SciELO - Scientific Electronic Library Online

 
vol.16 número2Voronovskaya type asymptotic expansions for multivariate quasi-interpolation neural network operators índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Revista

Articulo

Indicadores

Links relacionados

  • En proceso de indezaciónCitado por Google
  • No hay articulos similaresSimilares en SciELO
  • En proceso de indezaciónSimilares en Google

Compartir


Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.16 no.2 Temuco  2014

http://dx.doi.org/10.4067/S0719-06462014000200001 

 

Pseudo-Almost Periodic and Pseudo-Almost Automorphic Solutions to Some Evolution Equations Involving Theoretical Measure Theory

 

Toka Diagana1, Khalil Ezzinbi2, Mohsen Miraoui3

1Howard University, 2441 6th Street N.W., Washington, D.C. 20059, USA. tdiagana@howard.edu
2Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, BP 2390, Marrakesh, Maroc
.
3Institut supérieur des Etudes technologiques de Kairouan, Rakkada-3191 Kairouan, Tunisie.


ABSTRACT
Motivated by the recent works by the first and the second named authors, in this paper we introduce the notion of doubly-weighted pseudo-almost periodicity (respectively, doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basic properties of these new spaces are studied. To illustrate our work, we study, under Acquistapace–Terreni conditions and exponential dichotomy, the existence of - pseudo almost periodic (respectively, -pseudo almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces. A few illustrative examples will be discussed at the end of the paper.

Keywords and Phrases: Evolution family; exponential dichotomy; Acquistapace–Terreni conditions; pseudo-almost periodic; pseudo-almost automorphic; evolution equation; nonautonomous
equation; doubly-weighted pseudo-almost periodic; doubly-weighted pseudo-almost automorphy; -pseudo-almost periodicity; -pseudo-almost automorphy; neutral systems; positive measure.

2010 AMS Mathematics Subject Classification: 34C27; 34K14; 34K30; 35B15; 43A60;


RESUMEN
Motivado por los trabajos recientes del primer y segundo autor, en este artículo introducimos la noción de seudo-casi periodicidad con doble peso (seudo-casi automorfía con doble peso respectivamente) usando Teoría de la Medida. Se estudian las propiedades básicas de estos espacios nuevos. Para ilustrar nuestro trabajo, bajo las condiciones de Acquistapace-Terreni y dicotomía exponencial estudiamos la existencia de soluciones (respectivamente, seudo-casi periódicas seudo-casi automórficas) para algunas ecuaciones parciales de evolución autónomas en espacios de Banach. Algunos ejemplos ilustrativos se discutirán al final del artículo.


 

