American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A nonlocal Weickert type PDE applied to multi-frame super-resolution
  • EECT Home
  • This Issue
  • Next Article
    Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"
doi: 10.3934/eect.2021008

Almost periodic type functions and densities

Marko Kostić ,

Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia

* Corresponding author: Marko Kostić

Received  July 2020 Published  January 2021

Fund Project: The author is supported by grant no. 451-03-68/2020/14/200156, Ministry of Science and Technological Development, Republic of Serbia

In this paper, we introduce and analyze the notions of $ \odot_{g} $-almost periodicity and Stepanov $ \odot_{g} $-almost periodicity for functions with values in complex Banach spaces. In order to do that, we use the recently introduced notions of lower and upper (Banach) $ g $-densities. We also analyze uniformly recurrent functions, generalized almost automorphic functions and apply our results in the qualitative analysis of solutions of inhomogeneous abstract integro-differential inclusions. We present plenty of illustrative examples, results of independent interest, questions and unsolved problems.

Keywords: $ \odot_{g} $-Almost periodic functions, uniformly recurrent functions, almost automorphic functions, lower and upper (Banach) densities, abstract integro-differential equations.
Mathematics Subject Classification: Primary: 42A75, 35B15; Secondary: 47D06.
Citation: Marko Kostić. Almost periodic type functions and densities. Evolution Equations & Control Theory, doi: 10.3934/eect.2021008
References:
[1]

S. Abbas, A note on Weyl pseudo almost automorphic functions and their properties, Math. Sci. (Springer), 6 (2012), 5 pp. doi: 10.1186/2251-7456-6-29.  Google Scholar

[2]

B. Basit, Some problems concerning different types of vector valued almost periodic functions, Dissertationes Math., 338 (1995), 26 pp.  Google Scholar

[3]

B. Basit and H. Güenzler, On spectral criteria for solutions of evolution equations and comments on reduced spectra, Far East J. Math. Sci. (FJMS), 65 (2012), 273-288.   Google Scholar

[4]

M. V. Bebutov, On dynamical systems in the space of continuous functions, Byull. Moskov. Gos. Univ. Mat., 2 (1940), 1-52.   Google Scholar

[5]

A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.  Google Scholar

[6]

H. Bohr, Zur theorie der fastperiodischen Funktionen Ⅰ; Ⅱ; Ⅲ, Acta Math., 45 (1924), 29–127; H6 (1925), 101–214; H5 (1926), 237–281. doi: 10.1007/BF02395468.  Google Scholar

[7]

H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947.  Google Scholar

[8]

L. I. Danilov, The uniform approximation of recurrent functions and almost recurrent functions, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 4 (2013), 36-54.   Google Scholar

[9]

J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar

[10]

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Cham, 2013. doi: 10.1007/978-3-319-00849-3.  Google Scholar

[11]

H.-S. Ding, J. Liang and T.-J. Xiao, Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations, Appl. Anal., 88 (2009), 1079-1091.  doi: 10.1080/00036810903156164.  Google Scholar

[12]

H.-S. Ding, W. Long and G. M. N'Guérékata, Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients, J. Comput. Anal. Appl., 13 (2011), 231-242.   Google Scholar

[13]

H.-S. Ding and S.-M. Wan, Asymptotically almost automorphic solutions of differential equations with piecewise constant argument, Open Math., 15 (2017), 595-610.  doi: 10.1515/math-2017-0051.  Google Scholar

[14]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Text in Mathematics, vol. 272, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.  Google Scholar

[15]

K.-J. Engel and R. Nagel, One–Parameter Semigroups for Linear Evolution Equations, Springer–Verlag, New York, 2000.  Google Scholar

[16]

A. M. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

[17]

A. M. Fink, Extensions of almost automorphic sequences, J. Math. Anal. Appl., 27 (1969), 519-523.  doi: 10.1016/0022-247X(69)90132-2.  Google Scholar

[18]

A. M. Fink, Almost Periodic Points in Topological Transformation Semi-groups, Ph.D thesis, Iowa State University, Digital Repository (1960), 44 pp.  Google Scholar

[19]

A. Geroldinger and I. Z. Ruzsa, Combinatorial Number Theory and Additive Group Theory, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8962-8.  Google Scholar

[20]

G. Grekos, V. Toma and J. Tomanová, A note on uniform or Banach density, Ann. Math. Blaise Pascal, 17 (2010), 153-163.  doi: 10.5802/ambp.280.  Google Scholar

[21]

G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4757-4482-8.  Google Scholar

[22]

G. M. N'Guérékata, Topics in Almost Automorphy, Springer–Verlag, New York, 2005.  Google Scholar

[23]

G. M. N'Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., 68 (2008), 2658-2667.  doi: 10.1016/j.na.2007.02.012.  Google Scholar

[24]

A. Haraux, Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes, J. Differential Equations, 49 (1983), 473-483.  doi: 10.1016/0022-0396(83)90008-6.  Google Scholar

