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Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative
Almost periodic type functions and densities
Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, 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.
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. |
[2] |
B. Basit, Some problems concerning different types of vector valued almost periodic functions, Dissertationes Math., 338 (1995), 26 pp. |
[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.
|
[4] |
M. V. Bebutov,
On dynamical systems in the space of continuous functions, Byull. Moskov. Gos. Univ. Mat., 2 (1940), 1-52.
|
[5] |
A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955. |
[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. |
[7] |
H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947. |
[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.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[15] |
K.-J. Engel and R. Nagel, One–Parameter Semigroups for Linear Evolution Equations, Springer–Verlag, New York, 2000. |
[16] |
A. M. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974. |
[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. |
[18] |
A. M. Fink, Almost Periodic Points in Topological Transformation Semi-groups, Ph.D thesis, Iowa State University, Digital Repository (1960), 44 pp. |
[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. |
[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. |
[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. |
[22] |
G. M. N'Guérékata, Topics in Almost Automorphy, Springer–Verlag, New York, 2005. |
[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. |
[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. |
[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. |
[26] |
H. R. Henríquez,
On Stepanov-almost periodic semigroups and cosine functions of operators, J. Math. Anal. Appl., 146 (1990), 420-433.
|
[27] |
E. Hille, Functional Analysis and Semi-Groups, American Mathematical Society, New York, 1948. |
[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. |
[29] |
M. Kostić, Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations, De Gruyter, Berlin, 2019.
doi: 10.1515/9783110641851. |
[30] |
M. Kostić, Chaos for Linear Operators and Abstract Differential Equations, Nova Science Publishers Inc., New York, 2020. |
[31] |
M. Kostić,
${\mathcal F}$-Hypercyclic operators on Fréchet spaces, Publ. Inst. Math. (Beograd) (N.S.), 106 (2019), 1-18.
doi: 10.2298/pim1920001k. |
[32] |
M. Kostić, Quasi-asymptotically almost periodic functions and applications, Bull. Braz. Math. Soc., New Series, (2020).
doi: 10.1007/s00574-020-00197-7. |
[33] |
B. M. Levitan, Počti-periodičeskie funkcii, (Russian) [Almost Periodic Functions] |
[34] |
B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, London, 1982.
![]() ![]() |
[35] |
P. Ribenboim,
Density results on families of Diophantine equations with finitely many solutions, Enseign. Math. (2), 39 (1993), 3-23.
|
[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. |
[37] |
W. A. Veech,
Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.
doi: 10.2307/2373071. |
[38] |
W. A. Veech,
On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.
doi: 10.2307/1970363. |
[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. |
[40] |
S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Research Notes in Math., vol.126, Pitman, Boston, MA, 1985. |
[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. |
[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. |
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. |
[2] |
B. Basit, Some problems concerning different types of vector valued almost periodic functions, Dissertationes Math., 338 (1995), 26 pp. |
[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.
|
[4] |
M. V. Bebutov,
On dynamical systems in the space of continuous functions, Byull. Moskov. Gos. Univ. Mat., 2 (1940), 1-52.
|
[5] |
A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955. |
[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. |
[7] |
H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947. |
[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.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[15] |
K.-J. Engel and R. Nagel, One–Parameter Semigroups for Linear Evolution Equations, Springer–Verlag, New York, 2000. |
[16] |
A. M. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974. |
[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. |
[18] |
A. M. Fink, Almost Periodic Points in Topological Transformation Semi-groups, Ph.D thesis, Iowa State University, Digital Repository (1960), 44 pp. |
[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. |
[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. |
[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. |
[22] |
G. M. N'Guérékata, Topics in Almost Automorphy, Springer–Verlag, New York, 2005. |
[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. |
[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. |
[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. |
[26] |
H. R. Henríquez,
On Stepanov-almost periodic semigroups and cosine functions of operators, J. Math. Anal. Appl., 146 (1990), 420-433.
|
[27] |
E. Hille, Functional Analysis and Semi-Groups, American Mathematical Society, New York, 1948. |
[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. |
[29] |
M. Kostić, Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations, De Gruyter, Berlin, 2019.
doi: 10.1515/9783110641851. |
[30] |
M. Kostić, Chaos for Linear Operators and Abstract Differential Equations, Nova Science Publishers Inc., New York, 2020. |
[31] |
M. Kostić,
${\mathcal F}$-Hypercyclic operators on Fréchet spaces, Publ. Inst. Math. (Beograd) (N.S.), 106 (2019), 1-18.
doi: 10.2298/pim1920001k. |
[32] |
M. Kostić, Quasi-asymptotically almost periodic functions and applications, Bull. Braz. Math. Soc., New Series, (2020).
doi: 10.1007/s00574-020-00197-7. |
[33] |
B. M. Levitan, Počti-periodičeskie funkcii, (Russian) [Almost Periodic Functions] |
[34] |
B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, London, 1982.
![]() ![]() |
[35] |
P. Ribenboim,
Density results on families of Diophantine equations with finitely many solutions, Enseign. Math. (2), 39 (1993), 3-23.
|
[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. |
[37] |
W. A. Veech,
Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.
doi: 10.2307/2373071. |
[38] |
W. A. Veech,
On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.
doi: 10.2307/1970363. |
[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. |
[40] |
S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Research Notes in Math., vol.126, Pitman, Boston, MA, 1985. |
[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. |
[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. |
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