# Pseudo almost periodic and automorphic mild solutions to nonautonomous neutral partial evolution equations

Abdelkarim-Nidal Akdad; Khalil Ezzinbi; Lotti Souden

Nonautonomous Dynamical Systems (2015)

- Volume: 2, Issue: 1, page 12-30, electronic only
- ISSN: 2353-0626

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topAbdelkarim-Nidal Akdad, Khalil Ezzinbi, and Lotti Souden. "Pseudo almost periodic and automorphic mild solutions to nonautonomous neutral partial evolution equations." Nonautonomous Dynamical Systems 2.1 (2015): 12-30, electronic only. <http://eudml.org/doc/270854>.

@article{Abdelkarim2015,

abstract = {In this work, 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 μ-pseudo almost periodic and μ-pseudo almost automorphic mild solutions for some nonautonomous evolution equations with Stepanov μ-pseudo almost periodic terms. An example is shown to illustrate our results.},

author = {Abdelkarim-Nidal Akdad, Khalil Ezzinbi, Lotti Souden},

journal = {Nonautonomous Dynamical Systems},

keywords = {Neutral equation; semigroup; mild solution; Measure theory; ergodicity; μ-pseudo almost periodic functions; μ-pseudo almost automorphic functions; Sp − μ-pseudo almost periodic functions; Sp − μ-pseudo almost automorphic functions; Banach fixed point theorem; neutral equation; measure theory; -pseudo almost periodic functions; -pseudo almost automorphic functions; -pseudo almost periodic functions; -pseudo almost automorphic functions},

language = {eng},

number = {1},

pages = {12-30, electronic only},

title = {Pseudo almost periodic and automorphic mild solutions to nonautonomous neutral partial evolution equations},

url = {http://eudml.org/doc/270854},

volume = {2},

year = {2015},

}

TY - JOUR

AU - Abdelkarim-Nidal Akdad

AU - Khalil Ezzinbi

AU - Lotti Souden

TI - Pseudo almost periodic and automorphic mild solutions to nonautonomous neutral partial evolution equations

JO - Nonautonomous Dynamical Systems

PY - 2015

VL - 2

IS - 1

SP - 12

EP - 30, electronic only

AB - In this work, 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 μ-pseudo almost periodic and μ-pseudo almost automorphic mild solutions for some nonautonomous evolution equations with Stepanov μ-pseudo almost periodic terms. An example is shown to illustrate our results.

LA - eng

KW - Neutral equation; semigroup; mild solution; Measure theory; ergodicity; μ-pseudo almost periodic functions; μ-pseudo almost automorphic functions; Sp − μ-pseudo almost periodic functions; Sp − μ-pseudo almost automorphic functions; Banach fixed point theorem; neutral equation; measure theory; -pseudo almost periodic functions; -pseudo almost automorphic functions; -pseudo almost periodic functions; -pseudo almost automorphic functions

UR - http://eudml.org/doc/270854

ER -

## References

top- [1] C. Zhang, Integration of vector-valued pseudo-almost periodic functions, Proc. Am. Math. Soc. 121(1), (1994), 167-174. Zbl0818.42003
- [2] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl. 151, (1994), 62-76.
- [3] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl. 192, (1995), 543-561. Zbl0826.34040
- [4] C. Corduneanu, Almost Periodic Functions, Wiley, New York, 1968.
- [5] S. Bochner; Continuous mappings of almost automorphic and almost periodic functions, Proc. Nat. Acad. Sci. USA, 52, (1964), 907-910. [Crossref] Zbl0134.30102
- [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. [WoS] Zbl1209.35011
- [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. Zbl1177.34077
- [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. Zbl1266.43004
- [9] J. Blot, P. Cieutat and K. Ezzinbi: Measure theory and pseudo almost automorphic functions, new developements and applications Applicable Analysis, (2011), 1-29. [WoS]
- [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. Zbl1244.34092
- [11] M. Frechet, Sur le théorème ergodique de Birkhoff, C. R. Math. Acad. Sci. Paris 213, (1941), 607-609 (in French). Zbl67.0231.02
- [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. Zbl1047.35078
- [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. Zbl1167.34023
- [14] J. Liang, T.J. Xiao, J. Zhang, Decomposition of weighted pseudo almost periodic functions, Nonlinear Anal. 73, (10), (2010), 3456-3461. Zbl1198.43004
- [15] T.Diagana: Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations Nonlinear Analysis. 69, (2008), 4277-4285. Zbl1169.34330
- [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. Zbl1184.43005
- [17] T. Diagana: stepanov -like pseudo almost periodicity and its application to some nonautonomes differential equation, Commun. Math. Anal.3, (2007), 9-18.
- [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. Zbl1209.34074
- [20] T. Diagana: Weighted pseudo-almost periodic solutions to some differential equations Nonlinear Analysis. 68, (2008), 2250- 2260. Zbl1131.42006
- [21] T. Diagana, Existence of p-almost automorphic mild solution to some abstract differential equations, Int. J. Evol. Equ. 1, (2005), 57-67. Zbl1083.35052
- [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. [WoS][Crossref] Zbl1154.46023
- [23] H. Lee and H. Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear Anal. 69, (2008), 2158-2166. Zbl1162.34063
- [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. Zbl1165.34418
- [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. Zbl0646.34006
- [27] G.M. N’Guérékata, A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Analysis. TMA, 68, (2008), 2658-2667. [Crossref] Zbl1140.34399
- [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. Zbl1248.34004
- [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. Zbl1173.42308
- [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. Zbl1170.34344
- [32] Z. Hu, Z. Jin Stepanov-like pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations Nonlinear Analysis, 75, (2012), 244-252. Zbl1236.42006
- [33] Z. Hu, Z. Jin Stepanov-like pseudo almost automorphic mild solutions to nonautonomous evolution equations Nonlinear Analysis, 71, (2009), 2349-2360. Zbl1172.34038
- [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. Zbl1162.34063
- [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. Zbl1175.34076

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