Trajectory Controllability of Semilinear Differential Evolution Equations with Impulses and Delay

Affiliation(s)

College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, P. R. China.

College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, P. R. China.

ABSTRACT

This paper researches trajectory controllability of semilinear differential evolution equations with impulses and delay. The main techniques in our paper rely on the fixed point theorem and monotone operator theory. In the end of the paper, an example is given to explain our main result.

Cite this paper

M. Bin and Y. Liu, "Trajectory Controllability of Semilinear Differential Evolution Equations with Impulses and Delay,"*Open Journal of Applied Sciences*, Vol. 3 No. 1, 2013, pp. 37-43. doi: 10.4236/ojapps.2013.31B1008.

M. Bin and Y. Liu, "Trajectory Controllability of Semilinear Differential Evolution Equations with Impulses and Delay,"

References

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[2] A. Anguraj and M. Mallika Arjunan, “Existence Results for an Impulsive Neutral Integro-differential Equations in Banach Spaces,” Nonlinear Study, Vol. 16, No. 1, 2009, pp. 33-48.

[3] D. D. Bainov and P. S. Simeonov, “Im-pulsive Differential Equations: Periodic Solutions and Applica-tions,” Longman Scientific and Technical Group, England, 1993.

[4] M. Benchohra J. Henderson and S. K. Ntouyas, “Existence Results for Impulsive Multivalued Semilinear Neutral Functional Inclusions in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 263, No. 2, 2001, pp. 763-780. doi:10.1006/jmaa.2001.7663

[5] T. A. Burton and Colleen Kirk, “A Fixed Point Theorem of Krasnoselskiii-Schaefer Type,” Mathematische Nachrichten, Vol. 189, No. 1, 1998, pp. 23-31. doi:10.1002/mana.19981890103

[6] S. Carl and S. Heikkila, “Fixed Point Theory in Ordered Sets and Applications,” Springer, New York, Dordrecht, Heidelberg, London, 2010.

[7] D. N. Chalishajar, R. K. George, A. K. Nandakumaran and F. S. Acharya, “Trajectory Controllability of Nonlinear Integro-differential System,” Journal of Frankin Insitute., Vol. 347, No. 7, 2010, pp. 1065-1075. doi:10.1016/j.jfranklin.2010.03.014

[8] L. Chen and G. Li, “Approximate Controllability of Impulsive Differential Equations with Nonlocal Conditions,” International Journal of Nonlinear Science, Vol. 10, 2010, pp. 438-446.

[9] Z. Fan, “Impulsive Problems for Semilinear Differential Equations with Nonlocal Conditions,” Nonlinear Analysis, Vol. 72, No. 2, 2010, pp. 1104-1109. doi:10.1016/j.na.2009.07.049

[10] Z. Fan and G. Li, “Existence Results for Semilinear Differential Equations with Nonlocal and Impulsive Conditions,” Journal of Functional Analysis, Vol. 258, No. 5, 2010, pp. 1709-1727. doi:10.1016/j.jfa.2009.10.023

[11] E. Hernandez, M. Pierri and G. Goncalves, “Existence Results for an Impulsive Abstract Partial Differential Equation with State-dependent Delay,” Computer Mathematic Application, Vol. 52, No. 3-4, 2006, pp. 411-420. doi:10.1016/j.camwa.2006.03.022

[12] E. Hernandez, M. M. Rabello and H. Henriaquez, “Existence of Solutions for Impulsive Partial Neutral Functional Differential Equations,” Journal of Mathematic Analysis and Application, Vol. 331, No. 2, 2007, pp. 1135-1158. doi:10.1016/j.jmaa.2006.09.043

[13] S. Ji and S. Wen, “Nonlocal Cauchy Problem for Impulsive Differential Equations in Banach Spaces,” International Journal of Nonlinear Science, Vol. 10, 2010, pp. 88-95.

[14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific, Singapore, 1989.

[15] M. Li, M. Wang and F. Zhang, “Controllability of Impulsive Functional Differential Systems in Banach Spaces,” Chaos, Solitons and Fractals, Vol. 29, No. 1, 2006, pp. 175-181. doi:10.1016/j.chaos.2005.08.041

[16] Z. H. Liu and X. W. Li, “On the Controllability of Impulsive Fractional Evolution Inclusions in Banach Spaces,” Journal of Optimizition Theory and Application, Vol. 156, No. 1, 2013, pp. 167–182. doi:10.1007/s10957-012-0236-x

[17] A. M. Samoilenko and N. A. Perestyuk, “Impulsive Differential Equations,” World Scientific, Singapore, 1995.