References

[1] P. Acquistapace, F. Flandoli, B. Terreni, Initial boundary value problems and optimal control for nonautonomous parabolic systems, SIAM Journal on Control and Optimization, 29, (1991), 89-118.         [ Links ]
[2] P. Acquistapace, B. Terreni,A unified approach to abstract linear nonautonomous parabolic equations, Rendiconti del Seminario Matematico della Università di Padova, 78, (1987), 47- 107.         [ Links ]
[3] E. Ait Dads, P. Cieutat, K. Ezzinbi, The existence of pseudo-almost periodic solutions for some nonlinear differential equations in Banach spaces Nonlinear Analysis: Theory, Methods and Applications, Volume 69, Issue 4, 15 August 2008, Pages 1325-1342.         [ Links ]
[4] E. Ait Dads, K. Ezzinbi, and O. Arino, Pseudo almost periodic solutions for some differential equations in Banach space, Nonlinear Analysis Theory Methods Appl. 28, (7), (1997), 1141- 1155.         [ Links ]
[5] E. Ait Dads and K. Ezzinbi, Pseudo almost periodic solutions of some delay differential equations, Journal of Mathematical Analysis and Applications, 201, (287), (1996), 840-850.         [ Links ]
[6] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Berlin, 1995.         [ Links ]
[7] B. Amir and L. Maniar, Composition of pseudo-almost periodic functions and cauchy problems with perator of nondense domain. Annales Mathématiques Blaise Pascal, 6, (1999), no. 1, 1-11.         [ Links ]
[8] M. Baroun, S. Boulite, T. Diagana, L. Maniar, Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations, Journal of Mathematical Analysis and Applications, 349, (2009), 74-84.         [ Links ]
[9] M. Baroun, S. Boulite, G. M. N’Guérékata, L. Maniar, Almost automorphy of semilinear parabolic evolution equations, Electronic Journal of Differential Equations, (2008), no. 60, 1-9.         [ Links ]
[10] C. J. K. Batty, W. Hutter, F. Räbiger, Almost periodicity of mild solutions of inhomogeneous Cauchy problems, Journal Differential Equations, 156, (1999), 309-327.         [ Links ]
[11] J. Blot, G. M. Mophou, G. M. N’Guérékata, and D. Pennequin, Weighted pseudo-almost automorphic functions and applications to abstract differential equations. Nonlinear Anal. 71(2009), nos. 3-4, pp. 903–909.         [ Links ]
[12] 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.         [ Links ]
[13] J. Blot, P. Cieutat and K. Ezzinbi, Measure theory and pseudo almost automorphic functions: New developments and applications, Nonlinear Analysis, 75, (2012), 2426-2447.         [ Links ]
[14] S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proceedings of the National Academy of Sciences of the United States of America, 52, (1964), 907-910.         [ Links ]
[15] H. Bohr, Almost periodic functions. Chelsea Publishing Company, New York, 1947.         [ Links ]
[16] N. Boukli-Hacene and K. Ezzinbi, Weighted pseudo almost periodic solutions for some partial functional differential equations Nonlinear Analysis: Theory, Methods and Applications, Volume 71, Issue 9, 1 November 2009, 3612-3621.         [ Links ]
[17] N. Boukli-Hacene, K. Ezzinbi, Weighted pseudo-almost automorphic solutions for partial functional differential equations, Nonlinear Analysis: Real World Applications, 12, (1), (2010), 562-570.         [ Links ]
[18] S. Boulite, L. Maniar, G. M. N’Guérékata, Almost automorphic solutions for hyperbolic semilinear evolution equations, Semigroup Forum, 71, (2005), 231-240.         [ Links ]
[19] Y-K. Chang, Z-H. Zhao, J.J. Nieto, Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to semi-linear differential equations in Hilbert spaces, Revista Matemática Complutense, 24, No. 2,(2011), 421-438.         [ Links ]
[20] P. Cieutat, S. Fatajou, G.M. N’Guérékata, Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations, Applicable Analysis, 89, (1), (2010), 11-27.         [ Links ]
[21] C. Corduneanu,Almost Periodic Functions, Wiley, New York, 1968 (Reprinted, Chelsea, New York, 1989).         [ Links ]
[22] T. Diagana, G. N’Guérékata, N. Van Minh, Almost automorphic solutions of evolution equations, Proc. Amer. Math. Soc., 132, (2004), 3289-3298.         [ Links ]
[23] T. Diagana, C.M. Mahop, G.M. N’Guérékata, and B. Toni, Existence and uniqueness of pseudo-almost periodic solutions to some classes of semilinear differential equations and applications, Nonlinear Analysis Theory Methods Appl. 64, (11), (2006), 2442-2453.         [ Links ]
[24] T. Diagana, Existence and uniqueness of pseudo-almost periodic solutions to some classes of partial evolution equations, Nonlinear Analysis Theory Methods Appl. 66, (2), (2007), 384-395.         [ Links ]
[25] T. Diagana, Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations, Journal of Mathematical Analysis and Applications, 350, (2009), 18-28.         [ Links ]
[26] T. Diagana, Existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations, Nonlinear Analysis, 74, (2011), 600-615.         [ Links ]
[27] T. Diagana, Pseudo Almost Periodic Functions in Banach Spaces, Nova Science Publishers, Inc., New York, 2007.         [ Links ]
[28] T. Diagana, Pseudo almost periodic solutions to some differential equations, Nonlinear Analysis Theory Methods Appl. 60, (7), (2005), 1277-1286.         [ Links ]
[29] T. Diagana, Weighted pseudo almost periodic functions and applications C.R.A.S, 343, (10), (2006), 643-646.         [ Links ]
[30] T. Diagana, Doubly-weighted pseudo almost periodic functions. Afr. Diaspora J. Math. 12 (2011), no. 1, 121–136.         [ Links ]
