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Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations. (English) Zbl 1194.47047
The present paper discusses the existence and uniqueness of an almost automorphic (a weighted pseudo almost automorphic) mild solution to a class of semilinear evolution equations x ' (t)=A(t)x(t)+f(t,x(t)) in a Banach space. The main results are Theorems 3.2 and 4.2. However, Theorem 3.2 can be seen from [H.-S. Ding, W. Long and G. M. N’Guérékata, Nonlinear Anal., Theory Methods Appl. 70, No. 12 (A), 4158–4164 (2009; Zbl 1161.43301)]. Moreover, the authors use the Banach contraction mapping principle to obtain the conclusion in the proof of Theorem 4.2; thus, the completeness of the space WPAA(R,ρ) is needed. But from Lemma 2.10 in [J. Blot, G. M. Mophou, G. M. N’Guérékata and D. Pennequin, Nonlinear Anal., Theory Methods Appl. 71, No. 3–4 (A), 903–909 (2009, Zbl 1177.34077)], one only knows that the space WPAA(R,ρ) is a Banach space if ρU b . Actually, to the best of the reviewer’s knowledge, there is no proof in the literature that says that WPAA(R,ρ) is complete in the case when ρ is not necessarily bounded. On the other hand, when ρ is bounded, Theorem 4.2 is known from [T.-J. Xiao, X.-X. Zhu and J. Liang, Nonlinear Anal., Theory Methods Appl. 70, No. 11 (A), 4079–4085 (2009, Zbl 1175.34076)] since WPAA(R,ρ)=PAA(X) in this case.
MSC:
47D06One-parameter semigroups and linear evolution equations
34G10Linear ODE in abstract spaces
References:
[1]Acquistapace, P.: Evolution operators and strong solution of abstract linear parabolic equations, Differential integral equations 1, 433-457 (1998) · Zbl 0723.34046
[2]Acquistapace, P.; Terreni, B.: A unified approach to abstract linear parabolic equations, Rend. sem. Mat. univ. Padova 78, 47-107 (1987) · Zbl 0646.34006 · doi:numdam:RSMUP_1987__78__47_0
[3]Blot, J.; Mophou, G. M.; N’guérékata, G. M.; Pennequin, D.: Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear anal. 71, 903-909 (2009) · Zbl 1177.34077 · doi:10.1016/j.na.2008.10.113
[4]Boukli-Hacene, N.; Ezzinbi, K.: Weighted pseudo almost periodic solutions for some partial functional differential equations, Nonlinear anal. 71, 3612-3621 (2009) · Zbl 1175.34101 · doi:10.1016/j.na.2009.02.022
[5]Diagana, T.: Weighted pseudo almost periodic functions and applications, C. R. Acad. sci. Paris ser. I 343, No. 10, 643-646 (2006) · Zbl 1112.43005 · doi:10.1016/j.crma.2006.10.008
[6]Diagana, T.: Weighted pseudo almost periodic solutions to some differential equations, Nonlinear anal. 68, 2250-2260 (2008) · Zbl 1131.42006 · doi:10.1016/j.na.2007.01.054
[7]Diagana, T.: Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations, J. math. Anal. appl. 350, 18-28 (2009) · Zbl 1167.34023 · doi:10.1016/j.jmaa.2008.09.041
[8]Diagana, T.; Henriquez, H. R.; Hernández, E. M.: Almost automorphic mild solutions to some partial neutral functional – differential equations and applications, Nonlinear anal. 69, 1485-1493 (2008) · Zbl 1162.34062 · doi:10.1016/j.na.2007.06.048
[9]Ezzinbi, K.; Fatajou, S.; N’guérékata, G. M.: Pseudo almost automorphic solutions to some neutral partial functional differential equations in Banach spaces, Nonlinear anal. 70, 1641-1647 (2009) · Zbl 1165.34418 · doi:10.1016/j.na.2008.02.039
[10]Ezzinbi, K.; N’guérékata, G. M.: Massera type theorem for almost automorphic solutions of functional differential equations of neutral type, J. math. Anal. appl. 316, 707-721 (2006) · Zbl 1122.34052 · doi:10.1016/j.jmaa.2005.04.074
[11]Ezzinbi, K.; N’guérékata, G. M.: Almost automorphic solutions for some partial functional differential equations, J. math. Anal. appl. 328, No. 1, 344-358 (2007) · Zbl 1121.34081 · doi:10.1016/j.jmaa.2006.05.036
[12]Goldstein, J. A.; N’guérékata, G. M.: Almost automorphic solutions of semilinear evolution equations, Proc. amer. Math. soc. 133, 2401-2408 (2005) · Zbl 1073.34073 · doi:10.1090/S0002-9939-05-07790-7
[13]Liang, J.; N’guérékata, G. M.; Xiao, Ti-Jun; Zhang, Jun: Some properties of pseudo-almost automorphic functions and applications to abstract differential equations, Nonlinear anal. 70, 2731-2735 (2009) · Zbl 1162.44002 · doi:10.1016/j.na.2008.03.061
[14]Liang, J.; Zhang, J.; Xiao, T. J.: Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. math. Anal. appl. 340, 1493-1499 (2008) · Zbl 1134.43001 · doi:10.1016/j.jmaa.2007.09.065
[15]N’guérékata, G. M.: Almost automorphic and almost periodic functions in abstract spaces, (2001)
[16]N’guérékata, G. M.: Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup forum 69, 80-86 (2004) · Zbl 1077.47058 · doi:10.1007/s00233-003-0021-0
[17]N’guérékata, G. M.: Topics in almost automorphy, (2005)
[18]Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983)
[19]Xiao, T. -J.; Liang, J.; Zhang, J.: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup forum 76, No. 3, 518-524 (2008) · Zbl 1154.46023 · doi:10.1007/s00233-007-9011-y
[20]Xiao, T. -J.; Zhu, X. -X.; Liang, J.: Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear anal. 70, 4079-4085 (2009) · Zbl 1175.34076 · doi:10.1016/j.na.2008.08.018
[21]Zhang, L.; Xu, Y.: Weighted pseudo-almost periodic solutions of a class of abstract differential equations, Nonlinear anal. 71, 3705-3714 (2009) · Zbl 1173.34041 · doi:10.1016/j.na.2009.02.032