Further properties of Stepanov--Orlicz almost periodic functions

Yousra Djabri; Fazia Bedouhene; Fatiha Boulahia

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 3, page 363-382
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Djabri, Yousra, Bedouhene, Fazia, and Boulahia, Fatiha. "Further properties of Stepanov--Orlicz almost periodic functions." Commentationes Mathematicae Universitatis Carolinae 61.3 (2020): 363-382. <http://eudml.org/doc/297253>.

@article{Djabri2020,
abstract = {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.},
author = {Djabri, Yousra, Bedouhene, Fazia, Boulahia, Fatiha},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Bohr almost periodic; Bochner transform; Stepanov--Orlicz almost periodic function; semilinear evolution equations; Nemytskii operator},
language = {eng},
number = {3},
pages = {363-382},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Further properties of Stepanov--Orlicz almost periodic functions},
url = {http://eudml.org/doc/297253},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Djabri, Yousra
AU - Bedouhene, Fazia
AU - Boulahia, Fatiha
TI - Further properties of Stepanov--Orlicz almost periodic functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 3
SP - 363
EP - 382
AB - 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.
LA - eng
KW - Bohr almost periodic; Bochner transform; Stepanov--Orlicz almost periodic function; semilinear evolution equations; Nemytskii operator
UR - http://eudml.org/doc/297253
ER -

References

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  1. Albrycht J., The theory of Marcinkiewic–Orlicz spaces, Rozprawy Mat. 27 (1962), 56 pages. MR0139935
  2. Amerio L., Prouse G., Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold, New York, Ont.-Melbourne, 1971. MR0275061
  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. Zbl1133.42002MR2275292
  4. Andres J., Pennequin D., 10.1080/10236198.2011.587813, J. Difference Equ. Appl. 18 (2012), no. 10, 1665–1682. MR2979829DOI10.1080/10236198.2011.587813
  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. MR2910769
  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. 
  7. Bugajewski D., Nawrocki A., 10.5186/aasfm.2017.4250, Ann. Acad. Sci. Fenn., Math. 42 (2017), no. 2, 809–836. MR3701650DOI10.5186/aasfm.2017.4250
  8. Chen S., Geometry of Orlicz Spaces, Dissertationes Math. (Rozprawy Mat.), 356, 1996. MR1410390
  9. Cichoń M., Metwali M. M. A., 10.1016/j.jmaa.2011.09.013, J. Math. Anal. Appl. 387 (2012), no. 1, 419–432. MR2845761DOI10.1016/j.jmaa.2011.09.013
  10. Corduneanu C., Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, 22, Interscience Publishers, John Wiley, New York, 1968. MR0481915
  11. Dads A. E. H., Es-Sebbar B., Ezzinbi K., Ziat M., 10.1002/mma.4145, Math. Methods Appl. Sci. 40 (2017), no. 7, 2377–2397. MR3636701DOI10.1002/mma.4145
  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. MR1639154
  13. Diagana T., 10.1016/j.na.2007.10.051, Nonlinear Anal. 69 (2008), no. 12, 4277–4285. MR2467232DOI10.1016/j.na.2007.10.051
  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. MR3104964
  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. MR2807574
  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. MR0856581
  17. Hu Z., Boundedness and Stepanov's almost periodicity of solutions, Electron. J. Differential. Equations 2005 (2005), no. 35, 7 pages. MR2135246
  18. Hu Z., Mingarelli A. B., 10.1007/s10231-008-0066-5, Ann. Mat. Pura Appl. (4) 187 (2008), no. 4, 719–736. MR2413376DOI10.1007/s10231-008-0066-5
  19. Hudzik H., Uniform convexity of Musielak–Orlicz spaces with Luxemburg's norm, Comment. Math. Prace Mat. 23 (1983), no. 1, 21–32. MR0709167
  20. Kasprzak P., Nawrocki A., Signerska-Rynkowska J., 10.1016/j.jde.2017.10.025, J. Differential Equations 264 (2018), no. 4, 2495–2537. MR3737845DOI10.1016/j.jde.2017.10.025
  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). 
  22. Kozlowski W. M., Modular Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 122, Marcel Dekker, New York, 1988. Zbl0718.41049MR1474499
  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. MR0482102
  24. Levitan B. M., Zhikov V. V., Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, 1982. Zbl0499.43005MR0690064
  25. Luxemburg W. A. J., Banach Function Spaces, PhD. Dissertation, Delft University of Technology, Delft, 1955. Zbl0162.44701MR0072440
  26. Musielak J., Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, 1034, Springer, Berlin, 1983. Zbl0557.46020MR0724434
  27. Pankov A. A., 10.1007/978-94-011-9682-6_5, Mathematics and Its Applications (Soviet Series), 55, Kluwer Academic Publishers Group, Dordrecht, 1990. MR1120781DOI10.1007/978-94-011-9682-6_5
  28. Radová L., Theorems of Bohr–Neugebauer-type for almost-periodic differential equations, Math. Slovaca 54 (2004), no. 2, 191–207. Zbl1068.34042MR2074215
  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. MR0407409
  30. Stepanoff W., 10.1007/BF01206623, Math. Ann. 95 (1926), no. 1, 473–498 (German). MR1512290DOI10.1007/BF01206623
  31. Stoiński S., Almost periodic functions in the Lebesgue measure, Comment. Math. (Prace Mat.) 34 (1994), 189–198. MR1325086
  32. Zaidman S., 10.4153/CMB-1971-097-5, Canad. Math. Bull. 14 (1971), 551–554. MR0310382DOI10.4153/CMB-1971-097-5

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