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Stepanov-like pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations. (English) Zbl 1236.42006
Authors’ abstract: We obtain new existence and uniqueness theorems of pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations \begin{aligned} {d\over dt}[u(t)+ f(t, u(t))]= A(t)[u(t)- f(t,u(t))]+ g(t,u(t)),\quad & t\in\mathbb{R},\\ {d\over dt} [u(t)+ f(t,Bu(t))]= A(t)[u(t)+ f(t,Bu(t))]+ g(t,Cu(t)),\quad & t\in\mathbb{R},\end{aligned} assuming that $$A(t)$$ satisfy “Acquistapace-Terreni” conditions, the evolution family generated by $$A(t)$$ has exponential dichotomy, $$R(\lambda_0, A(\cdot))$$ is almost periodic, $$B$$, $$C$$ are densely defined closed linear operators, $$f$$, $$g$$ are Lipschitz with respect to the second argument uniformyl in the first argument, $$f$$ is pseudo almost periodic in the first argument, $$g$$ is Stepanov-like pseudo almost periodic in the first argument for $$p> 1$$ and jointly continuous. To illustrate our abstract results, two examples are given.

##### MSC:
 42A75 Classical almost periodic functions, mean periodic functions 47D06 One-parameter semigroups and linear evolution equations
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##### References:
 [1] Diagana, T., Stepanov-like pseudo almost periodicity and its applications to some nonautonomous differential equations, Nonlinear anal. TMA, 69, 4277-4285, (2008) · Zbl 1169.34330 [2] Diagana, T., Stepanov-like pseudo almost periodic functions and their applications to differential equations, Commun. math. anal., 3, 1, 9-18, (2007) · Zbl 1286.44007 [3] Hu, Z.R.; Jin, Z., Stepanov-like peudo almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay, Nonlinear anal. TMA, 71, 5381-5391, (2009) · Zbl 1173.42308 [4] Acquistapace, P.; Terreni, B., A unified approach to abstract linear parabolic equations, Rend. sem. mat. univ. Padova, 78, 47-107, (1987) · Zbl 0646.34006 [5] Maniar, L.; Schnaubelt, R., Almost periodicity of inhomogeneous parabolic evolution equations, (), 299-318 · Zbl 1047.35078 [6] Ding, H.S.; Liang, J.; N’Guérékata, G.M.; Xiao, T.J., Pseudo almost periodicity to some nonautonomous evolution equations with delay, Nonlinear anal. TMA, 67, 1412-1418, (2007) · Zbl 1122.34345 [7] Ding, H.S.; Liang, J.; N’Guérékata, G.M.; Xiao, T.J., Mild pseudo almost periodic solutions of nonautonomous semilinear evolution equations, Math. comput. modelling, 45, 579-584, (2007) · Zbl 1165.34387 [8] Xiao, T.J.; Zhu, X.X.; Liang, J., Pseudo almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear anal. TMA, 70, 4079-4085, (2009) · Zbl 1175.34076 [9] Acquistapace, P., Evolution operators and strong solution of abstract linear parabolic equations, Differential integral equations, 1, 433-457, (1988) · Zbl 0723.34046 [10] Yagi, A.; Terreni, B., Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II, Funkcial. ekvac., 33, 139-150, (1990) · Zbl 0706.35060 [11] Yagi, A., Abstract quasilinear evolution equations of parabolic type in Banach spaces, Boll. unione mat. ital. B, 5, 7, 341-368, (1991) · Zbl 0851.35060 [12] Engel, K.J.; Nagel, R., () [13] Coppel, W.A., Dichotomies in stability theory, (1978), Springer-Verlag · Zbl 0376.34001 [14] Henry, D., Geometric theory of semilinear probolic equations, (1981), Springer-Verlag [15] Chicone, C.; Latushkin, Y., Evolution semigroups in dynamical systems and differential equations, (1999), Proc. Amer. Math. Soc. · Zbl 0970.47027 [16] Diagana, T.; N’Guérékata, G.M., Pseudo almost periodic mild solutions to hyperbolic evolution equations in abstract intermediate Banach spaces, Appl. anal., 85, 6, 769-780, (2006) · Zbl 1103.34051 [17] Diagana, T.; Mahop, C.M.; N’Guérékata, G.M.; Toni, B., Existence and uniqueness of pseudo almost periodic solutions to some classes of semilinear differential equations and applications, Nonlinear anal. TMA, 64, 2442-2453, (2006) · Zbl 1102.34043 [18] Diagana, T., Pseudo almost periodic solutions to some differential equations, Nonlinear anal. TMA, 60, 1277-1286, (2005) · Zbl 1061.34040 [19] Cuevas, C.; Pinto, M., Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain, Nonlinear anal. TMA, 45, 73-83, (2001) · Zbl 0985.34052 [20] Li, H.X.; Huang, F.L.; Li, J.Y., Composition of pseudo almost-periodic functions and semilinear differential equations, J. math. anal. appl., 255, 436-446, (2001) · Zbl 1047.47030 [21] Amir, B.; Maniar, L., Composition of pseudo almost periodic functions and Cauchy problems with operator of nondense domain, Ann. math. blaise Pascal, 6, 1-11, (1999) · Zbl 0941.34059 [22] Ait Dads, E.; Ezzinbi, K.; Arino, O., Pseudo almost periodic solution for some differential equation in a Banach space, Nonlinear anal. TMA, 28, 1141-1155, (1997) · Zbl 0874.34041 [23] N’Guérékata, G.M.; Pankov, A., Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear anal. TMA, 68, 2658-2667, (2008) · Zbl 1140.34399 [24] Lee, H.; Alkahby, H., Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear anal. TMA, 69, 2158-2166, (2008) · Zbl 1162.34063 [25] Diagana, T., Existence and uniqueness of pseudo almost periodic solutions to some classes of partial evolution equations, Nonlinear anal. TMA, 66, 384-395, (2007) · Zbl 1105.35304 [26] Zhang, C.Y., Pseudo almost periodic solutions of some differential equations, J. math. anal. appl., 151, 62-76, (1994) · Zbl 0796.34029 [27] Zhang, C.Y., Pseudo almost periodic solutions of some differential equations II, J. math. anal. appl., 192, 543-561, (1995) · Zbl 0826.34040 [28] Zhang, C.Y., Integration of vector-valued pseudo almost periodic functions, Proc. amer. math. soc., 121, 167-174, (1994) · Zbl 0818.42003 [29] C.Y. Zhang, Pseudo Almost Periodic Functions and Their Applications, Thesis, the University of Western Ontario, 1992. [30] Diagana, T.; Hernández, E.M., Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional differential equations and applications, J. math. anal. appl., 327, 776-791, (2007) · Zbl 1123.34060 [31] Hernández, E.M.; Henriquez, H., Pseudo almost periodic solutions for non-autonomous neutral differential equations with unbounded delay, Nonlinear anal. RWA, 9, 430-437, (2008) · Zbl 1143.35382 [32] Ait Dads, E.; Ezzinbi, K., Pseudo almost periodic solutions of some delay differential equations, Math. anal. appl., 201, 840-850, (1996) · Zbl 0858.34055 [33] Zhao, Z.H.; Chang, Y.K.; Li, W.S., Asymptotically almost periodic, almost periodic and pseudo-almost periodic mild solutions for neutral differential equations, Nonlinear anal. RWA, 11, 3037-3044, (2010) · Zbl 1205.34088 [34] N’Guérékata, G.M., Almost automorphic functions and almost periodic functions in abstract spaces, (2001), Kluwer Academic/Plenum New York, London, Moscow · Zbl 1001.43001 [35] Pankov, A., Bounded and almost periodic solutions of nonlinear operator differential equations, (1990), Kluwer Dordrecht · Zbl 0712.34001 [36] Fink, A.M., () [37] Diagana, T., Weighted pseudo almost periodic functions and applications, C. R. math., 343, 10, 643-646, (2006) · Zbl 1112.43005 [38] Agarwal, Ravi P.; Diagana, T.; Hernández, E.M., Weighted pseudo-almost periodic solutions to some partial neutral functional differential equations and applications, J. nonlinear convex anal., 8, 397-415, (2007) · Zbl 1155.35104 [39] Agarwal, Ravi P.; de Andrade, B.; Cuevas, C., Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear anal. RWA, 11, 3532-3554, (2010) · Zbl 1248.34004 [40] Diagana, T.; Mophou, G.M.; N’Guérékata, G.M., Existence of weighted pseudo-almost periodic solutions to some classes of differential equations with $$S^p$$-weighted pseudo-almost periodic coefficients, Nonlinear anal. TMA, 72, 430-438, (2010) · Zbl 1184.43005 [41] Chen, X.X; Hu, X.Y., Weighted pseudo almost periodic solutions of neutral functional differential equations, Nonlinear anal. RWA, 12, 601-610, (2011) · Zbl 1210.34095 [42] Liang, J.; Xiao, T.J; Zhang, J., Decomposition of weighted pseudo-almost periodic functions, Nonlinear anal. TMA, 73, 3456-3461, (2010) · Zbl 1198.43004 [43] Diagana, T., The existence of a weighted Mean for almost periodic functions, Nonlinear anal. TMA, 74, 4269-4273, (2011) · Zbl 1222.43010
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