Banach Journal of Mathematical Analysis

Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations

Marko Kostić

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Abstract

The main purpose of this article is to investigate Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations and inclusions. The class of asymptotically Weyl-almost periodic functions that we introduce here seems not to have been considered elsewhere, even in the scalar-valued case. We analyze the Weyl-almost periodic and asymptotically Weyl-almost periodic properties of convolution products and various types of degenerate solution operator families subgenerated by multivalued linear operators.

Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 64-90.

Dates
Received: 16 February 2018
Accepted: 20 May 2018
First available in Project Euclid: 25 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1540454496

Digital Object Identifier
doi:10.1215/17358787-2018-0016

Mathematical Reviews number (MathSciNet)
MR3894064

Zentralblatt MATH identifier
07002032

Subjects
Primary: 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
Secondary: 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

Keywords
Weyl-almost periodic functions asymptotically Weyl-almost periodic functions abstract Volterra integro-differential equations

Citation

Kostić, Marko. Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations. Banach J. Math. Anal. 13 (2019), no. 1, 64--90. doi:10.1215/17358787-2018-0016. https://projecteuclid.org/euclid.bjma/1540454496


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References

  • [1] S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron. J. Differential Equations 2011, no. 9.
  • [2] S. Abbas, A note on Weyl pseudo almost automorphic functions and their properties, Math. Sci. (Springer) 6 (2012), no. 29.
  • [3] R. P. Agarwal, B. de Andrade, and C. Cuevas, On type of periodicity and ergodicity to a class of fractional order differential equations, Adv. Differential Equations 2010, no. 179750.
  • [4] L. Amerio and G. Prouse, Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold, New York, 1971.
  • [5] J. Andres, A. M. Bersani, and R. F. Grande, Hierarchy of almost-periodic function spaces, Rend. Mat. Appl. (7) 26 (2006), no. 2, 121–188.
  • [6] J. Andres, A. M. Bersani, and K. Leśniak, On some almost-periodicity problems in various metrics, Acta Appl. Math. 65 (2001), no. 1–3, 35–57.
  • [7] W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monagr. Math. 96, Birkhäuser, Basel, 2001.
  • [8] E. G. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. dissertation, Eindhoven University of Technology, Eindhoven, Netherlands, 2001.
  • [9] A. S. Besicovitch, Almost Periodic Functions, Dover, New York, 1955.
  • [10] D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi, New York, 2009.
  • [11] F. Chérif, A various types of almost periodic functions on Banach spaces, I, Int. Math. Forum 6 (2011), no. 17–20, 921–952.
  • [12] F. Chérif, A various types of almost periodic functions on Banach spaces, II, Int. Math. Forum 6 (2011), no. 17–20, 953–985.
  • [13] R. Cross, Multivalued Linear Operators, Pure Appl. Math. 213, Marcel Dekker, New York, 1998.
  • [14] B. de Andrade and C. Lizama, Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl. 382 (2011), no. 2, 761–771.
  • [15] T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Cham, 2013.
  • [16] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Math. 2004, Springer, Berlin, 2010.
  • [17] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Pure Appl. Math. 215, Marcel Dekker, New York, 1999.
  • [18] H. R. Henríquez, On Stepanov-almost periodic semigroups and cosine functions of operators, J. Math. Anal. Appl. 146 (1990), no. 2, 420–433.
  • [19] Y. Hino, T. Naito, N. V. Minh, and J. S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces, Stab. Control: Theory, Methods Appl. 15, Taylor and Francis, London, 2002.
  • [20] M. Kostić, Generalized Semigroups and Cosine Functions, Posebna Izdanja 23, Mat. Inst. SANU, Belgrade, 2011.
  • [21] M. Kostić, Abstract Volterra Integro-Differential Equations, CRC Press, Boca Raton, Fla., 2015.
  • [22] M. Kostić, Almost periodicity of abstract Volterra integro-differential equations, Adv. Oper. Theory 2 (2017), no. 3, 353–382.
  • [23] M. Kostić, Existence of generalized almost periodic and asymptotic almost periodic solutions to abstract Volterra integro-differential equations, Electron. J. Differential Equations 2017, no. 239, 1–30.
  • [24] M. Kostić, On Besicovitch-Doss almost periodic solutions of abstract Volterra integro-differential equations, Novi Sad J. Math. 47 (2017), no. 2, 187–200.
  • [25] M. Kostić, On generalized $C^{(n)}$-almost periodic solutions of abstract Volterra integro-differential equations, Novi Sad J. Math. 48 (2018), no. 1, 73–91.
  • [26] M. Kostić, Perturbation results for abstract degenerate Volterra integro-differential equations, J. Fract. Calc. Appl. 9 (2018), no. 1, 137–152.
  • [27] M. Kostić, Almost Periodic Functions, Almost Automorphic Functions and Integro-differential Equations, in preparation.
  • [28] A. S. Kovanko, Sur la compacité des systèmes de fonctions presque périodiques généralisées de H. Weyl, C. R. (Doklady) Acad. Sci. URSS (N.S.) 43 (1944), 275–276.
  • [29] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982.
  • [30] J. Mu, Y. Zhoa, and L. Peng, Periodic solutions and $S$-asymptotically periodic solutions to fractional evolution equations, Discrete Dyn. Nat. Soc. 2017, no. 1364532.
  • [31] G. M. N’Guérékata, Almost Automorphic and Almost Periodic Functions, Kluwer Academic/Plenum, New York, 2001.
  • [32] F. Periago and B. Straub, A functional calculus for almost sectorial operators and applications to abstract evolution equations, J. Evol. Equ 2 (2002), no. 1, 41–68.
  • [33] R. Ponce and M. Warma, Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel, preprint, arXiv:1610.08750v1 [math.AP].
  • [34] J. Prüss, Evolutionary Integral Equations and Applications, Monogr. Math. 87, Birkhäuser, Basel, 1993.
  • [35] W. M. Ruess and W. H. Summers, Asymptotic almost periodicity and motions of semigroups of operators, Linear Algebra Appl. 84 (1986), 335–351.
  • [36] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Derivatives and Integrals: Theory and Applications, Gordon and Breach, Yverdon, 1993.
  • [37] W. von Wahl, Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen, Nachr. Akad. Wiss. Göttingen II: Math. Phys. Kl. 1983 11 (1972), 231–258.
  • [38] R.-N. Wang, D.-N. Chen, and T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations 252 (2012), no. 1, 202–235.
  • [39] S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Res. Notes Math. 126, Pitman, Boston, 1985.