| Title:
|
Further properties of Stepanov--Orlicz almost periodic functions (English) |
| Author:
|
Djabri, Yousra |
| Author:
|
Bedouhene, Fazia |
| Author:
|
Boulahia, Fatiha |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
363-382 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We revisit the concept of Stepanov--Orlicz almost periodic functions introduced by Hillmann in terms of Bochner transform. Some structural properties of these functions are investigated. A particular attention is paid to the Nemytskii operator between spaces of Stepanov--Orlicz almost periodic functions. Finally, we establish an existence and uniqueness result of Bohr almost periodic mild solution to a class of semilinear evolution equations with Stepanov--Orlicz almost periodic forcing term. (English) |
| Keyword:
|
Bohr almost periodic |
| Keyword:
|
Bochner transform |
| Keyword:
|
Stepanov--Orlicz almost periodic function |
| Keyword:
|
semilinear evolution equations |
| Keyword:
|
Nemytskii operator |
| MSC:
|
34C27 |
| MSC:
|
35B15 |
| MSC:
|
46E30 |
| idZBL:
|
Zbl 07286010 |
| idMR:
|
MR4186113 |
| DOI:
|
10.14712/1213-7243.2020.030 |
| . |
| Date available:
|
2020-11-27T07:44:40Z |
| Last updated:
|
2022-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148472 |
| . |
| Reference:
|
[1] Albrycht J.: The theory of Marcinkiewic–Orlicz spaces.Rozprawy Mat. 27 (1962), 56 pages. MR 0139935 |
| Reference:
|
[2] Amerio L., Prouse G.: Almost-Periodic Functions and Functional Equations.Van Nostrand Reinhold, New York, Ont.-Melbourne, 1971. MR 0275061 |
| Reference:
|
[3] Andres J., Bersani A. M., Grande R. F.: Hierarchy of almost-periodic function spaces.Rend. Mat. Appl. (7) 26 (2006), no. 2, 121–188. Zbl 1133.42002, MR 2275292 |
| Reference:
|
[4] Andres J., Pennequin D.: On Stepanov almost-periodic oscillations and their discretizations.J. Difference Equ. Appl. 18 (2012), no. 10, 1665–1682. MR 2979829, 10.1080/10236198.2011.587813 |
| Reference:
|
[5] Andres J., Pennequin D.: On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations.Proc. Amer. Math. Soc. 140 (2012), no. 8, 2825–2834. MR 2910769 |
| Reference:
|
[6] Bedouhene F., Challali N., Mellah O., Raynaud de Fitte P., Smaali M.: Almost periodic solution in distribution for stochastic differential equations with Stepanov almost periodic coefficients.available at arXiv: 1703.00282v3 [math.PR] (2017), 42 pages. |
| Reference:
|
[7] Bugajewski D., Nawrocki A.: 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), no. 2, 809–836. MR 3701650, 10.5186/aasfm.2017.4250 |
| Reference:
|
[8] Chen S.: Geometry of Orlicz Spaces.Dissertationes Math. (Rozprawy Mat.), 356, 1996. MR 1410390 |
| Reference:
|
[9] Cichoń M., Metwali M. M. A.: On quadratic integral equations in Orlicz spaces.J. Math. Anal. Appl. 387 (2012), no. 1, 419–432. MR 2845761, 10.1016/j.jmaa.2011.09.013 |
| Reference:
|
[10] Corduneanu C.: Almost Periodic Functions.Interscience Tracts in Pure and Applied Mathematics, 22, Interscience Publishers, John Wiley, New York, 1968. MR 0481915 |
| Reference:
|
[11] Dads A. E. H., Es-Sebbar B., Ezzinbi K., Ziat M.: Behavior of bounded solutions for some almost periodic neutral partial functional differential equations.Math. Methods Appl. Sci. 40 (2017), no. 7, 2377–2397. MR 3636701, 10.1002/mma.4145 |
| Reference:
|
[12] Danilov L. I.: On the uniform approximation of a function that is almost periodic in the sense of Stepanov.Izv. Vyssh. Uchebn. Zaved. Mat (1998), no. 5, 10–18. MR 1639154 |
| Reference:
|
[13] Diagana T.: Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations.Nonlinear Anal. 69 (2008), no. 12, 4277–4285. MR 2467232, 10.1016/j.na.2007.10.051 |
