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doi: 10.3934/eect.2021017
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Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation

Siqi Chen , Yong-Kui Chang , and  Yanyan Wei

School of Mathematics and Statistics, Xidian University, Xi'an 710071, Shaanxi, China

* Corresponding author: Yong-Kui Chang

Received  September 2020 Revised  February 2021 Early access April 2021

Fund Project: This work is partially supported by NSF of Shaanxi Province (2020JM-183)

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.

Keywords: Pseudo S-asymptotically ω-anti-periodicity, pseudo S-asymptotically Bloch type periodicity, damped evolution equations, nonlocal Cauchy condition.
Mathematics Subject Classification: Primary: 34K13, 58D25; Secondary: 34C25.
Citation: Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations & Control Theory, doi: 10.3934/eect.2021017
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003. doi: 10.1007/978-0-387-21593-8.  Google Scholar

[18]

M. F. Hasler and G. M. N'Guérékata, Bloch-periodic functions and some applications, Nonlinear Stud., 21 (2014), 21-30.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

M. Pierri and V. Rolnik, On pseudo $S$-asymptotically periodic functions, Bull. Aust. Math. Soc., 87 (2013), 238-254.  doi: 10.1017/S0004972712000950.  Google Scholar

[25]

S. Y. Ren, Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves, Springer-Verlag, New York, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003. doi: 10.1007/978-0-387-21593-8.  Google Scholar

[18]

M. F. Hasler and G. M. N'Guérékata, Bloch-periodic functions and some applications, Nonlinear Stud., 21 (2014), 21-30.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

M. Pierri and V. Rolnik, On pseudo $S$-asymptotically periodic functions, Bull. Aust. Math. Soc., 87 (2013), 238-254.  doi: 10.1017/S0004972712000950.  Google Scholar

[25]

S. Y. Ren, Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves, Springer-Verlag, New York, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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