×

zbMATH — the first resource for mathematics

Pseudo almost periodic and automorphic mild solutions to nonautonomous neutral partial evolution equations. (English) Zbl 1329.34105
Summary: We present a new concept of Stepanov weighted pseudo almost periodic and automorphic functions which is more generale than the classical one, and we obtain a new existence result of \(\mu\)-pseudo almost periodic and \(\mu\)-pseudo almost automorphic mild solutions for some nonautonomous evolution equations with Stepanov \(\mu\)-pseudo almost periodic terms. An example is shown to illustrate our results.
MSC:
34G20 Nonlinear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
47A45 Canonical models for contractions and nonselfadjoint linear operators
47D06 One-parameter semigroups and linear evolution equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
37C60 Nonautonomous smooth dynamical systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] C. Zhang, Integration of vector-valued pseudo-almost periodic functions, Proc. Am. Math. Soc. 121(1), (1994), 167-174. · Zbl 0818.42003
[2] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl. 151, (1994), 62-76. · Zbl 0796.34029
[3] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl. 192, (1995), 543-561. · Zbl 0826.34040
[4] C. Corduneanu, Almost Periodic Functions, Wiley, New York, 1968. · Zbl 0175.09101
[5] S. Bochner; Continuous mappings of almost automorphic and almost periodic functions, Proc. Nat. Acad. Sci. USA, 52, (1964), 907-910. · Zbl 0134.30102
[6] H.X. Li, L.L. Li, Stepanov-like pseudo almost periodicity and semilinear differential equations with uniform continuity, Reaults. Math. 59, (2011), 43-61. · Zbl 1209.35011
[7] J. Blot, G.M. Mophou, G.M. NGuérékata and D. Pennequin, Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Analysis, Theory, Methods and Applications Vol 71, Issue 3-4, (2009), 903-909.
[8] J. Blot, P. Cieutat and K. Ezzinbi: New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications Applicable Analysis, (2011), 1-34. · Zbl 1266.43004
[9] J. Blot, P. Cieutat and K. Ezzinbi: Measure theory and pseudo almost automorphic functions, new developements and applications Applicable Analysis, (2011), 1-29.
[10] M. Damak, K. Ezzinbi and L. Souden, Weighted pseudo-almost periodic solutions for some neutral partial functional differential equations, Vol. 2012, No. 47, (2012), 1-13. · Zbl 1244.34092
[11] M. Frechet, Sur le théorème ergodique de Birkhoff, C. R. Math. Acad. Sci. Paris 213, (1941), 607-609 (in French). · JFM 67.0231.02
[12] L.Maniar, R. Schnaubelt, Almost periodicity of inhomogeneous parabolic evolution equations , in: Lecture Notes in Pure and Appl. Math. vol. 234, Dekker, New york, 2003, 299-318. · Zbl 1047.35078
[13] T.Diagana: Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations. J. Math. Anal. Appl. 350, (2009), 18-28. · Zbl 1167.34023
[14] J. Liang, T.J. Xiao, J. Zhang, Decomposition of weighted pseudo almost periodic functions, Nonlinear Anal. 73, (10), (2010), 3456-3461. · Zbl 1198.43004
[15] T.Diagana: Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations Nonlinear Analysis. 69, (2008), 4277-4285. · Zbl 1169.34330
[16] T. Diagana, Giséle M. Mophou and Gaston M. N’Guérékata: Existence of weighted pseudo almost periodic solutions to some classes of differential equations with Sp-weighted pseudo almost periodic coefficients Nonlinear Analysis. 72, (2010), 430- 438. · Zbl 1184.43005
[17] T. Diagana: stepanov -like pseudo almost periodicity and its application to some nonautonomes differential equation, Commun. Math. Anal.3, (2007), 9-18. · Zbl 1286.44007
[18] T. Diagana: Weighted pseudo almost periodic functions and applications C.R.A.S, 343, (10), (2006), 643-646.
[19] T. Diagana: Existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations Nonlinear Analysis. 74, (2011), 600-615. · Zbl 1209.34074
[20] T. Diagana: Weighted pseudo-almost periodic solutions to some differential equations Nonlinear Analysis. 68, (2008), 2250- 2260. · Zbl 1131.42006
[21] T. Diagana, Existence of p-almost automorphic mild solution to some abstract differential equations, Int. J. Evol. Equ. 1, (2005), 57-67. · Zbl 1083.35052
[22] T.J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic solution to semilinear differential equations in Banach spaces, Semigroup Forum 76, (2008), 518-524. · Zbl 1154.46023
[23] H. Lee and H. Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear Anal. 69, (2008), 2158-2166. · Zbl 1162.34063
[24] K. Ezzinbi, S. Fatajou, G.M. N’Guérékata, Pseudo almost automorphic solutions to some neutral partial functional differential equations in Banach spaces, Nonlinear Anal. TMA 70, (2009), 1641-1647. · Zbl 1165.34418
[25] G.M. N’Guérékata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005.
[26] P. Acquistapace, B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Sem. Mat. Univ. Padova. 78, (1987), 47-107. · Zbl 0646.34006
[27] G.M. N’Guérékata, A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Analysis. TMA, 68, (2008), 2658-2667. · Zbl 1140.34399
[28] R. P. Agarwal, Bruno de Andrade and Claudio Cuevas,Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations Nonlinear Analysis: RealWorld Applications. 11, (2010), 3532-3554. · Zbl 1248.34004
[29] Z.R. Hu, Z. Jin, Stepanov-like pseudo almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay, Nonlinear Anal. 71, (2009), 5381-5391. · Zbl 1173.42308
[30] K.J. Engel, R. Nagel, one parametr semigroups for linear evolution equations, in: Graduate texts in Mathematics, Springer- Verlag, 2000.
[31] M. Baroun, S. Boulite, G. M. N’Guérékata, L. Maniar, Almost automorphy of semilinear parabolic evolution equations, Electronic Journal of Differential Equations, No. 60 (2008), 1-9. · Zbl 1170.34344
[32] Z. Hu, Z. Jin Stepanov-like pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations Nonlinear Analysis, 75, (2012), 244-252. · Zbl 1236.42006
[33] Z. Hu, Z. Jin Stepanov-like pseudo almost automorphic mild solutions to nonautonomous evolution equations Nonlinear Analysis, 71, (2009), 2349-2360. · Zbl 1172.34038
[34] H. S. Dinga, J. Lianga, G. M. N’Guérékatab, T. J. Xiao Pseudo almost periodicity of some nonautonomous evolution equations with delay Nonlinear Analysis, 67, (2007), 1412-1418.
[35] H. Lee, H. Alkahby Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay Nonlinear Analysis, 69, (2008), 2158-2166. · Zbl 1162.34063
[36] T.J. Xiao, X.X. Zhu, J. Liang, Pseudo almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Anal. TMA 70 (2009), 4079-4085. · Zbl 1175.34076
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.