| Reference:
|
[14] Diagana T., Zitane M.: Stepanov-like pseudo-almost automorphic functions in Lebesgue spaces with variable exponents ${L}^{p(x)}$.Electron. J. Differential Equations 2013 (2013), No. 188, 20 pages. MR 3104964 |
| Reference:
|
[15] Ding H.-S., Long W., N'Guérékata G. M.: Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients.J. Comput. Anal. Appl. 13 (2011), no. 2, 231–242. MR 2807574 |
| Reference:
|
[16] Hillmann T. R.: Besicovitch–Orlicz spaces of almost periodic functions.Real and stochastic analysis, Wiley Ser. Probab. Math. Statist. Probab. Math. Statist., Wiley, 1986, 119–167. MR 0856581 |
| Reference:
|
[17] Hu Z.: Boundedness and Stepanov's almost periodicity of solutions.Electron. J. Differential. Equations 2005 (2005), no. 35, 7 pages. MR 2135246 |
| Reference:
|
[18] Hu Z., Mingarelli A. B.: Bochner's theorem and Stepanov almost periodic functions.Ann. Mat. Pura Appl. (4) 187 (2008), no. 4, 719–736. MR 2413376, 10.1007/s10231-008-0066-5 |
| Reference:
|
[19] Hudzik H.: Uniform convexity of Musielak–Orlicz spaces with Luxemburg's norm.Comment. Math. Prace Mat. 23 (1983), no. 1, 21–32. MR 0709167 |
| Reference:
|
[20] Kasprzak P., Nawrocki A., Signerska-Rynkowska J.: Integrate-and-fire models with an almost periodic input function.J. Differential Equations 264 (2018), no. 4, 2495–2537. MR 3737845, 10.1016/j.jde.2017.10.025 |
| Reference:
|
[21] Kourat H.: Caractérisation de quelques propriétés géométriques locales dans les espaces de type Musielak–Orlicz.PhD. Thesis, Mouloud Mammeri University of Tizi–Ouzou, Tizi–Ouzou, 2016 (French). |
| Reference:
|
[22] Kozlowski W. M.: Modular Function Spaces.Monographs and Textbooks in Pure and Applied Mathematics, 122, Marcel Dekker, New York, 1988. Zbl 0718.41049, MR 1474499 |
| Reference:
|
[23] Kufner A., John O., Fučík S.: Function Spaces.Monographs and Textsbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leyden, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977. MR 0482102 |
| Reference:
|
[24] Levitan B. M., Zhikov V. V.: Almost Periodic Functions and Differential Equations.Cambridge University Press, Cambridge, 1982. Zbl 0499.43005, MR 0690064 |
| Reference:
|
[25] Luxemburg W. A. J.: Banach Function Spaces.PhD. Dissertation, Delft University of Technology, Delft, 1955. Zbl 0162.44701, MR 0072440 |
| Reference:
|
[26] Musielak J.: Orlicz Spaces and Modular Spaces.Lecture Notes in Mathematics, 1034, Springer, Berlin, 1983. Zbl 0557.46020, MR 0724434 |
| Reference:
|
[27] Pankov A. A.: Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations.Mathematics and Its Applications (Soviet Series), 55, Kluwer Academic Publishers Group, Dordrecht, 1990. MR 1120781, 10.1007/978-94-011-9682-6_5 |
| Reference:
|
[28] Radová L.: Theorems of Bohr–Neugebauer-type for almost-periodic differential equations.Math. Slovaca 54 (2004), no. 2, 191–207. Zbl 1068.34042, MR 2074215 |
| Reference:
|
[29] Rao A. S.: On the Stepanov-almost periodic solution of a second-order operator differential equation.Proc. Edinburgh Math. Soc. (2) 19 (1974/75), 261–263. MR 0407409 |
| Reference:
|
[30] Stepanoff W.: Über einige Verallgemeinerungen der fast periodischen Funktionen.Math. Ann. 95 (1926), no. 1, 473–498 (German). MR 1512290, 10.1007/BF01206623 |
| Reference:
|
[31] Stoiński S.: Almost periodic functions in the Lebesgue measure.Comment. Math. (Prace Mat.) 34 (1994), 189–198. MR 1325086 |
| Reference:
|
[32] Zaidman S.: An existence result for Stepanoff almost-periodic differential equations.Canad. Math. Bull. 14 (1971), 551–554. MR 0310382, 10.4153/CMB-1971-097-5 |
| . |
