# zbMATH — the first resource for mathematics

Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay. (English) Zbl 1173.42308
Summary: We prove the invariance of Stepanov-like pseudo-almost periodic functions under bounded linear operators. Furthermore, we obtain existence and uniqueness theorems of pseudo-almost periodic mild solutions to evolution equations $$u{^{\prime}}(t)=A(t)u(t)+h(t)$$ and $$u^\prime (t)= A(t)u(t)+f(t,Bu(t))+\int_{-\infty}^t C(t,s)u(s)\text ds+F(t)$$ on $$\mathbb R$$, assuming that $$A(t)$$ satisfy “Acquistapace-Terreni” conditions, that the evolution family generated by $$A(t)$$ has exponential dichotomy, that $$R(\lambda _{0},A(\cdot))$$ is almost periodic, that $$B,C(t,s)_{t\geq s}$$ are bounded linear operators, that $$f$$ is Lipschitz with respect to the second argument uniformly in the first argument and that $$h, f, F$$ are Stepanov-like pseudo-almost periodic for $$p>1$$ and continuous. To illustrate our abstract result, a concrete example is given.

##### MSC:
 42A75 Classical almost periodic functions, mean periodic functions 42A85 Convolution, factorization for one variable harmonic analysis 44A35 Convolution as an integral transform
Full Text:
##### 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 there applications to differential equations, Comm. math. anal., 3, 1, 9-18, (2007) · Zbl 1286.44007 [3] Acquistapace, P.; Terreni, B., A unified approach to abstract linear parabolic equations, Rend. sem. mat. univ. Padova, 78, 47-107, (1987) · Zbl 0646.34006 [4] Maniar, L.; Schnaubelt, R., Almost periodicity of inhomogeneous parabolic evolution equations, (), 299-318 · Zbl 1047.35078 [5] 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 [6] 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 [7] 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 [8] Acquistapace, P., Evolution operators and strong solution of abstract linear parabolic equations, Differential integral equations, 1, 433-457, (1988) · Zbl 0723.34046 [9] 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 [10] Yagi, A., Abstract quasilinear evolution equations of parabolic type in Banach spaces, Boll. un. mat. ital. B, 5, 7, 341-368, (1991) · Zbl 0851.35060 [11] Engel, K.J.; Nagel, R., () [12] Coppel, W.A., Dichotomies in stability theory, (1978), Springer-Verlag · Zbl 0376.34001 [13] Henry, D., Geometric theory of semilinear parabolic equations, (1981), Springer-Verlag · Zbl 0456.35001 [14] Chicone, C.; Latushkin, Y., Evolution semigroups in dynamical systems and differential equations, Proc. amer. math. soc., (1999) · Zbl 0970.47027 [15] Burton, T.A.; Zhang, B., Periodic solutions of abstract differential equations with infinite delay, J. differential equations, 90, 357-396, (1991) · Zbl 0760.34060 [16] Diagana, T., Pseudo almost periodic solutions to some differential equations with infinite delay, Electron. J. differential equations, 2006, 79, 1-10, (2006) · Zbl 1117.34075 [17] 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 [18] 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 [19] 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 [20] 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 [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] Diagana, T., Pseudo almost periodic solutions to some differential equations, Nonlinear anal. TMA, 60, 1277-1286, (2005) · Zbl 1061.34040 [24] 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 [25] Lee, H.; Alkahby, H., Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear anal., 69, 2158-2166, (2008) · Zbl 1162.34063 [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] Ait Dads, E.; Ezzinbi, K., Pseudo almost periodic solutions of some delay differential equations, Math. anal. appl., 201, 840-850, (1996) · Zbl 0858.34055 [31] 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 [32] 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 [33] 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 [34] Fatajou, S.; Van Minh, N.; N’Guérékata, G.M.; Pankov, A., Stepanov-like almost automorphic solutions for some nonautonomous evolution equations, Electron. J. differential equations, 2007, 121, 1-11, (2007) · Zbl 1146.47029 [35] Diagana, T.; N’Guérékata, G., Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. anal., 86, 6, 723-733, (2007) · Zbl 1128.43006 [36] 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 [37] Pankov, A., Bounded and almost periodic solutions of nonlinear operator differential equations, (1990), Kluwer Dordrecht · Zbl 0712.34001 [38] Fink, A.M., ()
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.