For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.
References
[1]
Bochner, S. (1964) Continuous Mappings of Almost Automorphic and Almost Periodic Functions. Proceedings of the National Academy of Sciences of the United States of America, 52, 907-910. https://doi.org/10.1073/pnas.52.4.907
[2]
Bochner, S. (1961) Uniform Convergence of Monotone Sequences of Functions. Proceedings of the National Academy of Sciences of the United States of America, 47, 582-585. https://doi.org/10.1073/pnas.47.4.582
[3]
Bochner, S. (1962) A New Approach to Almost Periodicity. Proceedings of the National Academy of Sciences of the United States of America, 48, 2039-2043. https://doi.org/10.1073/pnas.48.12.2039
[4]
Bochner, S. and Von Neumann, J. (1935) On Compact Solutions of Operational-Differential Equations. I. Annals of Mathematics, 36, 255-291. https://doi.org/10.2307/1968678
[5]
N’Guérékata, G. (2001) Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Academic/Plenum Publishers, New York. https://doi.org/10.1007/978-1-4757-4482-8
[6]
Schah, S.M. and Wiener, J. (1983) Advanced Differential Equations with Piecewise Constant Argument Deviations. International Journal of Mathematics and Mathematical Sciences, 6, 671-703. https://doi.org/10.1155/S0161171283000599
[7]
Wiener, J. (1999) A Second-Order Delay Differential Equation with Multiple Periodic Solutions. Journal of Mathematical Analysis and Applications, 229, 6596-676. https://doi.org/10.1006/jmaa.1998.6196
[8]
Wiener, J. and Debnath, L. (1997) Boundary Value Problems for the Diffusion Equation with Piecewise Continuous Time Delay. International Journal of Mathematics and Mathematical Sciences, 20, 187-195. https://doi.org/10.1155/S0161171297000239
[9]
Wiener, J. and Debnath, L. (1995) A Survey of Partial Differential Equations with Piecewise Continuous Arguments. International Journal of Mathematics and Mathematical Sciences, 18, 209-228. https://doi.org/10.1155/S0161171295000275
[10]
Wiener, J. and Lakshmikantham, V. (1999) Excitability of a Second-Order Delay Differential Equation. Nonlinear Analysis, 38, 1-11. https://doi.org/10.1016/S0362-546X(98)00245-4
[11]
Wiener, J. (1999) Generalized Solutions of Functional Differential Equations. World Scientific, Singapore.
[12]
Dimbour, W. and Manou-Abi, S. (2018) Asymptotically ω-Periodic Function in the Stepanov Sense and Its Applications for an Advanced Differential Equation with Piecewise Constant Argument in a Banach Space. Mediterranean Journal of Mathematics, 15, Article No. 25. https://doi.org/10.1007/s00009-018-1071-6
[13]
Dimbour, W. and Manou-Abi, S. (2018) S-Asymptotically ω-Periodic Solution for a Nonlinear Differential Equation with Piecewise Constant Argument via S-Asymptotically ω-Periodic Functions in the Stepanov Sense. Journal of Nonlinear Systems and Applications, 7, 14-20.
[14]
Dimbour, W. (2011) Almost Automorphic Solutions for Differential Equations with Piecewise Constant Argument in a Banach Space. Nonlinear Analysis, 74, 2351-2357. https://doi.org/10.1016/j.na.2010.11.038
[15]
Van-Minh, N. and Tat Dat, T. (2007) On the Almost Automorphy of Bounded Solutions of Differential Equations with Piecewise Constant Argument. Journal of Mathematical Analysis and Applications, 326, 165-178. https://doi.org/10.1016/j.jmaa.2006.02.079
[16]
Akhmet, M.U. (2007) On the Reduction Principle for Differential Equations with Piecewise Constant Argument of Generalized Type. Journal of Mathematical Analysis and Applications, 336, 646-663. https://doi.org/10.1016/j.jmaa.2007.03.010
[17]
Akhmet, M.U. (2008) Stability of Differential Equations with Piecewise Constant Arguments of Generalized Type. Nonlinear Analysis, 68, 794-803. https://doi.org/10.1016/j.na.2006.11.037
[18]
Chiu, K. and Pinto, M. (2010) Periodic Solutions of Differential Equations with a General Piecewise Constant Argument and Applications. The Electronic Journal of Qualitative Theory of Differential Equations, No. 46, 1-19. https://doi.org/10.14232/ejqtde.2010.1.46
[19]
Chávez, A., Castillo, S. and Pinto, M. (2014) Discontinuous Almost Automorphic Functions and Almost Automorphic Solutions for Differential Equations with Piecewise Constant Argument. Electronic Journal of Differential Equations, No. 56, 1-13. https://doi.org/10.14232/ejqtde.2014.1.75
[20]
Dieudonné, J. (1960) Foundations of Modern Analysis. Academic Press Inc., New York.
