| Title:
|
Remotely $c$-almost periodic type functions in ${\mathbb{R}}^{n}$ (English) |
| Author:
|
Kostić, Marco |
| Author:
|
Kumar, Vipin |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
58 |
| Issue:
|
2 |
| Year:
|
2022 |
| Pages:
|
85-104 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely $c$-almost periodic functions in ${\mathbb{R}}^{n},$ slowly oscillating functions in ${\mathbb{R}}^{n},$ and further analyze the recently introduced class of quasi-asymptotically $c$-almost periodic functions in ${\mathbb{R}}^{n}.$ We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations and the ordinary differential equations. (English) |
| Keyword:
|
remotely $c$-almost periodic functions in ${\mathbb{R}}^{n}$ |
| Keyword:
|
slowly oscillating functions in ${\mathbb{R}}^{n}$ |
| Keyword:
|
quasi-asymptotically $c$-almost periodic functions in ${\mathbb{R}}^{n}$ |
| Keyword:
|
abstract Volterra integro-differential equations |
| Keyword:
|
Richard-Chapman ordinary differential equation with external perturbation |
| MSC:
|
42A75 |
| MSC:
|
43A60 |
| MSC:
|
47D99 |
| idZBL:
|
Zbl 07547203 |
| idMR:
|
MR4448485 |
| DOI:
|
10.5817/AM2022-2-85 |
| . |
| Date available:
|
2022-05-16T10:30:36Z |
| Last updated:
|
2022-08-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150423 |
| . |
| Reference:
|
[1] Alvarez, E., Castillo, S., Pinto, M.: $(\omega ,c)$-pseudo periodic functions, first order Cauchy problem and Lasota-Wazewska model with ergodic and unbounded oscillating production of red cells.Bound. Value Probl. 106 (2019), 1–20. MR 3974664 |
| Reference:
|
[2] Alvarez, E., Castillo, S., Pinto, M.: $(\omega ,c)$-asymptotically periodic functions, first-order Cauchy problem, and Lasota-Wazewska model with unbounded oscillating production of red cells.Math. Methods Appl. Sci. 43 (2020), 305–319. MR 4044240, 10.1002/mma.5880 |
| Reference:
|
[3] Alvarez, E., Gómez, A., Pinto, M.: $(\omega ,c)$-periodic functions and mild solution to abstract fractional integro-differential equations.Electron. J. Qual. Theory Differ. Equ. 16 (2018), 1–8. MR 3789407, 10.14232/ejqtde.2018.1.16 |
| Reference:
|
[4] Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems.Birkhäuser, Springer Basel AG, 2001. MR 2798103 |
| Reference:
|
[5] Besicovitch, A.S.: Almost Periodic Functions.Dover Publ., New York, 1954. MR 0068029 |
| Reference:
|
[6] Blot, J., Cieutat, P., N’Guérékata, G.M.: S-asymptotically $\omega $-periodic functions and applications to evolution equations.-asymptotically $\omega $-periodic functions and applications to evolution equations, African Diaspora J. Math. 12 (2011), 113–121. MR 2847308 |
| Reference:
|
[7] Brindle, D.: $S$-asymptotically $\omega $-periodic functions and sequences and applications to evolution equations.Ph.D. thesis, Morgan State University, 2019, https://mdsoar.org/handle/11603/17680. MR 4071576 |
| Reference:
|
[8] Chang, Y.-K., Wei, Y.: $S$-asymptotically Bloch type periodic solutions to some semi-linear evolution equations in Banach spaces.Acta Math. Sci. 41B (2021), 413–425. MR 4206851, 10.1007/s10473-021-0206-1 |
| Reference:
|
[9] Chávez, A., Khalil, K., Kostić, M., Pinto, M.: Multi-dimensional almost periodic type functions and applications.submitted 2020, arXiv:2012.00543. |
| Reference:
|
[10] Chávez, A., Khalil, K., Kostić, M., Pinto, M.: Multi-dimensional almost automorphic type functions and applications.Bull. Braz. Math. Soc. (N.S.) (2022), 1–51, https/doi.org/10.1007/s00574-022-00284-x. MR 4458789, 10.1007/s00574-022-00284-x |
