×

Pseudo almost automorphic mild solutions to non-autonomous differential equations in the “strong topology”. (English) Zbl 1515.34063

Summary: This paper established some results on the existence of pseudo almost automorphic mild solutions to non-autonomous linear and semi-linear differential equations using exponential dichotomy of the evolution family and Bi-pseudo almost automorphy of Green’s function in the “strong topology”. In our results, the assumption for Green’s function “Bi-almost automorphic” is weaken as “Bi-pseudo almost automorphic” in the “strong topology”. Moreover, we consider the semi-linear differential equations under some compact conditions instead of Lipschitz conditions.

MSC:

34G20 Nonlinear differential equations in abstract spaces
37C60 Nonautonomous smooth dynamical systems
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
47N20 Applications of operator theory to differential and integral equations
34B27 Green’s functions for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Acquistapace, P., Evolution operators and strong solutions of abstract linear parabolic equations, Differ. Integr. Equ., 1, 4, 433-457 (1988) · Zbl 0723.34046
[2] Acquistapace, P.; Terreni, B., A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78, 47-107 (1987) · Zbl 0646.34006
[3] Bochner, S., Uniform convergence of monotone sequences of functions, Proc. Natl. Acad. Sci. USA, 47, 4, 582-585 (1961) · Zbl 0103.05304 · doi:10.1073/pnas.47.4.582
[4] Bochner, S., A new approach to almost periodicity, Proc. Natl. Acad. Sci. USA, 48, 12, 2039-2043 (1962) · Zbl 0112.31401 · doi:10.1073/pnas.48.12.2039
[5] Bochner, S., Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl. Acad. Sci. USA, 52, 4, 907-910 (1964) · Zbl 0134.30102 · doi:10.1073/pnas.52.4.907
[6] Bochner, S., Neumann, J.V.: On compact solutions of operational-differential equations. i, Ann. Math. 255-291 (1935) · JFM 61.0442.01
[7] Cao, JF; Huang, ZT; N’Guérékata, GM, Existence of asymptotically almost automorphic mild solutions for nonautonomous semilinear evolution equations, Electron. J. Differ. Equ., 2018, 37, 1-16 (2018) · Zbl 1397.34101
[8] Chicone, CC; Latushkin, Y., Evolution Semigroups in Dynamical Systems and Differential Equations (1999), Providence: American Mathematical Society, Providence · Zbl 0970.47027 · doi:10.1090/surv/070
[9] Coppel, WA, Dichotomies in Stability Theory (2006), Berlin: Springer, Berlin · Zbl 0376.34001
[10] Coronel, A.; Maulén, C.; Pinto, M.; Sepúlveda, D., Almost automorphic delayed differential equations and lasota-Wazewska model, Discrete Contin. Dyn. Syst., 37, 4, 1959-1977 (2017) · Zbl 1368.34081 · doi:10.3934/dcds.2017083
[11] Diagana, T., Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces (2013), Cham: Springer International Publishing AG, Cham · Zbl 1279.43010 · doi:10.1007/978-3-319-00849-3
[12] Ding, HS; Wan, SM, Asymptotically almost automorphic solutions of differential equations with piecewise constant argument, Open Math., 15, 1, 595-610 (2017) · Zbl 1367.34093 · doi:10.1515/math-2017-0051
[13] Ding, HS; Liang, J.; N’Guérékata, GM; Xiao, TJ, Pseudo-almost periodicity of some nonautonomous evolution equations with delay, Nonlinear Anal., 67, 5, 1412-1418 (2007) · Zbl 1122.34345 · doi:10.1016/j.na.2006.07.026
[14] Dollard, JD; Friedman, CN, Product Integration with Application to Differential Equations (1984), New York: Cambridge University Press, New York · doi:10.1017/CBO9781107340701
[15] Liang, J.; Zhang, J.; Xiao, TJ, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl., 340, 2, 1493-1499 (2008) · Zbl 1134.43001 · doi:10.1016/j.jmaa.2007.09.065
[16] N’Guérékata, GM, Sur les solutions presqu’automorphes d’équations différentielles abstraites, Ann. Sci. Math. Québec, 5, 1, 69-79 (1981) · Zbl 0503.34035
[17] N’Guérékata, GM, Almost Periodic and Almost Automorphic Functions in Abstract Spaces (2021), Cham: Springer International Publishing AG, Cham · Zbl 1479.43001 · doi:10.1007/978-3-030-73718-4
[18] Salah, MB; Ezzinbi, K.; Rebey, A., Pseudo-almost periodic and pseudo-almost automorphic solutions to evolution equations in Hilbert spaces, Mediterr. J. Math., 13, 2, 703-717 (2016) · Zbl 1338.43007 · doi:10.1007/s00009-014-0510-2
[19] Xiao, TJ; Liang, J.; Zhang, J., Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum, 76, 3, 518-524 (2008) · Zbl 1154.46023 · doi:10.1007/s00233-007-9011-y
[20] Xiao, TJ; Zhu, XX; Liang, J., Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Anal., 70, 11, 4079-4085 (2009) · Zbl 1175.34076 · doi:10.1016/j.na.2008.08.018
[21] Zhu, HL; Liao, FF, Almost automorphic solutions of non-autonomous differential equations, Bull. Iran. Math. Soc., 44, 1, 205-223 (2018) · Zbl 1409.34043 · doi:10.1007/s41980-018-0015-z
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.