A Note on Almost Anti-Periodic Functions in Banach Spaces

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DOI: 10.46793/KgJMat2002.287K


The main aim of this note is to introduce the notion of an almost anti-periodic function in Banach space. We prove some characterizations for this class of functions, investigating also its relationship with the classes of anti-periodic functions and almost periodic functions in Banach spaces.


Almost anti-periodic functions, almost periodic functions, anti-periodic functions, Bohr transform, Banach spaces.


[1]   W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics 96, Birkhäuser Verlag, Basel, 2001.

[2]   H. Bart and S. Goldberg, Characterizations of almost periodic strongly continuous groups and semigroups, Math. Ann. 236 (1978), 105–116.

[3]    A. S. Besicovitch, Almost Periodic Functions, Dover Publications Inc. New York, 1954.

[4]   S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. USA 48 (1962), 2039–2043.

[5]   Y. Q. Chen, Anti-periodic solutions for semilinear evolution equations, J. Math. Anal. Appl. 315 (2006), 337–348.

[6]   T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, 2013.

[7]   W. Dimbour and V. Valmorin, Asymptotically antiperiodic solutions for a nonlinear differential equation with piecewise constant argument in a Banach space, Appl. Math. 7 (2016), 1726–1733.

[8]   A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Chapman and Hall/CRC Pure and Applied Mathematics, New York, 1998.

[9]   G. M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Acad. Publ. Dordrecht, 2001.

[10]   G. M. N’Guérékata and V. Valmorin, Antiperiodic solutions of semilinear integrodifferential equations in Banach spaces, Appl. Math. Comput. 218 (2012), 1118–1124.

[11]   M. F. Hasler and G. M. N’Guérékata, Bloch-periodic functions and some applications, Nonlinear Stud. 21 (2014), 21–30.

[12]   H. R. Henríquez, On Stepanov-almost periodic semigroups and cosine functions of operators, J. Math. Anal. Appl. 146 (1990), 420–433.

[13]   G. Mophou, G. M. N’Guérékata and V. Valmorin, Asymptotic behavior of mild solutions of some fractional functional integro-differential equations, Afr. Diaspora J. Math. 16 (2013), 70–81.

[14]   M. Kostić, Almost Periodic Functions, Almost Automorphic Functions and Integro-Differential Equations, Book Manuscript, 2017.

[15]    M. Kostić, Existence of generalized almost periodic and asymptotic almost periodic solutions to abstract Volterra integro-differential equations, Electron. J. Differential Equations 2017(239) (2017), 1–30.

[16]   M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, London, 1982.

[17]   J. Liu and L. Zhang, Existence of anti-periodic (differentiable) mild solutions to semilinear differential equations with nondense domain, Springerplus 5(704) (2016), DOI 10.1186/s40064-016-2315-1

[18]    J. H. Liu, X. Q. Song and L. T. Zhang, Existence of anti-periodic mild solutions to semilinear nonautonomous evolution equations, J. Math. Anal. Appl. 425 (2015), 295–306.