| Reference:
|
[11] Cuevas, C., de Souza, J.C.: Existence of S-asymptotically $\omega $-periodic solutions for fractional order functional integro-differential equations with infinite delay.Nonlinear Anal. 72 (2010), 1683–1689. MR 2577568, 10.1016/j.na.2009.09.007 |
| Reference:
|
[12] Diagana, T.: Almost automorphic type and almost periodic type functions in abstract spaces.Springer-Verlag, New York, 2013. MR 3098423 |
| Reference:
|
[13] Dimbour, W., Manou-Ab, S.M.: Asymptotically $\omega $-periodic functions in the Stepanov sense and its application for an advanced differential equation with piecewise constant argument in a Banach space.Mediterranean J. Math. 15 (1) (2018), 18 pp., https://doi.org/10.1007/s00009-018-1071-61660-5446/18/010001-18. MR 3750300, 10.1007/s00009-018-1071-61660-5446/18/010001-18 |
| Reference:
|
[14] Fink, A.M.: Almost periodic differential equations.Springer-Verlag, Berlin, 1974. MR 0460799 |
| Reference:
|
[15] Friedman, A.: Partial differential equations of parabolic type.Prentice Hall, Englewood Cliffs, NJ, 1964, xiv+347 pp. MR 0181836 |
| Reference:
|
[16] Garcia, S.R.: Remembering Donald Sarason (1933-2017).Notices Amer. Math. Soc. 65 (2018), 195–200. MR 3751316 |
| Reference:
|
[17] Haraux, A., Souplet, P.: An example of uniformly recurrent function which is not almost periodic.J. Fourier Anal. Appl. 10 (2004), 217–220. MR 2054309, 10.1007/s00041-004-8012-4 |
| Reference:
|
[18] Henríquez, H.R., Pierri, M., Táboas, P.: On S-asymptotically $\omega $-periodic functions on Banach spaces and applications.-asymptotically $\omega $-periodic functions on Banach spaces and applications, J. Math. Appl. Anal. 343 (2008), 1119–1130. MR 2417129, 10.1016/j.jmaa.2008.02.023 |
| Reference:
|
[19] Khalladi, M.T., Kostić, M., Pinto, M., Rahmani, A., Velinov, D.: Generalized $c$-almost periodic functions and applications.Bull. Int. Math. Virtual Inst. 11 (2021), 283–293. MR 4187070 |
| Reference:
|
[20] Khalladi, M.T., Kostić, M., Rahmani, A., Pinto, M., Velinov, D.: $c$-almost periodic type functions and applications.Nonauton. Dyn. Syst. 7 (2020), 176–193. MR 4185794, 10.1515/msds-2020-0111 |
| Reference:
|
[21] Kostić, M.: Multi-dimensional $c$-almost periodic type functions and applications.Electron. J. Differential Equations, in press, aXiv:2012.15735. MR 4252727 |
| Reference:
|
[22] Kostić, M.: Generalized semigroups and cosine functions.Mathematical Institute SANU, Belgrade, 2011. MR 2790849 |
| Reference:
|
[23] Kostić, M.: Almost periodic and almost automorphic type solutions to integro-differential equations.W. de Gruyter, Berlin, 2019. MR 3931753 |
| Reference:
|
[24] Kostić, M.: Generalized $c$-almost periodic type functions in ${\mathbb{R}}^{n}$.Arch. Math. (Brno) 57 (2021), 221–253. MR 4346112, 10.5817/AM2021-4-221 |
| Reference:
|
[25] Kostić, M.: Multi-dimensional $(\omega , c)$-periodic type functions and applications.Nonauton. Dyn. Syst. 8 (2021), 136–151. MR 4252727, 10.1515/msds-2020-0130 |
| Reference:
|
[26] Kostić, M.: Quasi-asymptotically almost periodic functions and applications.Bull. Braz. Math. Soc. (N.S.) 52 (2021), 183–212. MR 4253402, 10.1007/s00574-020-00197-7 |
| Reference:
|
[27] Kostić, M.: Almost periodic functions and densities.Evol. Equ. Control Theory 11 (2) (2022), 457–486. MR 4376332, 10.3934/eect.2021008 |
| Reference:
|
[28] Kostić, M.: Selected topics in almost periodicity.W. de Gruyter, Berlin, 2022. |
| Reference:
|
