ASYMPTOTIC PERIODIC SOLUTIONS TO A CLASS OF NON-DENSELY DEFINED DIFFERENTIAL EQUATIONS WITH INFINITE DELAY
Main Article Content
Abstract
In this paper, we prove a Massera-type theorem for the asymptotic periodicity of bounded solutions to non-densely defined differential equations with infinite delay, using the circular spectral theory of functions.
Keywords
Non-densely, infinite delay, asymptotic periodicity.
Article Details
References
[1] A. Afoukal, K. Ezzinbi (2025), Bohr–Neugebauer Type Result for a Class of Nondensely Nonautonomous Linear Partial Functional Diferential Equations with Infnite Delay, Differential Equations and Dynamical Systems, 33, 227-259.
[2] Burton (1985), Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press; Orlando, Florida.
[3] C. Cuevas, M. Pinto (2001), Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain, Nonlinear Anal. , 45, 73-83.
[4] G. Da Prato, E. Sinestrari (1987), Differential operators with nondense domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4, 285-344.
[5] B. De Andrade, C. Cuevas (2010), S-asymptotically omega-periodic and asymptotically omega-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal. , 72, 3190-3208.
[6] B. De Andrade, J.H. Liu (2002), Periodic solutions for non densely defined delay evolution equations, Appl. Math. Stoch. Anal., 15, 105-114.
[7] F. Guhring, F. Rabiger (1999), Asymptotic properties of mild solutions for nonautonomous evolution equations with applications to retarded differential equations, Journal of Abstract and Applied Analysis, 4, 169-194.
[8] J. Hale, J. Kato (1978), Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21, 11-41.
[9] D. Henry (1981), Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. Springer-Verlag, Berlin-New York.
[10] Y. Hino, S. Murakami, T. Naito (1991), Functional Differential Equations with Infinite Delay, Springer.
[11] N.T. Huy, N.Q. Dang (2016), Existence, uniqueness and conditional stability of periodic solutions to evolution equations, J. Math. Anal. Appl., 433, 1190-1203.
[12] N. D. Huy, L. A. Minh, V. T. Luong, N. N. Vien (2025), Asymptotic Periodic Solutions of Differential Equations with Infinite Delay, Bull. Malays. Math. Sci. Soc. 48:186.
[13] K-J. Engel, R. Nagel (2000), One-parameter semigroups for linear evolution equations, Graduate Texts in Math., Springer-Verlag.
[14] J.H. Liu, G.M. N’Guerekata, N.V. Minh (2008), Topics on Stability and Periodicity in Abstract Differential Equations. Series on Concrete and Applicable Mathematics, World Scientific Publishing, Singapore.
[15] V. T. Luong, D. V. Loi, N. V. Minh, H. Matsunaga (2022), A Massera theorem for asymptotic periodic solutions of periodic evolution equations, J. Differential Equations, 329, 371-394.
[16] V. T. Luong, N. H. Tri, N. V. Minh (2021), Asymptotic behavior of solutions of periodic linear partial functional differential equations on the half line, J. Differential Equations, 296, 1-14.
[17] L. A. Minh, N. N. Vien (2023), Circular Spectrum And Asymptotic Periodic Solutions To A Class Of Non-Densely Defined Evolution Equations, Commun. Korean Math. Soc., 38, 1153-1162.
[18] N.V. Minh, H. Matsunaga, N.D. Huy, V. T. Luong (2022), A Katznelson-Tzafriri type theorem for difference equations and applications, Proc. Amer. Math. Soc., 150, 1105-1114.
[19] N. V. Minh, G. N'Guerekata, S. Siegmund (2009), Circular spectrum and bounded solutions of periodic evolution equations, J. Differential Equations, 246, 3089-3108.
[20] S. Murakami, T. Naito, N. V. Minh (2004), Massera’s theorem for almost periodic solutions of functional differential equations, J. Math. Soc. Japan, 56 (1):247-268.
[2] Burton (1985), Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press; Orlando, Florida.
[3] C. Cuevas, M. Pinto (2001), Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain, Nonlinear Anal. , 45, 73-83.
[4] G. Da Prato, E. Sinestrari (1987), Differential operators with nondense domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4, 285-344.
[5] B. De Andrade, C. Cuevas (2010), S-asymptotically omega-periodic and asymptotically omega-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal. , 72, 3190-3208.
[6] B. De Andrade, J.H. Liu (2002), Periodic solutions for non densely defined delay evolution equations, Appl. Math. Stoch. Anal., 15, 105-114.
[7] F. Guhring, F. Rabiger (1999), Asymptotic properties of mild solutions for nonautonomous evolution equations with applications to retarded differential equations, Journal of Abstract and Applied Analysis, 4, 169-194.
[8] J. Hale, J. Kato (1978), Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21, 11-41.
[9] D. Henry (1981), Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. Springer-Verlag, Berlin-New York.
[10] Y. Hino, S. Murakami, T. Naito (1991), Functional Differential Equations with Infinite Delay, Springer.
[11] N.T. Huy, N.Q. Dang (2016), Existence, uniqueness and conditional stability of periodic solutions to evolution equations, J. Math. Anal. Appl., 433, 1190-1203.
[12] N. D. Huy, L. A. Minh, V. T. Luong, N. N. Vien (2025), Asymptotic Periodic Solutions of Differential Equations with Infinite Delay, Bull. Malays. Math. Sci. Soc. 48:186.
[13] K-J. Engel, R. Nagel (2000), One-parameter semigroups for linear evolution equations, Graduate Texts in Math., Springer-Verlag.
[14] J.H. Liu, G.M. N’Guerekata, N.V. Minh (2008), Topics on Stability and Periodicity in Abstract Differential Equations. Series on Concrete and Applicable Mathematics, World Scientific Publishing, Singapore.
[15] V. T. Luong, D. V. Loi, N. V. Minh, H. Matsunaga (2022), A Massera theorem for asymptotic periodic solutions of periodic evolution equations, J. Differential Equations, 329, 371-394.
[16] V. T. Luong, N. H. Tri, N. V. Minh (2021), Asymptotic behavior of solutions of periodic linear partial functional differential equations on the half line, J. Differential Equations, 296, 1-14.
[17] L. A. Minh, N. N. Vien (2023), Circular Spectrum And Asymptotic Periodic Solutions To A Class Of Non-Densely Defined Evolution Equations, Commun. Korean Math. Soc., 38, 1153-1162.
[18] N.V. Minh, H. Matsunaga, N.D. Huy, V. T. Luong (2022), A Katznelson-Tzafriri type theorem for difference equations and applications, Proc. Amer. Math. Soc., 150, 1105-1114.
[19] N. V. Minh, G. N'Guerekata, S. Siegmund (2009), Circular spectrum and bounded solutions of periodic evolution equations, J. Differential Equations, 246, 3089-3108.
[20] S. Murakami, T. Naito, N. V. Minh (2004), Massera’s theorem for almost periodic solutions of functional differential equations, J. Math. Soc. Japan, 56 (1):247-268.