June  2017, 10(3): 463-473. doi: 10.3934/dcdss.2017022

Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

* Corresponding author: Yongkun Li

Received  November 2015 Revised  December 2016 Published  February 2017

Fund Project: The first author is supported by the National Natural Sciences Foundation of China under Grant 11361072.

We first propose a concept of almost periodic functions in the sense of Stepanov on time scales. Then, we consider a class of neutral functional dynamic equations with Stepanov-almost periodic terms on time scales in a Banach space. By means of the contraction mapping principle, we establish some criteria for the existence and uniqueness of almost periodic solutions for this class of dynamic equations on time scales. Finally, we give an example to illustrate the effectiveness of our results.

Citation: Yongkun Li, Pan Wang. Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 463-473. doi: 10.3934/dcdss.2017022
References:
[1]

B. Amir and L. Maniar, Composition of pseudo almost periodic functions and Cauchy problems with operator of non dense domain, Ann. Math. Blaise Pascal, 6 (1999), 1-11.  doi: 10.5802/ambp.110.

[2]

L. Amerio and G. Prouse, Almost-periodic Functions and Functional Differential Equations, Van Nostrand-Reinhold, New York, 1971.

[3]

J. Andres and D. Pennequin, On Stepanov almost-periodic oscillations and their discretizations, J. Differ. Equ. Appl., 18 (2012), 1665-1682.  doi: 10.1080/10236198.2011.587813.

[4]

J. Andres and D. Pennequin, On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations, Proc. Am. Math. Soc., 140 (2012), 2825-2834.  doi: 10.1090/S0002-9939-2012-11154-2.

[5]

S. Bochner, Beitrage zur Theorie der fastperiodischen Funktionen, Ⅰ, Mathematische Annalen, 96 (1927), 119-147.  doi: 10.1007/BF01209156.

[6]

S. Bochner, Abstrakte fastperiodische Funktionen, Acta Mathematica, 61 (1933), 149-184.  doi: 10.1007/BF02547790.

[7]

S. Bochner, Fastperiodische Lösungen der Wellengleichung, Acta Mathematica, 62 (1933), 227-237.  doi: 10.1007/BF02393605.

[8]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh$ \ddot{\mathrm{a}} $user, Boston, 2001. doi: 10.1007/978-1-4612-0201-1.

[9]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh$ \ddot{\mathrm{a}} $user, Boston, 2003. doi: 10.1007/978-1-4612-0201-1.

[10]

H. Bohr, Zur Theorie der fastperiodischen Funktionen, Ⅰ, Acta Mathematica, 45 (1925), 29-127.  doi: 10.1007/BF02395468.

[11]

H. Bohr, Zur Theorie der fastperiodischen Funktionen, Ⅱ, Acta Mathematica, 46 (1925), 101-214.  doi: 10.1007/BF02543859.

[12]

A. Cabada and D. R. Vivero, Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral; application to the calculus of Δ-antiderivatives, Math. Comput. Modelling, 43 (2006), 194-207.  doi: 10.1016/j.mcm.2005.09.028.

[13]

X. X. Chen and F. X. Lin, Almost periodic solutions of neutral functional differential equations, Nonlinear Anal. Real World Appl., 11 (2010), 1182-1189.  doi: 10.1016/j.nonrwa.2009.02.010.

[14]

J. Favard, Sur les équations différentielles á coefficients présquepériodiques, Acta Mathematica, 51 (1927), 31-81.  doi: 10.1007/BF02545660.

[15]

J. Favard, Leçons sur les fonctions presque-périodiques, Paris, Gauthier-Villars, 1933.

[16]

A. M. Fink, Almost Periodic Differential Equations, Springer, Berlin, 1974.

[17]

Z. Hu, Boundedness and Stepanov's almost periodicity of solutions, Electron. J. Differ. Equ., 2005 (2005), 1-7. 

[18]

Z. Hu and A. B. Mingarelli, Bochner's theorem and Stepanov almost periodic functions, Ann. Mat. Pura Appl., 187 (2008), 719-736.  doi: 10.1007/s10231-008-0066-5.

[19]

M. N. Islam and Y. N. Raffoul, Periodic solutions of neutral nonlinear system of differential equations with functional delay, J. Math. Anal. Appl., 331 (2007), 1175-1186.  doi: 10.1016/j.jmaa.2006.09.030.

