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Almost periodic solutions for second order dynamic equations on time scales. (English) Zbl 1417.34099
Summary: We firstly introduce the concept and the properties of \(C^m\) almost periodic functions on time scales, which generalizes the concept of almost periodic functions on time scales and the concept of \(C^{(n)}\)-almost periodic functions. Secondly, we consider the existence and uniqueness of almost periodic solutions for second order dynamic equations on time scales by Schauder’s fixed point theorem and contracting mapping principle. At last, we obtain alternative theorems for second order dynamic equations on time scales.
MSC:
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34N05 Dynamic equations on time scales or measure chains
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[1] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results in Mathematics, 18, 1-2, 18-56, (1990) · Zbl 0722.39001
[2] Amster, P.; Rogers, C.; Tisdell, C. C., Existence of solutions to boundary value problems for dynamic systems on time scales, Journal of Mathematical Analysis and Applications, 308, 2, 565-577, (2005) · Zbl 1085.34013
[3] Atici, F. M.; Cabada, A.; Chyan, C. J.; Kaymakçalan, B., Nagumo type existence results for second-order nonlinear dynamic BVPS, Nonlinear Analysis: Theory, Methods & Applications, 60, 2, 209-220, (2005) · Zbl 1072.34015
[4] Benchohra, M.; Ntouyas, S. K.; Ouahab, A., Existence results for second order boundary value problem of impulsive dynamic equations on time scales, Journal of Mathematical Analysis and Applications, 296, 1, 65-73, (2004) · Zbl 1060.34017
[5] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An introduction with Applications, (2001), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 0978.39001
[6] Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales, (2003), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 1025.34001
[7] Bohr, H., Almost Periodic Functions, (1951), New York, NY, USA: Chelsea, New York, NY, USA
[8] Bochner, S., Continuous mappings of almost automorphic and almost periodic functions, Proceedings of the National Academy of Sciences of the United States of America, 52, 907-910, (1964) · Zbl 0134.30102
[9] Fink, A. M., Almost Periodic Differential Equations. Almost Periodic Differential Equations, Lecture Notes in Mathematics, 377, (1974), Berlin, Germany: Springer, Berlin, Germany · Zbl 0325.34039
[10] N’Guérékata, G. M., Almost Automorphic and Almost Periodic Functions in Abstract Spaces, (2001), Amsterdam, The Netherlands: Kluwer Academic Publishers, Amsterdam, The Netherlands · Zbl 1001.43001
[11] Shen, W.; Yi, Y., Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows. Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, Memoirs of the American Mathematical Society, 647, (1998) · Zbl 0913.58051
[12] Guan, Y.; Wang, K., Translation properties of time scales and almost periodic functions, Mathematical and Computer Modelling, 57, 5-6, 1165-1174, (2013)
[13] Li, Y.; Wang, C., Almost periodic functions on time scales and applications, Discrete Dynamics in Nature and Society, 2011, (2011) · Zbl 1232.26055
[14] Li, Y.; Wang, C., Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales, Abstract and Applied Analysis, 2011, (2011) · Zbl 1223.34125
[15] Kulenović, M. R. S.; Ljubović, C., Necessary and sufficient conditions for the oscillation of a second order linear differential equation, Mathematische Nachrichten, 213, 1, 105-115, (2000) · Zbl 0958.34028
[16] Kulenović, M. R. S.; Hadžiomerspahić, S., Existence of nonoscillatory solution of second order linear neutral delay equation, Journal of Mathematical Analysis and Applications, 228, 2, 436-448, (1998) · Zbl 0919.34067
[17] Şahiner, Y., Oscillation of second-order delay differential equations on time scales, Nonlinear Analysis: Theory, Methods & Applications, 63, 5-7, e1073-e1080, (2005) · Zbl 1224.34294
[18] Saker, S. H., Oscillation criteria of second-order half-linear dynamic equations on time scales, Journal of Computational and Applied Mathematics, 177, 2, 375-387, (2005) · Zbl 1082.34032
[19] Torres, P. J., Weak singularities may help periodic solutions to exist, Journal of Differential Equations, 232, 1, 277-284, (2007) · Zbl 1116.34036
[20] Zhang, M., Periodic solutions of Liénard equations with singular forces of repulsive type, Journal of Mathematical Analysis and Applications, 203, 1, 254-269, (1996) · Zbl 0863.34039
[21] del Pino, M. A.; Manásevich, R. F., Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity,, Journal of Differential Equations, 103, 2, 260-277, (1993) · Zbl 0781.34032
[22] Kevrekidis, P. G.; Drossinos, Y., Nonlinearity from linearity: the Ermakov-Pinney equation revisited, Mathematics and Computers in Simulation, 74, 2-3, 196-202, (2007) · Zbl 1116.34007
[23] Lei, J.; Li, X.; Zhang, M.; Yan, P., Twist character of the least amplitude periodic solution of the forced pendulum, SIAM Journal on Mathematical Analysis, 35, 4, 844-867, (2003) · Zbl 1189.37064
[24] Adamczak, M., C-almost periodic functions, Commentationes Mathematicae, 37, 1-12, (1997) · Zbl 0896.42004
[25] Bugajewski, D.; N’Guérékata, G. M., On some classes of almost periodic functions in abstract spaces, International Journal of Mathematics and Mathematical Sciences, 2004, 61, 3237-3247, (2004) · Zbl 1070.42003
[26] Zhang, J.; Fan, M.; Zhu, H., Necessary and sufficient criteria for the existence of exponential dichotomy on time scales, Computers & Mathematics with Applications, 60, 8, 2387-2398, (2010) · Zbl 1205.34138
[27] Zhang, J.; Fan, M.; Zhu, H., Existence and roughness of exponential dichotomies of linear dynamic equations on time scales, Computers & Mathematics with Applications, 59, 8, 2658-2675, (2010) · Zbl 1193.34186
[28] Cabada, A.; Vivero, D. R., Criterions for absolute continuity on time scales, Journal of Difference Equations and Applications, 11, 11, 1013-1028, (2005) · Zbl 1081.39011
[29] Martins, N.; Torres, D. F. M., L’Hôpital-type rules for monotonicity with application to quantum calculus, International Journal of Mathematics and Computation, 10, 11, 99-106, (2011)
[30] Lloyd, N. G., Degree Theory, (1978), Cambridge, UK: Cambridge University Press, Cambridge, UK
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