[25]

A. Haraux and P. Souplet, An example of uniformly recurrent function which is not almost periodic, J. Fourier Anal. Appl., 10 (2004), 217-220.  doi: 10.1007/s00041-004-8012-4.  Google Scholar

[26]

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

[27]

E. Hille, Functional Analysis and Semi-Groups, American Mathematical Society, New York, 1948.  Google Scholar

[28]

D. Ji and Y. Lu, Stepanov-like pseudo almost automorphic solution to a parabolic evolution equation, Adv. Difference Equ., 341 (2015), 17 pp. doi: 10.1186/s13662-015-0667-4.  Google Scholar

[29]

M. Kostić, Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations, De Gruyter, Berlin, 2019. doi: 10.1515/9783110641851.  Google Scholar

[30]

M. Kostić, Chaos for Linear Operators and Abstract Differential Equations, Nova Science Publishers Inc., New York, 2020. Google Scholar

[31]

M. Kostić, ${\mathcal F}$-Hypercyclic operators on Fréchet spaces, Publ. Inst. Math. (Beograd) (N.S.), 106 (2019), 1-18.  doi: 10.2298/pim1920001k.  Google Scholar

[32]

M. Kostić, Quasi-asymptotically almost periodic functions and applications, Bull. Braz. Math. Soc., New Series, (2020). doi: 10.1007/s00574-020-00197-7.  Google Scholar

[33]

B. M. Levitan, Počti-periodičeskie funkcii, (Russian) [Almost Periodic Functions]  Google Scholar

[34] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, London, 1982.   Google Scholar
[35]

P. Ribenboim, Density results on families of Diophantine equations with finitely many solutions, Enseign. Math. (2), 39 (1993), 3-23.   Google Scholar

[36]

A. M. Samoilenko and S. I. Trofimchuk, Unbounded functions with almost periodic differences, Ukrainian Math. J., 43 (1991), 1306-1309.  doi: 10.1007/BF01061818.  Google Scholar

[37]

W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.  doi: 10.2307/2373071.  Google Scholar

[38]

W. A. Veech, On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.  doi: 10.2307/1970363.  Google Scholar

[39]

R. Xie and C. Zhang, Space of $\omega$-periodic limit functions and its applications to an abstract Cauchy problem, J. Function Spaces, vol. 2015, Art. ID 953540, 10 pp. doi: 10.1155/2015/953540.  Google Scholar

[40]

S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Research Notes in Math., vol.126, Pitman, Boston, MA, 1985.  Google Scholar

[41]

C. Zhang, Ergodicity and asymptotically almost periodic solutions of some differential equations, Int. J. Math. Math. Sci., 25 (2001), 787-800.  doi: 10.1155/S016117120100429X.  Google Scholar

[42]

H. Y. Zhao and M. Fečkan, Pseudo almost periodic solutions of an iterative equation with variable coefficients, Miskolc Math. Notes, 18 (2017), 515-524.  doi: 10.18514/MMN.2017.2047.  Google Scholar

show all references

References:
[1]

S. Abbas, A note on Weyl pseudo almost automorphic functions and their properties, Math. Sci. (Springer), 6 (2012), 5 pp. doi: 10.1186/2251-7456-6-29.  Google Scholar

[2]

B. Basit, Some problems concerning different types of vector valued almost periodic functions, Dissertationes Math., 338 (1995), 26 pp.  Google Scholar

[3]

B. Basit and H. Güenzler, On spectral criteria for solutions of evolution equations and comments on reduced spectra, Far East J. Math. Sci. (FJMS), 65 (2012), 273-288.   Google Scholar

[4]

M. V. Bebutov, On dynamical systems in the space of continuous functions, Byull. Moskov. Gos. Univ. Mat., 2 (1940), 1-52.   Google Scholar

[5]

A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.  Google Scholar

[6]

H. Bohr, Zur theorie der fastperiodischen Funktionen Ⅰ; Ⅱ; Ⅲ, Acta Math., 45 (1924), 29–127; H6 (1925), 101–214; H5 (1926), 237–281. doi: 10.1007/BF02395468.  Google Scholar

[7]

H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947.  Google Scholar

[8]

L. I. Danilov, The uniform approximation of recurrent functions and almost recurrent functions, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 4 (2013), 36-54.   Google Scholar

[9]

J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar

[10]

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Cham, 2013. doi: 10.1007/978-3-319-00849-3.  Google Scholar

[11]

H.-S. Ding, J. Liang and T.-J. Xiao, Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations, Appl. Anal., 88 (2009), 1079-1091.  doi: 10.1080/00036810903156164.  Google Scholar

[12]

H.-S. Ding, W. Long and G. M. N'Guérékata, Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients, J. Comput. Anal. Appl., 13 (2011), 231-242.   Google Scholar

[13]

H.-S. Ding and S.-M. Wan, Asymptotically almost automorphic solutions of differential equations with piecewise constant argument, Open Math., 15 (2017), 595-610.  doi: 10.1515/math-2017-0051.  Google Scholar

