zbMATH — the first resource for mathematics

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
Full Text:
References:
 [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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.