[20]

B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math., 186 (2006), 391-415.  doi: 10.1016/j.cam.2005.02.011.

[21]

Y. K. Li and B. Li, Existence and exponential stability of positive almost periodic solution for Nicholson's blowflies models on time scales, SpringerPlus, 5 (2016), p1096.  doi: 10.1186/s40064-016-2700-9.

[22]

Y. K. Li and C. Wang, Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales, Abstr. Appl. Anal., 2011 (2011), Article ID 341520, 22 pp. doi: 10.1155/2011/341520.

[23]

Y. K. Li, L. L. Zhao and L. Yang, $ C^1 $-almost periodic solutions of BAM neural networks with time-varying delays on time scales, The Scientific World J., 2015 (2015), Article ID 727329, 15 pp. doi: 10.1155/2015/727329.

[24]

Md. Maqbul, Almost periodic solutions of neutral functional differential equations with Stepanov-almost periodic terms, Electron. J. Differ. Equ., 2011 (2011), 1-9. 

[25]

Md. Maqbul and D. Bahuguna, Almost periodic solutions for Stepanov-almost periodic differential equations, Differ. Equ. Dyn. Syst., 22 (2014), 251-264.  doi: 10.1007/s12591-013-0172-8.

[26]

V. V. Stepanov, Uber einige Verallgemeinerungen der fastperiodischen Funktionen, Mathematische Annalen, 95 (1926), 437-498. 

[27]

Y.-H. Su and Z. S. Feng, A non-autonomous Hamiltonian system on time scales, Nonlinear Anal., 75 (2012), 4126-4136.  doi: 10.1016/j.na.2012.03.003.

[28]

Y.-H. Su and Z. S. Feng, Homoclinic orbits and periodic solutions for a class of Hamiltonian systems on time scales, J. Math. Anal. Appl., 411 (2014), 37-62.  doi: 10.1016/j.jmaa.2013.08.068.

[29]

C. Wang and Y. K. Li, Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales, Ann. Pol. Math., 108 (2013), 225–240, Available from: http://eudml.org/doc/280802. doi: 10.4064/ap108-3-3.

[30]

L. Yang and Y. K. Li, Existence and global exponential stability of almost periodic solutions for a class of delay duffing equations on time scales, Abstr. Appl. Anal., 2014 (2014), Article ID 857161, 8 pp. doi: 10.1155/2014/857161.

[31]

H. ZhouZ. F. Zhou and W. Jiang, Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales, Neurocomputing, 157 (2015), 223-230.  doi: 10.1016/j.neucom.2015.01.013.

show all references

References:
[1]

B. Amir and L. Maniar, Composition of pseudo almost periodic functions and Cauchy problems with operator of non dense domain, Ann. Math. Blaise Pascal, 6 (1999), 1-11.  doi: 10.5802/ambp.110.

[2]

L. Amerio and G. Prouse, Almost-periodic Functions and Functional Differential Equations, Van Nostrand-Reinhold, New York, 1971.

[3]

J. Andres and D. Pennequin, On Stepanov almost-periodic oscillations and their discretizations, J. Differ. Equ. Appl., 18 (2012), 1665-1682.  doi: 10.1080/10236198.2011.587813.

[4]

J. Andres and D. Pennequin, On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations, Proc. Am. Math. Soc., 140 (2012), 2825-2834.  doi: 10.1090/S0002-9939-2012-11154-2.

[5]

S. Bochner, Beitrage zur Theorie der fastperiodischen Funktionen, Ⅰ, Mathematische Annalen, 96 (1927), 119-147.  doi: 10.1007/BF01209156.

[6]

S. Bochner, Abstrakte fastperiodische Funktionen, Acta Mathematica, 61 (1933), 149-184.  doi: 10.1007/BF02547790.

[7]

S. Bochner, Fastperiodische Lösungen der Wellengleichung, Acta Mathematica, 62 (1933), 227-237.  doi: 10.1007/BF02393605.

[8]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh$ \ddot{\mathrm{a}} $user, Boston, 2001. doi: 10.1007/978-1-4612-0201-1.

[9]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh$ \ddot{\mathrm{a}} $user, Boston, 2003. doi: 10.1007/978-1-4612-0201-1.