[14]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Text in Mathematics, vol. 272, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.  Google Scholar

[15]

K.-J. Engel and R. Nagel, One–Parameter Semigroups for Linear Evolution Equations, Springer–Verlag, New York, 2000.  Google Scholar

[16]

A. M. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

[17]

A. M. Fink, Extensions of almost automorphic sequences, J. Math. Anal. Appl., 27 (1969), 519-523.  doi: 10.1016/0022-247X(69)90132-2.  Google Scholar

[18]

A. M. Fink, Almost Periodic Points in Topological Transformation Semi-groups, Ph.D thesis, Iowa State University, Digital Repository (1960), 44 pp.  Google Scholar

[19]

A. Geroldinger and I. Z. Ruzsa, Combinatorial Number Theory and Additive Group Theory, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8962-8.  Google Scholar

[20]

G. Grekos, V. Toma and J. Tomanová, A note on uniform or Banach density, Ann. Math. Blaise Pascal, 17 (2010), 153-163.  doi: 10.5802/ambp.280.  Google Scholar

[21]

G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4757-4482-8.  Google Scholar

[22]

G. M. N'Guérékata, Topics in Almost Automorphy, Springer–Verlag, New York, 2005.  Google Scholar

[23]

G. M. N'Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., 68 (2008), 2658-2667.  doi: 10.1016/j.na.2007.02.012.  Google Scholar

[24]

A. Haraux, Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes, J. Differential Equations, 49 (1983), 473-483.  doi: 10.1016/0022-0396(83)90008-6.  Google Scholar

[25]

A. Haraux and P. Souplet, An example of uniformly recurrent function which is not almost periodic, J. Fourier Anal. Appl., 10 (2004), 217-220.  doi: 10.1007/s00041-004-8012-4.  Google Scholar

[26]

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

[27]

E. Hille, Functional Analysis and Semi-Groups, American Mathematical Society, New York, 1948.  Google Scholar

[28]

D. Ji and Y. Lu, Stepanov-like pseudo almost automorphic solution to a parabolic evolution equation, Adv. Difference Equ., 341 (2015), 17 pp. doi: 10.1186/s13662-015-0667-4.  Google Scholar

[29]

M. Kostić, Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations, De Gruyter, Berlin, 2019. doi: 10.1515/9783110641851.  Google Scholar

[30]

M. Kostić, Chaos for Linear Operators and Abstract Differential Equations, Nova Science Publishers Inc., New York, 2020. Google Scholar

[31]

M. Kostić, ${\mathcal F}$-Hypercyclic operators on Fréchet spaces, Publ. Inst. Math. (Beograd) (N.S.), 106 (2019), 1-18.  doi: 10.2298/pim1920001k.  Google Scholar

[32]

M. Kostić, Quasi-asymptotically almost periodic functions and applications, Bull. Braz. Math. Soc., New Series, (2020). doi: 10.1007/s00574-020-00197-7.  Google Scholar

[33]

B. M. Levitan, Počti-periodičeskie funkcii, (Russian) [Almost Periodic Functions]  Google Scholar

[34] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, London, 1982.   Google Scholar
[35]

P. Ribenboim, Density results on families of Diophantine equations with finitely many solutions, Enseign. Math. (2), 39 (1993), 3-23.   Google Scholar

[36]

A. M. Samoilenko and S. I. Trofimchuk, Unbounded functions with almost periodic differences, Ukrainian Math. J., 43 (1991), 1306-1309.  doi: 10.1007/BF01061818.  Google Scholar

[37]

W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.  doi: 10.2307/2373071.  Google Scholar

[38]

W. A. Veech, On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.  doi: 10.2307/1970363.  Google Scholar

[39]

R. Xie and C. Zhang, Space of $\omega$-periodic limit functions and its applications to an abstract Cauchy problem, J. Function Spaces, vol. 2015, Art. ID 953540, 10 pp. doi: 10.1155/2015/953540.  Google Scholar

[40]

S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Research Notes in Math., vol.126, Pitman, Boston, MA, 1985.  Google Scholar

[41]

C. Zhang, Ergodicity and asymptotically almost periodic solutions of some differential equations, Int. J. Math. Math. Sci., 25 (2001), 787-800.  doi: 10.1155/S016117120100429X.  Google Scholar

[42]

H. Y. Zhao and M. Fečkan, Pseudo almost periodic solutions of an iterative equation with variable coefficients, Miskolc Math. Notes, 18 (2017), 515-524.  doi: 10.18514/MMN.2017.2047.  Google Scholar

[1]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136
[2]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
[3]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[4]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[5]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[6]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[7]

Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2479-2497. doi: 10.3934/dcdsb.2020191
[8]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192
[9]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[10]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[11]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[12]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065
[13]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181
[14]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153
[15]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[16]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[17]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[18]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
[19]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409
[20]

Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (18)
  • HTML views (44)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences