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Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation
School of Mathematics and Statistics, Xidian University, Xi'an 710071, Shaanxi, China |
This paper is mainly concerned with the existence of pseudo S-asymptotically Bloch type periodic solutions to damped evolution equations in Banach spaces. Some existence results for classical Cauchy conditions and nonlocal Cauchy conditions are established through properties of pseudo S-asymptotically Bloch type periodic functions and regularized families. The obtained results show that for each pseudo S-asymptotically Bloch type periodic input forcing disturbance, the output mild solutions to reference equations remain pseudo S-asymptotically Bloch type periodic.
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Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl., 382 (2011), 761-771.
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J. Cao and Z. Huang,
Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions, Open Math., 16 (2018), 792-805.
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Y. K. Chang and Y. Wei,
Pseudo $S$-asymptotically Bloch type periodicity with applications to some evolution equations, Z. Anal. Anwend., 40 (2021), 33-50.
doi: 10.4171/ZAA/1671. |
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Y. K. Chang and Y. Wei,
$S$-asymptotically Bloch type periodic solutions to some semi-linear evolution equations in Banach spaces, Acta Math. Sci. Ser. B (Engl. Ed.), 41 (2021), 413-425.
doi: 10.1007/s10473-021-0206-1. |
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Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 1-16.
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C. Cuevas and J. C. de Souza,
Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal., 72 (2010), 1683-1689.
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Solutions of second order abstract retarded functional differential equations on the line, J. Nonlinear Convex Anal., 12 (2011), 225-240.
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G. C. Gorain,
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H. R. Henríquez, M. Pierri and P. Táboas,
On $S$-asymptotically $\omega$-periodic functions on Banach spaces and applications, J. Math. Anal. Appl., 343 (2008), 1119-1130.
doi: 10.1016/j.jmaa.2008.02.023. |
| [21] |
C. Lizama and S. Rueda,
Nonlocal integrated solutions for a class of abstract evolution equations, Acta Appl. Math., 164 (2019), 165-183.
doi: 10.1007/s10440-018-00231-3. |
| [22] |
E. R. Oueama-Guengai and G. M. N'Guérékata,
On $S$-asymptotically $\omega$-periodic and Bloch periodic mild solutions to some fractional differential equations in abstract spaces, Math. Meth. Appl. Sci., 41 (2018), 9116-9122.
doi: 10.1002/mma.5062. |
| [23] |
M. Pierri,
On $S$-asymptotically $\omega$-periodic functions and applications, Nonliner Anal., 75 (2012), 651-661.
doi: 10.1016/j.na.2011.08.059. |
| [24] |
M. Pierri and V. Rolnik,
On pseudo $S$-asymptotically periodic functions, Bull. Aust. Math. Soc., 87 (2013), 238-254.
doi: 10.1017/S0004972712000950. |
| [25] |
S. Y. Ren, Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves, Springer-Verlag, New York, 2006. |
| [26] |
Z. Xia, D. Wang, C. F. Wen and J. C. Yao,
Pseudo asymptotically periodic mild solutions of semilinear functional integro-differential equations in Banach spaces, Math. Meth. Appl. Sci., 40 (2017), 7333-7355.
doi: 10.1002/mma.4533. |
| [27] |
M. Yang and Q. R. Wang,
Pseudo asymptotically periodic solutions for fractional integro-differential neutral equations, Sci. China Math., 62 (2019), 1705-1718.
doi: 10.1007/s11425-017-9222-2. |
show all references
References:
| [1] |
B. de Andrade and C. Lizama,
Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl., 382 (2011), 761-771.
doi: 10.1016/j.jmaa.2011.04.078. |
| [2] |
B. de Andrade, C. Cuevas, C. Silva and H. Soto,
Asymptotic periodicity for flexible structural systems and applications, Acta Appl. Math., 143 (2016), 105-164.
doi: 10.1007/s10440-015-0032-3. |
| [3] |
M. Benchohra and M. S. Souid,
$L^1$-Solutions for implicit fractional order differential equations with nonlocal conditions, Filomat, 30 (2016), 1485-1492.
doi: 10.2298/FIL1606485B. |
| [4] |
I. Benedetti, V. Obukhovskii and V. Taddei,
Evolution fractional differential problems with impulses and nonlocal conditions, Discrete Contin. Dyn. Syst.-S, 13 (2020), 1899-1919.
doi: 10.3934/dcdss.2020149. |
| [5] |
D. Brindle and G. M. N'Guérékata, $S$-asymptotically $\omega$-periodic mild solutions to fractional differential equations, Electron. J. Diff. Equ., 2020 (2020), 12pages. |
| [6] |
S. K. Bose and G. C. Gorain,
Stability of the boundary stablisized damped wave equation $y''+\lambda y''' = c^2(\Delta y+\mu\Delta y')$ in a bounded domain in $\mathbb{R}^n$, Indian J. Math., 40 (1998), 1-15.
|
| [7] |
S. K. Bose and G. C. Gorain,
Exact controllability and boundary stablization of torsional virations of an internally damped flexible space structure, J. Optim. Theory Appl., 99 (1998), 423-442.
doi: 10.1023/A:1021778428222. |
| [8] |
L. Byszewski and V. Lakshmikantham,
Theorem about the existence and uniqueness of a solutions of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal., 40 (1991), 11-19.
doi: 10.1080/00036819008839989. |
| [9] |
J. Cao and Z. Huang,
Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions, Open Math., 16 (2018), 792-805.
doi: 10.1515/math-2018-0068. |
| [10] |
Y. K. Chang and Y. Wei,
Pseudo $S$-asymptotically Bloch type periodicity with applications to some evolution equations, Z. Anal. Anwend., 40 (2021), 33-50.
doi: 10.4171/ZAA/1671. |
| [11] |
Y. K. Chang and Y. Wei,
$S$-asymptotically Bloch type periodic solutions to some semi-linear evolution equations in Banach spaces, Acta Math. Sci. Ser. B (Engl. Ed.), 41 (2021), 413-425.
doi: 10.1007/s10473-021-0206-1. |
| [12] |
P. Chen, X. Zhang and Y. Li,
Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 1-16.
doi: 10.1007/s10883-018-9423-x. |
| [13] |
C. Cuevas and J. C. de Souza,
Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal., 72 (2010), 1683-1689.
doi: 10.1016/j.na.2009.09.007. |
| [14] |
C. Cuevas and H. Henríquez,
Solutions of second order abstract retarded functional differential equations on the line, J. Nonlinear Convex Anal., 12 (2011), 225-240.
|
| [15] |
K. Deng,
Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630-637.
doi: 10.1006/jmaa.1993.1373. |
| [16] |
G. C. Gorain,
Boundary stablization of nonlinear vibrations of a flexible structure in a bounded domain in $\mathbb{R}^N$, J. Math. Anal. Appl., 319 (2006), 635-650.
doi: 10.1016/j.jmaa.2005.06.031. |
| [17] |
A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
doi: 10.1007/978-0-387-21593-8. |
| [18] |
M. F. Hasler and G. M. N'Guérékata,
Bloch-periodic functions and some applications, Nonlinear Stud., 21 (2014), 21-30.
|
| [19] |
H. Gao, K. Wang, F. Wei and X. Ding,
Massera-type theorem and asymptotically periodic Logisitc equations, Nonlinear Anal. RWA, 7 (2006), 1268-1283.
doi: 10.1016/j.nonrwa.2005.11.008. |
| [20] |
H. R. Henríquez, M. Pierri and P. Táboas,
On $S$-asymptotically $\omega$-periodic functions on Banach spaces and applications, J. Math. Anal. Appl., 343 (2008), 1119-1130.
doi: 10.1016/j.jmaa.2008.02.023. |
| [21] |
C. Lizama and S. Rueda,
Nonlocal integrated solutions for a class of abstract evolution equations, Acta Appl. Math., 164 (2019), 165-183.
doi: 10.1007/s10440-018-00231-3. |
| [22] |
E. R. Oueama-Guengai and G. M. N'Guérékata,
On $S$-asymptotically $\omega$-periodic and Bloch periodic mild solutions to some fractional differential equations in abstract spaces, Math. Meth. Appl. Sci., 41 (2018), 9116-9122.
doi: 10.1002/mma.5062. |
| [23] |
M. Pierri,
On $S$-asymptotically $\omega$-periodic functions and applications, Nonliner Anal., 75 (2012), 651-661.
doi: 10.1016/j.na.2011.08.059. |
| [24] |
M. Pierri and V. Rolnik,
On pseudo $S$-asymptotically periodic functions, Bull. Aust. Math. Soc., 87 (2013), 238-254.
doi: 10.1017/S0004972712000950. |
| [25] |
S. Y. Ren, Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves, Springer-Verlag, New York, 2006. |
| [26] |
Z. Xia, D. Wang, C. F. Wen and J. C. Yao,
Pseudo asymptotically periodic mild solutions of semilinear functional integro-differential equations in Banach spaces, Math. Meth. Appl. Sci., 40 (2017), 7333-7355.
doi: 10.1002/mma.4533. |
| [27] |
M. Yang and Q. R. Wang,
Pseudo asymptotically periodic solutions for fractional integro-differential neutral equations, Sci. China Math., 62 (2019), 1705-1718.
doi: 10.1007/s11425-017-9222-2. |
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