (2021). MR 4404311 | Zbl 1506.47001
[5] G. Bruno, A. Pankov: On convolution operators in the spaces of almost periodic functions and $L^p$ spaces. Z. Anal. Anwend. 19 (2000), 359-367. DOI 10.4171/ZAA/955 | MR 1768997 | Zbl 0972.47036
[6] D. Bugajewski: On the structure of solution sets of differential and integral equations, and the Perron integral. Proceedings of the Prague Mathematical Conference 1996. Icaris, Prague (1996), 47-51. MR 1703455 | Zbl 0966.34041
[7] D. Bugajewski: On the Volterra integral equation and the Henstock-Kurzweil integral. Math. Pannonica 9 (1998), 141-145. MR 1620430 | Zbl 0906.45005
[8] D. Bugajewski, K. Kasprzak, A. Nawrocki: Asymptotic properties and convolutions of some almost periodic functions with applications. Ann. Mat. Pura Appl. (4) 202 (2023), 1033-1050. DOI 10.1007/s10231-022-01270-2 | MR 4576930 | Zbl 1512.42008
[9] D. Bugajewski, A. Nawrocki: Some remarks on almost periodic functions in view of the Lebesgue measure with applications to linear differential equations. Ann. Acad. Sci. Fenn., Math. 42 (2017), 809-836. DOI 10.5186/aasfm.2017.4250 | MR 3701650 | Zbl 1372.42003
[10] D. Bugajewski, S. Szufla: On the Aronszajn property for differential equations and the Denjoy integral. Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 25 (1995), 61-69. MR 1384852 | Zbl 0854.34005
[11] H. Burkill: Almost periodicity and non-absolutely integrable functions. Proc. Lond. Math. Soc., II. Ser. 53 (1951), 32-42. DOI 10.1112/plms/s2-53.1.32 | MR 0043251 | Zbl 0042.31901
[12] T. S. Chew, F. Flordeliza: On $x'=f(t,x)$ and Henstock-Kurzweil integrals. Differ. Integral Equ. 4 (1991), 861-868. DOI 10.57262/die/1371225020 | MR 1108065 | Zbl 0733.34004
[13] R. Henstock: Definitions of Riemann type of the variational integral. Proc. Lond. Math. Soc., III. Ser. 11 (1961), 402-418. DOI 10.1112/plms/s3-11.1.402 | MR 0132147 | Zbl 0099.27402
[14] J. Horváth: Topological Vector Spaces and Distributions. Vol. I. Addison-Wesley, Reading (1966). MR 0205028 | Zbl 0143.15101
[15] P. Kasprzak, A. Nawrocki, J. Signerska-Rynkowska: Integrate-and-fire models an almost periodic input function. J. Differ. Equations 264 (2018), 2495-2537. DOI 10.1016/j.jde.2017.10.025 | MR 3737845 | Zbl 1380.42006
[16] J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7 (1957), 418-449. DOI 10.21136/CMJ.1957.100258 | MR 0111875 | Zbl 0090.30002
[17] J. Kurzweil: Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions. Series in Real Analysis 11. World Scientific, Hackensack (2012). DOI 10.1142/7907 | MR 2906899 | Zbl 1248.34001
[18] Y. Meyer: Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling. Afr. Diaspora J. Math. 13 (2012), 1-45. MR 2876415 | Zbl 1242.52026
[19] B. K. Pal, S. N. Mukhopadhyay: Denjoy-Bochner almost periodic functions. J. Aust. Math. Soc., Ser. A 37 (1984), 205-222. DOI 10.1017/S1446788700022047 | MR 0749501 | Zbl 0552.42005
[20] P. Pych-Taberska: Approximation of almost periodic functions integrable in the Denjoy-Perron sense. Function Spaces. Teubner-Texte zur Mathematik 120. B. G. Teubner, Stuttgart (1991), 186-196. MR 1155174 | Zbl 0757.41029
[21] P. Pych-Taberska: On some almost periodic convolutions. Funct. Approximatio, Comment. Math. 20 (1992), 65-77. MR 1201717 | Zbl 0848.42009
[22] S. Saks: Theory of the Integral. Monografie Matematyczne 7. G. E. Stechert & Co., New York (1937). MR 0167578 | Zbl 0017.30004
[23] Š. Schwabik: The Perron integral in ordinary differential equations. Differ. Integral Equ. 6 (1993), 863-882. DOI 10.57262/die/1370032239 | MR 1222306 | Zbl 0784.34006
[24] S. Stoiński: Almost periodic function in the Lebesgue measure. Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 34 (1994), 189-198. MR 1325086 | Zbl 0835.42009
[25] S. Stoiński: Almost Periodic Functions. Scientific Publisher AMU, Poznań (2008). (In Polish.)
[26] C. Swartz: Introduction to Gauge Integrals. World Scientific, Singapore (2001). DOI 10.1142/4361 | MR 1845270 | Zbl 0982.26006
Affiliations: Dariusz Bugajewski (corresponding author), Adam Nawrocki, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland, e-mail: ddbb@amu.edu.pl, adam.nawrocki@amu.edu.pl
