# American Institute of Mathematical Sciences

December  2011, 4(6): 1553-1564. doi: 10.3934/dcdss.2011.4.1553

## Backward problems of nonlinear dynamical equations on time scales

 1 Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China 2 Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025 3 Department of Mathematics, Guizou University, Guiyang, Guizhou Province

Received  February 2009 Revised  November 2009 Published  December 2010

In this paper, the backward problem of nonlinear dynamical equations on time scales is considered. Introducing the reasonable weak solution of the nonlinear backward problem, the existence of weak solution for nonlinear dynamical equation on time scales and its properties are presented.
Citation: Yunfei Peng, X. Xiang, W. Wei. Backward problems of nonlinear dynamical equations on time scales. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1553-1564. doi: 10.3934/dcdss.2011.4.1553
##### References:
 [1] M. Benchohra, J. Henderson and S. Ntouyas, "Impulsive Differential Equations and Inclusions," Hindawi Publishing Corporation, New York, 2006. doi: 10.1155/9789775945501.  Google Scholar [2] M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales," Birkhäuser, Boston, 2003.  Google Scholar [3] Yurong Gong and X. Xiang, A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales, Journal of Industrial and Management Optimization, 5 (2009), 1-13. doi: 10.3934/jimo.2009.5.1.  Google Scholar [4] G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285 (2003), 107-127. doi: 10.1016/S0022-247X(03)00361-5.  Google Scholar [5] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, "Dynamical Systems on Measure Chains," Kluwer Acadamic, Dordrecht, 1996.  Google Scholar [6] V. Lakshmikantham and S. Sivasundaram, Stability of moving invariant sets and uncertain dynamic systems on time scales, Computers and Mathematics with Applications, 36 (1998), 339-346. doi: 10.1016/S0898-1221(98)80034-5.  Google Scholar [7] H. Liu, X. Xiang and W. Wei, Existence and uniqueness of solutions for a class of the first order impulsive dynamic equations on time scales, DCDIS Proceeding, 3 (2005), 114-121.  Google Scholar [8] H. Liu and X. Xiang, A class of the first order impulsive dynamic equations on time scales, Nonlinear Analysis, 69 (2008), 2803-2811. doi: 10.1016/j.na.2007.08.052.  Google Scholar [9] Bryan P. Rynne, $L^2$ spaces and boundary value problems on time-scales, J. Mathe. Anal. Appl., 328 (2007), 1217-1236. doi: 10.1016/j.jmaa.2006.06.008.  Google Scholar [10] Christopher C. Tisdell and Atiya Zaidi, Basic qualitative and quantitative results for solutions to nonlinear dynamic equations on time scales with an application to economic modelling, Nonlinear Analsis, 68 (2008), 3504-3524. doi: 10.1016/j.na.2007.03.043.  Google Scholar [11] Da-Bin Wang, Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales, Computers and Mathematics with Applications, 56 (2008), 1496-1504. doi: 10.1016/j.camwa.2008.02.038.  Google Scholar

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##### References:
 [1] M. Benchohra, J. Henderson and S. Ntouyas, "Impulsive Differential Equations and Inclusions," Hindawi Publishing Corporation, New York, 2006. doi: 10.1155/9789775945501.  Google Scholar [2] M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales," Birkhäuser, Boston, 2003.  Google Scholar [3] Yurong Gong and X. Xiang, A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales, Journal of Industrial and Management Optimization, 5 (2009), 1-13. doi: 10.3934/jimo.2009.5.1.  Google Scholar [4] G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285 (2003), 107-127. doi: 10.1016/S0022-247X(03)00361-5.  Google Scholar [5] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, "Dynamical Systems on Measure Chains," Kluwer Acadamic, Dordrecht, 1996.  Google Scholar [6] V. Lakshmikantham and S. Sivasundaram, Stability of moving invariant sets and uncertain dynamic systems on time scales, Computers and Mathematics with Applications, 36 (1998), 339-346. doi: 10.1016/S0898-1221(98)80034-5.  Google Scholar [7] H. Liu, X. Xiang and W. Wei, Existence and uniqueness of solutions for a class of the first order impulsive dynamic equations on time scales, DCDIS Proceeding, 3 (2005), 114-121.  Google Scholar [8] H. Liu and X. Xiang, A class of the first order impulsive dynamic equations on time scales, Nonlinear Analysis, 69 (2008), 2803-2811. doi: 10.1016/j.na.2007.08.052.  Google Scholar [9] Bryan P. Rynne, $L^2$ spaces and boundary value problems on time-scales, J. Mathe. Anal. Appl., 328 (2007), 1217-1236. doi: 10.1016/j.jmaa.2006.06.008.  Google Scholar [10] Christopher C. Tisdell and Atiya Zaidi, Basic qualitative and quantitative results for solutions to nonlinear dynamic equations on time scales with an application to economic modelling, Nonlinear Analsis, 68 (2008), 3504-3524. doi: 10.1016/j.na.2007.03.043.  Google Scholar [11] Da-Bin Wang, Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales, Computers and Mathematics with Applications, 56 (2008), 1496-1504. doi: 10.1016/j.camwa.2008.02.038.  Google Scholar
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