Asymptotically periodic solutions of a partial differential equation with memory. (English) Zbl 1402.34077
In this work, there are established some conditions which ensure the existence of a mild solution to a specific type of semilinear differential equation in a Hilber space. As an application of the theoretical result, in the last part of this work, an example is included for the existence of \(S\)-asymptotically \(\omega\)-periodic mild solution for a particular system with integro-partial differential equations. To prove some results, the author uses Schaefer’s fixed point theorem and a new result about a Gronwall-type inequality.
Reviewer: Andrei Horvat-Marc (Baia Mare)
MSC:
| 34K30 | Functional-differential equations in abstract spaces |
| 34K13 | Periodic solutions to functional-differential equations |
| 45K05 | Integro-partial differential equations |
| 35R10 | Partial functional-differential equations |
Keywords:
partial differential equations with memory; \(S\)-asymptotically \(\omega\)-periodic in Stepanov sense; \(S\)-asymptotically \(\omega\)-periodic; decay properties; mild solutionReferences:
| [1] | Agarwal R. P., Cuevas C., Soto H.: Pseudo-almost periodic solutions of a class of semilinear fractional differential equations. J. Appl. Math. Comput. 37, 625-634 (2011) · Zbl 1368.34008 · doi:10.1007/s12190-010-0455-y |
| [2] | Agarwal R.P., Cuevas C., Soto H.: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Anal. Real World Appl. 11, 3532-3554 (2010) · Zbl 1248.34004 · doi:10.1016/j.nonrwa.2010.01.002 |
| [3] | Bochner S.: Beiträe zur theorie der fastperiodischen funktionen. Math. Ann. 96, 119-147 (1927) · JFM 52.0261.01 · doi:10.1007/BF01209156 |
| [4] | Baroun M., Boulite S., Diagana T., Maniar L.: Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations. J. Math. Anal. Appl. 349, 74-84 (2009) · Zbl 1153.35049 · doi:10.1016/j.jmaa.2008.08.034 |
| [5] | Caicedo C., Lizma C.: S-asymptotically \[{\omega}\] ω-periodic solutions for semilinear Volterra equations. Math. Methods Appl. Sci. 33, 1628-1636 (2010) · Zbl 1251.45007 |
| [6] | Corduneanu C.: Almost Periodic Functions. 2nd ed. Chelsea, New York (1989) · Zbl 0672.42008 |
| [7] | Diagana T.: Stepanov-like pseudo almost periodicity and its applications to some nonautonomous differential equations. Nonlinear Anal. 69, 4277-4285 (2008) · Zbl 1169.34330 · doi:10.1016/j.na.2007.10.051 |
| [8] | Ding H. S., Liang J., N’Guérékata G. M., Xiao T. J.: Pseudo almost periodicity of some nonautonomous evolution equations with delay. Nonlinear Anal. 67, 1412-1418 (2007) · Zbl 1122.34345 · doi:10.1016/j.na.2006.07.026 |
| [9] | Goldstein J. A., N’Guérékata G. M.: Almost automorphic solutions of semilinear evolution equations. Proc. Amer. Math. Soc. 133, 2401-2408 (2005) · Zbl 1073.34073 · doi:10.1090/S0002-9939-05-07790-7 |
| [10] | Goldstein J.A., N’Guérékata G. M.: Corrigendum on “Almost automorphic solutions of semilinear evolution equations”. Proc. Amer. Math. Soc. 140, 1111-1112 (2012) · Zbl 1239.34069 · doi:10.1090/S0002-9939-2011-11403-5 |
| [11] | B. He, J. Cao and B. Yang, Weighted Stepanov-like pseudo-almost automorphic mild solutions for semilinear fractional differential equations. Adv. Difference Equ. 2015 (2015), DOI:10.1186/s13662-015-0410-1. · Zbl 1351.34005 |
| [12] | Henríquez H. R.: Asymptotically periodic solutions of abstract differential equations. Nonlinear Anal. 80, 135-149 (2013) · Zbl 1266.34100 · doi:10.1016/j.na.2012.10.010 |
| [13] | Henríquez H. R., Pierri M., Taboas P.: Existence of S-asymptotically \[{\omega}\] ω-periodic solutions for abstract neutral functiional-differential equations. Bull. Aust. Math. Soc. 78, 365-382 (2008) · Zbl 1183.34122 · doi:10.1017/S0004972708000713 |
| [14] | Henríquez H. R., Pierri M., Taboas P.: On S-asymptotically \[{\omega}\] ω-periodic functions on Banach spaces and applications. J. Math. Anal. Appl. 343, 1119-1130 (2008) · Zbl 1146.43004 · doi:10.1016/j.jmaa.2008.02.023 |
| [15] | Nicola S. H. J., Pierri M.: A note on S-asymptotically periodic functions. Nonlinear Anal. Real World Appl. 10, 2937-2938 (2009) · Zbl 1163.42305 · doi:10.1016/j.nonrwa.2008.09.011 |
| [16] | Liang J., Zhang J., Xiao T. J.: Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J. Math. Anal. Appl. 340, 1493-1499 (2008) · Zbl 1134.43001 · doi:10.1016/j.jmaa.2007.09.065 |
| [17] | Lizma C., N’Guérékata G. M.: Bounded mild solutions for semilinear integro differential equations in Banach spaces. Integral Equations Operator Theory 68, 207-227 (2010) · Zbl 1209.45007 · doi:10.1007/s00020-010-1799-2 |
| [18] | Lizma C., Ponce R.: Almost automorphic solutions to abstract Volterra equations on the line. Nonlinear Aanl. 74, 3805-3814 (2011) · Zbl 1218.45012 · doi:10.1016/j.na.2011.03.029 |
| [19] | N’Guérékata G. M.: Topics in Almost Automorphy. Springer, New York (2005) · Zbl 1073.43004 |
| [20] | Prüss J.: Decay properties for the solutions of a partial differential equation with memory. Arch. Math. (Basel) 92, 158-173 (2009) · Zbl 1166.45007 · doi:10.1007/s00013-008-2936-x |
| [21] | Travis C.C., Webb G.F.: Existence, stability, and compactness in the \[{\alpha}\] α-norm for partial functional differential equations. Trans. Amer. Math. Soc. 240, 129-143 (1978) · Zbl 0414.34080 |
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