[31] T. Diagana, Weighted pseudo-almost periodic solutions to some differential equations, Nonlinear Analysis, 68, (2008), 2250-2260.         [ Links ]
[32] T. Diagana, Pseudo-almost automorphic solutions to some classes of nonautonomous partial evolution equations,Differential Equations et Applications, Volume 1, Number 4, (2009), 561- 582.         [ Links ]
[33] T. Diagana, Pseudo-almost periodic solutions for some classes of nonautonomous partial evolution equations. Journal of the Franklin Institute Vol. 348 (2011), pp. 2082–2098.         [ Links ]
[34] K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate texts in Mathematics, Springer Verlag, 1999.         [ Links ]
[35] K. Ezzinbi, G.M. N’Guérékata, Almost automorphic solutions for some partial functional differential equations, Journal of Mathematical Analysis and Applications, 328, (1), (2007), 344-358.         [ Links ]
[36] K. Ezzinbi and G.M. N’Guérékata,Almost automorphic solutions for some partial functional differential equations,Journal of Mathematical Analysis and Applications, 328, (1), (2007), 344-358.         [ Links ]
[37] K. Ezzinbi, V. Nelson, G.M. N’Guérékata, C(n)-almost automorphic solutions of some nonautonomous differential equations, Cubo, 10, (2), (2008), 61-74.         [ Links ]
[38] M. Fréchet, Sur le théorème ergodique de Birkhoff, Les Comptes Rendus Mathématique de l’Académie des sciences Paris, 213, (1941), 607-609 (in French).         [ Links ]
[39] J.A. Goldstein, G.M. N’Guérékata, Almost automorphic solutions of semilinear evolution equations, Proceedings of the America Mathematical Society, 133, (8), (2005), 2401-2408.         [ Links ]
[40] G. Gühring, F. Räbiger, Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations, Abstr. Appl. Anal. 4, (1999), 169-194.         [ Links ]
[41] G. Gühring, F. Räbiger, R. Schnaubelt, A characteristic equation for nonautonomous partial functional differential equations, Journal Differential Equations, 181, (2002), 439-462.         [ Links ]
[42] Y. Hino, S. Murakami, Almost automorphic solutions for abstract functional differential equations, Journal of Mathematical Analysis and Applications, 286, (2003), 741-752.         [ Links ]
[43] J. Liang, T.J. Xiao, and J. Zhang, Decomposition of weighted pseudo-almost periodic functions, Nonlinear Analysis Theory Methods Appl. 73,(2010), 3456-3461. Applicable Analysis 33 Downloaded by [Ezzinbi Khalil] at 02:02 16 November 2011.         [ Links ]
[44] J. Liang, G.M. N’Guérékata, T-J. Xiao, J. Zhang, Some properties of pseudo-almost automorphic functions and applications to abstract differentiel equations, Nonlinear Analysis, Theory, Methods and Applications, 70, (7), (2009), 2731-2735.         [ Links ]
[45] J. Liang, J. Zhang, T-J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, Journal of Mathematical Analysis and Applications, 340, (2), (2008), 1493-1499.         [ Links ]
[46] J. Liang, T-J. Xiao, J. Zhang, Decomposition of weighted pseudo-almost periodic functions, Nonlinear Analysis, Theory, Methods and Applications, 73, (10), (2010), 3456-3461.         [ Links ]
[47] J. H. Liu, G. M. N’Guérékata, N. V. Minh, Topics on stability and periodicity in abstract differential equations. Series on Concrete and Applicable Mathematics, Vol. 6,World Scientific, 2008.         [ Links ]
[48] L. Maniar and R. Schnaubelt, Almost periodicity of inhomogeneous parabolic evolution equations, in: Lecture Notes in Pure and Applied Mathematics, vol. 234, Dekker, New york, 2003, 299-318.         [ Links ]
[49] G.M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic Plenum Publishers, New York, Boston, Moscow, London, 2001.         [ Links ]
[50] G.M. N’Guérékata, Existence and uniqueness of almost automorphic mild solution to some semilinear abstract differential equations, Semigroup Forum 69, (2004), 80-86.         [ Links ]
[51] G. M. N’Guérékata ; Topics in Almost Automorphy, Springer, New York, Boston, Dordrecht, London, Moscow 2005.         [ Links ]
[52] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.         [ Links ]
[53] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, 1993.         [ Links ]
[54] L. Schwartz, Topologie Generale et Analyse Fonctionnelle, Hermann, Paris, 1976 (in French).         [ Links ]
[55] R. Schnaubelt, Sufficient conditions for exponential stability and dichotomy of evolution equations, Forum Math. 11 (1999) 543-566.         [ Links ]
[56] V Quôc Phóng, Stability and almost periodicity of trajectories of periodic processes, Journal Differential Equations, 115, (1995), 402-415.         [ Links ]
[57] T-J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum, 76, (3), (2008), 518-524.         [ Links ]
[58] T-J. Xiao, X. X. Zhu, J. Liang, Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Analysis, Theory,Methods and Applications, 70, (11), (2009), 4079-4085.         [ Links ]
[59] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations, Journal ofMathematical Analysis and Applications,151, (1994), 62-76.         [ Links ]
[60] C. Zhang, Pseudo Almost Periodic Type Functions and Ergodicity, Science Press, Kluwer Academic Publishers, Dordrecht, 2003.         [ Links ]
[61] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations II, Journal of Mathematical Analysis and Applications, 192, (1995), 543-561.


Received: January 2014. Revised: April 2014.