On asymptotic periodic solutions of fractional differential equations and applications. (English) Zbl 1527.34103
Consider the fractional differential equation
\[
D^{\alpha}_C u(t)=Au(t)+ f(t),\quad u(0)=x,\quad 0<\alpha\leq1.\tag{1}
\]
Here \(D^{\alpha}_C u(t)\) is the derivative of a function in the Caputo’s sense; \(A\) is a linear operator in a Banach space \(X\) that may be unbounded and \(f\) satisfies the property \(\lim_{t \rightarrow \infty} (f(t + 1) -f(t)) = 0\) which the authors call “asymptotic 1-periodicity”. Using the spectral theory of functions, the authors obtain analogues of the Katznelson-Tzafriri and Massera theorems. Specifically, they outline spectral conditions of the operator A that result in asymptotic mild solutions of equation (1) to be asymptotic 1-periodic, or alternatively, allow the existence of an asymptotic mild solution that is asymptotic 1-periodic.
Reviewer: Erdoğan Şen (Tekirdağ)
MSC:
| 34G10 | Linear differential equations in abstract spaces |
| 34A08 | Fractional ordinary differential equations |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34L05 | General spectral theory of ordinary differential operators |
References:
| [1] | Arendt, W., Almost periodic solutions of first- and second-order Cauchy problems, J. Differential Equations, 363-383 (1997) · Zbl 0879.34046 · doi:10.1006/jdeq.1997.3266 |
| [2] | Arendt, Wolfgang, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, xii+539 pp. (2011), Birkh\"{a}user/Springer Basel AG, Basel · Zbl 1226.34002 · doi:10.1007/978-3-0348-0087-7 |
| [3] | Baeumer, Boris, Brownian subordinators and fractional Cauchy problems, Trans. Amer. Math. Soc., 3915-3930 (2009) · Zbl 1186.60079 · doi:10.1090/S0002-9947-09-04678-9 |
| [4] | Baskakov, A. G., Harmonic and spectral analysis of power bounded operators and bounded semigroups of operators on a Banach space, Math. Notes. Mat. Zametki, 174-190 (2015) · doi:10.4213/mzm10285 |
| [5] | Batty, Charles J. K., Local spectra and individual stability of uniformly bounded \(C_0\)-semigroups, Trans. Amer. Math. Soc., 2071-2085 (1998) · Zbl 0893.47026 · doi:10.1090/S0002-9947-98-01919-9 |
| [6] | Bajlekova, Emilia Grigorova, Fractional evolution equations in Banach spaces, iv+107 pp. (2001), Eindhoven University of Technology, Eindhoven · Zbl 0989.34002 |
| [7] | Cl\'{e}ment, Ph., Schauder estimates for equations with fractional derivatives, Trans. Amer. Math. Soc., 2239-2260 (2000) · Zbl 0947.35023 · doi:10.1090/S0002-9947-00-02507-1 |
| [8] | Cuesta, Eduardo, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., 277-285 (2007) · Zbl 1163.45306 |
| [9] | Eidelman, Samuil D., Cauchy problem for fractional diffusion equations, J. Differential Equations, 211-255 (2004) · Zbl 1068.35037 · doi:10.1016/j.jde.2003.12.002 |
| [10] | Engel, Klaus-Jochen, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, xxii+586 pp. (2000), Springer-Verlag, New York · Zbl 0952.47036 |
| [11] | Applications of fractional calculus in physics, viii+463 pp. (2000), World Scientific Publishing Co., Inc., River Edge, NJ · Zbl 0998.26002 · doi:10.1142/9789812817747 |
| [12] | Hino, Yoshiyuki, Almost periodic solutions of differential equations in Banach spaces, Stability and Control: Theory, Methods and Applications, viii+250 pp. (2002), Taylor & Francis, London · Zbl 1026.34001 |
| [13] | Katznelson, Y., On power bounded operators, J. Funct. Anal., 313-328 (1986) · Zbl 0611.47005 · doi:10.1016/0022-1236(86)90101-1 |
| [14] | Keyantuo, Valentin, Existence, regularity and representation of solutions of time fractional wave equations, Electron. J. Differential Equations, Paper No. 222, 42 pp. (2017) · Zbl 1420.47025 |
| [15] | Kilbas, Anatoly A., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, xvi+523 pp. (2006), Elsevier Science B.V., Amsterdam · Zbl 1092.45003 |
| [16] | Lizama, Carlos, Mild solutions for abstract fractional differential equations, Appl. Anal., 1731-1754 (2013) · Zbl 1276.34004 · doi:10.1080/00036811.2012.698271 |
| [17] | Lizama, Carlos, Bounded mild solutions for semilinear integro differential equations in Banach spaces, Integral Equations Operator Theory, 207-227 (2010) · Zbl 1209.45007 · doi:10.1007/s00020-010-1799-2 |
| [18] | Luong, Vu Trong, A simple spectral theory of polynomially bounded solutions and applications to differential equations, Semigroup Forum, 456-476 (2021) · Zbl 1479.47090 · doi:10.1007/s00233-021-10165-2 |
| [19] | Luong, Vu Trong, A Massera theorem for asymptotic periodic solutions of periodic evolution equations, J. Differential Equations, 371-394 (2022) · Zbl 1505.34093 · doi:10.1016/j.jde.2022.05.010 |
| [20] | Massera, Jos\'{e} L., The existence of periodic solutions of systems of differential equations, Duke Math. J., 457-475 (1950) · Zbl 0038.25002 |
| [21] | Minh, Nguyen Van, Asymptotic behavior of individual orbits of discrete systems, Proc. Amer. Math. Soc., 3025-3035 (2009) · Zbl 1170.47004 · doi:10.1090/S0002-9939-09-09871-2 |
| [22] | Minh, Nguyen Van, A Katznelson-Tzafriri type theorem for difference equations and applications, Proc. Amer. Math. Soc., 1105-1114 (2022) · Zbl 1480.39002 · doi:10.1090/proc/15722 |
| [23] | Nguyen, Van Minh, Circular spectrum and bounded solutions of periodic evolution equations, J. Differential Equations, 3089-3108 (2009) · Zbl 1191.34077 · doi:10.1016/j.jde.2009.02.014 |
| [24] | Naito, Toshiki, New spectral criteria for almost periodic solutions of evolution equations, Studia Math., 97-111 (2001) · Zbl 0982.34074 · doi:10.4064/sm145-2-1 |
| [25] | van Neerven, Jan, The asymptotic behaviour of semigroups of linear operators, Operator Theory: Advances and Applications, xii+237 pp. (1996), Birkh\"{a}user Verlag, Basel · Zbl 0905.47001 · doi:10.1007/978-3-0348-9206-3 |
| [26] | Pr\"{u}ss, Jan, Evolutionary integral equations and applications, Monographs in Mathematics, xxvi+366 pp. (1993), Birkh\"{a}user Verlag, Basel · Zbl 0784.45006 · doi:10.1007/978-3-0348-8570-6 |
| [27] | Zhou, Yong, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 1063-1077 (2010) · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026 |
| [28] | Wang, Rong-Nian, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 202-235 (2012) · Zbl 1238.34015 · doi:10.1016/j.jde.2011.08.048 |
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