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Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications. (English) Zbl 1175.34076
The authors investigate the existence and uniqueness of pseudo-almost automorphic solutions to the following nonautonomous evolution equations in a Banach space \(X:\)
\[ x^{\prime}(t)=A(t)x(t)+f(t,x(t)),\;t\in\mathbb{R}, \]
\[ x^{\prime}(t)=A(t)x(t)+f(t,x(t-h)),\;t\in\mathbb{R}, \]
\[ x^{\prime}(t)=A(t)x(t)+f(t,x(t),\varkappa\left[ \alpha(t,x(t))\right] ),\;t\in\mathbb{R}. \]
They introduce a new concept of bi-almost automorphic functions, in order to study the existence of pseudo-almost automorphic solutions to the above equations. They also establish some new existence and uniqueness results for pseudo-almost automorphic mild solutions. As applications, one studies two heat equations with Dirichlet boundary conditions.

MSC:
34G20 Nonlinear differential equations in abstract spaces
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
35K05 Heat equation
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[1] Acquistapace, P., Evolution operators and strong solution of abstract parabolic equations, Differential integral equations, 1, 433-457, (1988) · 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
[3] Bochner, S., Uniform convergence of monotone sequences of functions, Proc. natl. acad. sci. USA, 47, 582-585, (1961) · Zbl 0103.05304
[4] Chicone, C.; Latushkin, Y., Evolution semigroups in dynamical systems and differential equations, (1999), Amer. Math. Soc. Providence, RI · Zbl 0970.47027
[5] T. Diagana, Stepanov-like pseudo almost periodicity and its applications to some nonautonomous differential equations, Nonlinear Anal., in press (doi:10.1016/j.na.2007.10.051) · Zbl 1169.34330
[6] Diagana, T.; Hernández, E.; Rabello, M., Pseudo almost periodic solutions to some non-autonomous neutral functional differential equations with unbounded delay, Math. comput. modelling, 45, 1241-1252, (2007) · Zbl 1133.34042
[7] Diagana, T.; Mahopa, C.M.; N’Guérékata, G.M., Pseudo-almost-periodic solutions to some semilinear differential equations, Math. comput. modelling, 43, 89-96, (2006) · Zbl 1096.34038
[8] Ding, H.S.; Liang, J.; N’Guérékata, G.M.; Xiao, T.J., Pseudo almost periodicity of some nonautonomous evolution equations with delay, Nonlinear anal. TMA, 67, 1412-1418, (2007) · Zbl 1122.34345
[9] Ezzinbi, K.; Fatajou, S.; N’Guérékata, G.M., \(C^n\)-almost automorphic solutions for partial neutral functional differential equations, Appl. anal., 86, 1127-1146, (2007) · Zbl 1153.34043
[10] Gal, C.G., Almost automorphic mild solutions to some semilinear abstract differential equations with deviated argument, J. integral equations appl., 17, 391-396, (2005) · Zbl 1104.43005
[11] Liang, J.; Zhang, J.; Xiao, T.J., Composition of pseudo almost automorphic functions, J. math. anal. appl., 340, 1493-1499, (2008) · Zbl 1134.43001
[12] Liu, James; N’Guérékata, G.M.; Van Minh, N., Almost automorphic solutions of second order evolution equations, Appl. anal., 84, 1173-1184, (2005) · Zbl 1085.34045
[13] Maniar, L.; Schnaubelt, R., Almost periodicity of inhomogeneous parabolic evolution equations, (), 299-318 · Zbl 1047.35078
[14] Van Minh, N.; Dat, T.T., On the almost automorphy of bounded solutions of differential equations with piecewise constant argument, J. math. anal. appl., 326, 165-178, (2007) · Zbl 1115.34068
[15] N’Guérékata, G.M., Almost automorphic and almost periodic functions in abstract space, (2001), Kluwer Academic/Plenum Publishers New York · Zbl 1001.43001
[16] Xiao, T.J.; Liang, J.; Zhang, J., Pseudo almost automorphic solutions to semilinear differential equations in Banach space, Semigroup forum, 76, 518-524, (2008) · Zbl 1154.46023
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