[29] Kovanko, A.S.: Sur la compacié des systémes de fonctions presque-périodiques généralisées de H. Wey.C.R. (Doklady) Ac. Sc. URSS 43 (1944), 275–276. MR 0012148 |
| Reference:
|
[30] Levitan, M.: Almost periodic functions.Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953, 396 pp. (in Russian). MR 0060629 |
| Reference:
|
[31] Maulén, C., Castillo, S., Kostić, M., Pinto, M.: Remotely almost periodic solutions of ordinary differential equations.J. Math. 2021 (2021), 9 pp., Article ID 9985454, 9 pp., https://doi.org/10.1155/2021/9985454. MR 4265296, 10.1155/2021/9985454 |
| Reference:
|
[32] N’Guérékata, G.M.: Almost automorphic and almost periodic functions in abstract spaces.Kluwer Acad. Publ. Dordrecht, 2001. MR 1880351 |
| Reference:
|
[33] Oueama-Guengai, E.R., N’Guérékata, G.M.: On $S$-asymptotically $\omega $-periodic and Bloch periodic mild solutions to some fractional differential equations in abstract spaces.Math. Methods Appl. Sci. 41 (2018), 9116–9122. MR 3897769, 10.1002/mma.5062 |
| Reference:
|
[34] Pankov, A.A.: Bounded and almost periodic solutions of nonlinear operator differential equations.Kluwer Acad. Publ., Dordrecht, 1990. MR 1120781 |
| Reference:
|
[35] Rabinovich, V.S., Roch, S.: The essential spectrum of Schrödinger operators on lattices.J. Phys. A 39 (2006), 8377–8394. MR 2238507, 10.1088/0305-4470/39/26/007 |
| Reference:
|
[36] Sarason, D.: Toeplitz operators with piecewise quasicontinuous symbols.Indiana Univ. Math. J. 26 (1977), 817–838. MR 0463968, 10.1512/iumj.1977.26.26066 |
| Reference:
|
[37] Sarason, D.: Remotely almost periodic functions.Contemp. Math. 32 (1984), 237–242. MR 0769512, 10.1090/conm/032/769512 |
| Reference:
|
[38] Sarason, D.: The Banach algebra of slowly oscillating functions.Houston J. Math. 33 (4) (2007), 1161–1182. MR 2350087 |
| Reference:
|
[39] Xie, R., Zhang, C.: Space of $\omega $-periodic limit functions and its applications to an abstract Cauchy problem.J. Function Spaces 2015 (2015), 10 pp., Article ID 953540, http://dx.doi.org/10.1155/2015/953540. MR 3326682, 10.1155/2015/953540 |
| Reference:
|
[40] Yang, F., Zhang, C.: Slowly oscillating solutions of a parabolic inverse problem: boundary value problems.Bound. Value Probl. 2010 (2010), 12 pp., Article ID 471491, doi:10.1155/2010/471491. MR 2753238, 10.1155/2010/471491 |
| Reference:
|
[41] Yang, F., Zhang, C.: Remotely almost periodic solutions to parabolic inverse problems.Taiwanese J. Math. 15 (2011), 43–57. MR 2780270, 10.11650/twjm/1500406160 |
| Reference:
|
[42] Zaidman, S.: Almost-periodic functions in abstract spaces.Pitman Research Notes in Math., vol. 126, Pitman, Boston, 1985. MR 0790316 |
| Reference:
|
[43] Zhang, C., Guo, Y.: Slowly oscillating solutions for differential equations with strictly monotone operator.J. Inequal. Appl. 2007 (2007), 9 pp., Article ID 60239, doi:10.1155/2007/60239. MR 2304457, 10.1155/2007/60239 |
| Reference:
|
[44] Zhang, C., Jiang, L.: Remotely almost periodic solutions of parabolic inverse problems.Nonlinear Anal. 65 (2006), 1613–1623. MR 2248688, 10.1016/j.na.2005.10.036 |
| Reference:
|
[45] Zhang, C., Jiang, L.: Remotely almost periodic solutions to systems of differential equations with piecewise constant argument.Appl. Math. Lett. 21 (2008), 761–768. MR 2436161, 10.1016/j.aml.2007.08.007 |
| Reference:
|
[46] Zhang, S., Piao, D.: Time remotely almost periodic viscosity solutions of Hamilton-Jacobi equations.ISRN Math. Anal. 2011 (2011), 13 pp., Article ID 415358, doi:10.5402/2011/415358. MR 2772323, 10.5402/2011/415358 |
| . |
