𝛼∼𝑙 relatively dense sets and weighted pseudo almost periodic phenomenon

Liang, Jin; N’Guérékata, G.; Xiao, Ti-Jun

doi:10.1090/proc/14818

$\alpha \sim l$ relatively dense sets and weighted pseudo almost periodic phenomenon
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by Jin Liang, G. M. N’Guérékata and Ti-Jun Xiao PDF

Proc. Amer. Math. Soc. 148 (2020), 687-696 Request permission

Abstract:

In this paper, we propose a new idea of studying the weighted pseudo almost periodic phenomenon. We introduce a new mathematical concept of $\alpha \sim l$ relatively dense set and show that the $\alpha \sim l$ relatively dense set is more useful than the classical relatively dense set due to Bohr or Bochner in treating the weighted pseudo almost periodic phenomenon. We prove the uniqueness of decomposition of the weighted pseudo almost periodic functions with values in Banach spaces and with weights in a new weight function set, and show that the new weight set is larger than all the existing weight function sets which ensures the uniqueness of such decomposition in literature. Moreover, we reveal several properties of the vector-valued weighted pseudo almost periodic functions, and provide a series of concrete examples and remarks for illustrating our theoretical results.

References

Ravi P. Agarwal, Bruno de Andrade, and Claudio Cuevas, Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 3532–3554. MR 2683811, DOI 10.1016/j.nonrwa.2010.01.002
Joël Blot, Philippe Cieutat, and Khalil Ezzinbi, New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications, Appl. Anal. 92 (2013), no. 3, 493–526. MR 3021274, DOI 10.1080/00036811.2011.628941
S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 2039–2043. MR 145283, DOI 10.1073/pnas.48.12.2039
Philippe Cieutat, Samir Fatajou, and Gaston M. N’Guérékata, Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations, Appl. Anal. 89 (2010), no. 1, 11–27. MR 2604277, DOI 10.1080/00036810903397503
C. Corduneanu, Almost periodicity in functional equations, Differential equations, chaos and variational problems, Progr. Nonlinear Differential Equations Appl., vol. 75, Birkhäuser, Basel, 2008, pp. 157–163. MR 2409102, DOI 10.1007/978-3-7643-8482-1_{1}2
Constantin Corduneanu, Almost periodic oscillations and waves, Springer, New York, 2009. MR 2460203, DOI 10.1007/978-0-387-09819-7
Toka Diagana, Weighted pseudo almost periodic functions and applications, C. R. Math. Acad. Sci. Paris 343 (2006), no. 10, 643–646 (English, with English and French summaries). MR 2271739, DOI 10.1016/j.crma.2006.10.008
Toka Diagana, Weighted pseudo-almost periodic solutions to some differential equations, Nonlinear Anal. 68 (2008), no. 8, 2250–2260. MR 2398647, DOI 10.1016/j.na.2007.01.054
Hui-Sheng Ding, Jin Liang, and Ti-Jun Xiao, Weighted pseudo almost automorphic functions and WPAA solutions to semilinear evolution equations, J. Math. Anal. Appl. 409 (2014), no. 1, 409–427. MR 3095050, DOI 10.1016/j.jmaa.2013.07.044
Zhenbin Fan, Jin Liang, and Ti-Jun Xiao, On Stepanov-like (pseudo) almost automorphic functions, Nonlinear Anal. 74 (2011), no. 8, 2853–2861. MR 2776533, DOI 10.1016/j.na.2011.01.007
A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799
B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. MR 690064
Jin Liang, L. Maniar, G. M. N’Guérékata, and Ti-Jun Xiao, Existence and uniqueness of $C^{(n)}$-almost periodic solutions to some ordinary differential equations, Nonlinear Anal. 66 (2007), no. 9, 1899–1910. MR 2304968, DOI 10.1016/j.na.2006.02.030
Jin Liang, Gaston M. N’Guérékata, Ti-Jun Xiao, and Jun Zhang, Some properties of pseudo-almost automorphic functions and applications to abstract differential equations, Nonlinear Anal. 70 (2009), no. 7, 2731–2735. MR 2499741, DOI 10.1016/j.na.2008.03.061
Jin Liang, Ti-Jun Xiao, and Jun Zhang, Decomposition of weighted pseudo-almost periodic functions, Nonlinear Anal. 73 (2010), no. 10, 3456–3461. MR 2680039, DOI 10.1016/j.na.2010.07.034
Gaston M. N’Guerekata, Almost automorphic and almost periodic functions in abstract spaces, Kluwer Academic/Plenum Publishers, New York, 2001. MR 1880351, DOI 10.1007/978-1-4757-4482-8
Gaston M. N’Guérékata and Alexander Pankov, Integral operators in spaces of bounded, almost periodic, and almost automorphic functions, Differential Integral Equations 21 (2008), no. 11-12, 1155–1176. MR 2482500
Wolfgang M. Ruess and Vũ Quốc Phóng, Asymptotically almost periodic solutions of evolution equations in Banach spaces, J. Differential Equations 122 (1995), no. 2, 282–301. MR 1355893, DOI 10.1006/jdeq.1995.1149
W. A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), 719–751. MR 187014, DOI 10.2307/2373071
Ti Jun Xiao and Jin Liang, Second order linear differential equations with almost periodic solutions, Acta Math. Sinica (N.S.) 7 (1991), no. 4, 354–359. A Chinese summary appears in Acta Math. Sinica 35 (1992), no. 5, 719–720. MR 1150329, DOI 10.1007/BF02594892
Tijun Xio and Jin Liang, Complete second order linear differential equations with almost periodic solutions, J. Math. Anal. Appl. 163 (1992), no. 1, 136–146. MR 1144711, DOI 10.1016/0022-247X(92)90283-J
Ti-Jun Xiao and Jin Liang, The Cauchy problem for higher-order abstract differential equations, Lecture Notes in Mathematics, vol. 1701, Springer-Verlag, Berlin, 1998. MR 1725643, DOI 10.1007/978-3-540-49479-9
Ti-Jun Xiao, Jin Liang, and Jun Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum 76 (2008), no. 3, 518–524. MR 2395199, DOI 10.1007/s00233-007-9011-y
Ti-Jun Xiao, Xing-Xing Zhu, and Jin Liang, Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Anal. 70 (2009), no. 11, 4079–4085. MR 2515324, DOI 10.1016/j.na.2008.08.018
S. Zaidman, Almost-periodic functions in abstract spaces, Research Notes in Mathematics, vol. 126, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 790316
Jun Zhang, Ti-Jun Xiao, and Jin Liang, Weighted pseudo almost-periodic functions and applications to semilinear evolution equations, Abstr. Appl. Anal. , posted on (2012), Art. ID 179525, 15. MR 2903805, DOI 10.1155/2012/179525