[10]

H. Bohr, Zur Theorie der fastperiodischen Funktionen, Ⅰ, Acta Mathematica, 45 (1925), 29-127.  doi: 10.1007/BF02395468.

[11]

H. Bohr, Zur Theorie der fastperiodischen Funktionen, Ⅱ, Acta Mathematica, 46 (1925), 101-214.  doi: 10.1007/BF02543859.

[12]

A. Cabada and D. R. Vivero, Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral; application to the calculus of Δ-antiderivatives, Math. Comput. Modelling, 43 (2006), 194-207.  doi: 10.1016/j.mcm.2005.09.028.

[13]

X. X. Chen and F. X. Lin, Almost periodic solutions of neutral functional differential equations, Nonlinear Anal. Real World Appl., 11 (2010), 1182-1189.  doi: 10.1016/j.nonrwa.2009.02.010.

[14]

J. Favard, Sur les équations différentielles á coefficients présquepériodiques, Acta Mathematica, 51 (1927), 31-81.  doi: 10.1007/BF02545660.

[15]

J. Favard, Leçons sur les fonctions presque-périodiques, Paris, Gauthier-Villars, 1933.

[16]

A. M. Fink, Almost Periodic Differential Equations, Springer, Berlin, 1974.

[17]

Z. Hu, Boundedness and Stepanov's almost periodicity of solutions, Electron. J. Differ. Equ., 2005 (2005), 1-7. 

[18]

Z. Hu and A. B. Mingarelli, Bochner's theorem and Stepanov almost periodic functions, Ann. Mat. Pura Appl., 187 (2008), 719-736.  doi: 10.1007/s10231-008-0066-5.

[19]

M. N. Islam and Y. N. Raffoul, Periodic solutions of neutral nonlinear system of differential equations with functional delay, J. Math. Anal. Appl., 331 (2007), 1175-1186.  doi: 10.1016/j.jmaa.2006.09.030.

[20]

B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math., 186 (2006), 391-415.  doi: 10.1016/j.cam.2005.02.011.

[21]

Y. K. Li and B. Li, Existence and exponential stability of positive almost periodic solution for Nicholson's blowflies models on time scales, SpringerPlus, 5 (2016), p1096.  doi: 10.1186/s40064-016-2700-9.

[22]

Y. K. Li and C. Wang, Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales, Abstr. Appl. Anal., 2011 (2011), Article ID 341520, 22 pp. doi: 10.1155/2011/341520.

[23]

Y. K. Li, L. L. Zhao and L. Yang, $ C^1 $-almost periodic solutions of BAM neural networks with time-varying delays on time scales, The Scientific World J., 2015 (2015), Article ID 727329, 15 pp. doi: 10.1155/2015/727329.

[24]

Md. Maqbul, Almost periodic solutions of neutral functional differential equations with Stepanov-almost periodic terms, Electron. J. Differ. Equ., 2011 (2011), 1-9. 

[25]

Md. Maqbul and D. Bahuguna, Almost periodic solutions for Stepanov-almost periodic differential equations, Differ. Equ. Dyn. Syst., 22 (2014), 251-264.  doi: 10.1007/s12591-013-0172-8.

[26]

V. V. Stepanov, Uber einige Verallgemeinerungen der fastperiodischen Funktionen, Mathematische Annalen, 95 (1926), 437-498. 

[27]

Y.-H. Su and Z. S. Feng, A non-autonomous Hamiltonian system on time scales, Nonlinear Anal., 75 (2012), 4126-4136.  doi: 10.1016/j.na.2012.03.003.

[28]

Y.-H. Su and Z. S. Feng, Homoclinic orbits and periodic solutions for a class of Hamiltonian systems on time scales, J. Math. Anal. Appl., 411 (2014), 37-62.  doi: 10.1016/j.jmaa.2013.08.068.

[29]

C. Wang and Y. K. Li, Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales, Ann. Pol. Math., 108 (2013), 225–240, Available from: http://eudml.org/doc/280802. doi: 10.4064/ap108-3-3.

[30]

L. Yang and Y. K. Li, Existence and global exponential stability of almost periodic solutions for a class of delay duffing equations on time scales, Abstr. Appl. Anal., 2014 (2014), Article ID 857161, 8 pp. doi: 10.1155/2014/857161.

[31]

H. ZhouZ. F. Zhou and W. Jiang, Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales, Neurocomputing, 157 (2015), 223-230.  doi: 10.1016/j.neucom.2015.01.013.

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