×

Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations. (English) Zbl 1408.43005

The paper investigates Weyl almost periodic and asymptotically Weyl almost periodic solutions of abstract Volterra integral equations and inclusions. The class of Weyl almost periodic functions is an extension of the class of the classical Stepanov almost periodic functions and it was defined by H. Weyl (1927).
The paper is organized as follows: Section 2 is devoted to the definitions of multivalued linear operators in Banach spaces, by repeating some things from the author’s previous work [“Perturbation results for abstract degenerate Volterra integro-differential equations”, J. Fract. Calc. Appl. 9, 137–152 (2018); http://fcag-egypt.com/Journals/JFCA/], as the \((a,k)\)-regularized C-resolvent families with subgenerator a multivalued linear operator \(\mathcal{A}\).
In Section 3, some facts from the theory of the Stepanov almost periodic and asymptotically almost periodic functions are presented. In Section 4, several new classes as the Weyl almost periodic and asymptotically Weyl almost periodic functions are introduced and their relationship is investigated.
In Section 5, the author gives information on the Weyl almost periodic properties of finite and infinite convolution products. For instance in Proposition 5.1 among others it is proved that if \((R(t))_{t\geq 0}\) is a strongly continuous family of linear operators from \(X\) to \(Y\) satisfying \(\|R\|_{L1}<+\infty\) and \(g:\mathbb{R}\to X\) is bounded and (equi)-Weyl almost periodic, then the function \(G(t):=\int_{-\infty}^tR(t-s)g(s)ds\), \(t\geq 0,\) is bounded and (equi)-Weyl almost periodic.
In Section 6, the author states (without proofs) some results concerning abstract Volterra integro-differential equations, while, in Section 7, some examples and applications to fractional inclusions with Weyl-Liapunov and Caputo type derivatives are presented.

MSC:

43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
47D06 One-parameter semigroups and linear evolution equations
45N05 Abstract integral equations, integral equations in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron. J. Differential Equations 2011, no. 9. · Zbl 1211.34096
[2] S. Abbas, A note on Weyl pseudo almost automorphic functions and their properties, Math. Sci. (Springer) 6 (2012), no. 29. · Zbl 1269.42003
[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. · Zbl 1194.34007
[4] L. Amerio and G. Prouse, Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold, New York, 1971. · Zbl 0215.15701
[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. · Zbl 1133.42002
[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. · Zbl 0997.34032
[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. · Zbl 0978.34001
[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. · Zbl 0065.07102
[10] D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi, New York, 2009. · Zbl 1222.34002
[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. · Zbl 0911.47002
[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. · Zbl 1221.35255
[15] T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Cham, 2013. · Zbl 1279.43010
[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. · Zbl 1215.34001
[17] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Pure Appl. Math. 215, Marcel Dekker, New York, 1999. · Zbl 0913.34001
[18] H. R. Henríquez, On Stepanov-almost periodic semigroups and cosine functions of operators, J. Math. Anal. Appl. 146 (1990), no. 2, 420–433. · Zbl 0719.47023
[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. · Zbl 1026.34001
[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. · Zbl 1370.35029
[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. · Zbl 1448.34134
[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. · Zbl 0499.43005
[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. · Zbl 1005.47015
[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. · Zbl 0616.47047
[36] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Derivatives and Integrals: Theory and Applications, Gordon and Breach, Yverdon, 1993. · Zbl 0818.26003
[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. · Zbl 0251.35052
[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. · Zbl 1238.34015
[39] S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Res. Notes Math. 126, Pitman, Boston, 1985. · Zbl 0648.42006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.