[18] V. Obukhovski and P. Zecca, “Controllability for Systems Governed by Semilinear Differential Inclusions in a Banach Space with a Noncompact Semigroup,” Nonlinear Analysis, Vol. 70, No. 9, 2009, pp. 3424-3436. doi:10.1016/j.na.2008.05.009

[19] C. Travis, G. Webb, “Existence and Stability for Partial Functional Differential Equations,” Transactions of American Mathematical Society, Vol. 200, 1974, pp. 395-418. doi:10.1090/S0002-9947-1974-0382808-3

[20] F. Wang, Z. H. Liu and J. Li, “Complete Controllability of Fractional Neutral Differential Systems in Abstract Space, Abstract and Applied Analysis,” Article ID 529025,2013, pp. 1-11.

[21] G. Webb, “An Abstract Semilinear Volterra Integrodifferential Equations,” Proceedings of American Math-ematic Society, Vol. 69, 1978, pp. 255-260. doi:10.1090/S0002-9939-1978-0467214-4

[22] R. Ye, “Existence of Solutions for Impulsive Partial Neutral Functional Differential Equation with Infinite Delay,” Nonlinear Analysis, Vol. 73, No. 1, 2010, pp. 155-162. doi:10.1016/j.na.2010.03.008

[1] A. Anguraj and M. Mallika Arjunan, “Existence and Uniqueness of Mild and Classical Solutions of Impulsive Evolution Equations,” Electronic Journal of Differential Equations, Vol. 111, 2005, pp. 1-8.

[2] A. Anguraj and M. Mallika Arjunan, “Existence Results for an Impulsive Neutral Integro-differential Equations in Banach Spaces,” Nonlinear Study, Vol. 16, No. 1, 2009, pp. 33-48.

[3] D. D. Bainov and P. S. Simeonov, “Im-pulsive Differential Equations: Periodic Solutions and Applica-tions,” Longman Scientific and Technical Group, England, 1993.

[4] M. Benchohra J. Henderson and S. K. Ntouyas, “Existence Results for Impulsive Multivalued Semilinear Neutral Functional Inclusions in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 263, No. 2, 2001, pp. 763-780. doi:10.1006/jmaa.2001.7663

[5] T. A. Burton and Colleen Kirk, “A Fixed Point Theorem of Krasnoselskiii-Schaefer Type,” Mathematische Nachrichten, Vol. 189, No. 1, 1998, pp. 23-31. doi:10.1002/mana.19981890103

[6] S. Carl and S. Heikkila, “Fixed Point Theory in Ordered Sets and Applications,” Springer, New York, Dordrecht, Heidelberg, London, 2010.

[7] D. N. Chalishajar, R. K. George, A. K. Nandakumaran and F. S. Acharya, “Trajectory Controllability of Nonlinear Integro-differential System,” Journal of Frankin Insitute., Vol. 347, No. 7, 2010, pp. 1065-1075. doi:10.1016/j.jfranklin.2010.03.014

[8] L. Chen and G. Li, “Approximate Controllability of Impulsive Differential Equations with Nonlocal Conditions,” International Journal of Nonlinear Science, Vol. 10, 2010, pp. 438-446.

[9] Z. Fan, “Impulsive Problems for Semilinear Differential Equations with Nonlocal Conditions,” Nonlinear Analysis, Vol. 72, No. 2, 2010, pp. 1104-1109. doi:10.1016/j.na.2009.07.049

[10] Z. Fan and G. Li, “Existence Results for Semilinear Differential Equations with Nonlocal and Impulsive Conditions,” Journal of Functional Analysis, Vol. 258, No. 5, 2010, pp. 1709-1727. doi:10.1016/j.jfa.2009.10.023

[11] E. Hernandez, M. Pierri and G. Goncalves, “Existence Results for an Impulsive Abstract Partial Differential Equation with State-dependent Delay,” Computer Mathematic Application, Vol. 52, No. 3-4, 2006, pp. 411-420. doi:10.1016/j.camwa.2006.03.022

[12] E. Hernandez, M. M. Rabello and H. Henriaquez, “Existence of Solutions for Impulsive Partial Neutral Functional Differential Equations,” Journal of Mathematic Analysis and Application, Vol. 331, No. 2, 2007, pp. 1135-1158. doi:10.1016/j.jmaa.2006.09.043

[13] S. Ji and S. Wen, “Nonlocal Cauchy Problem for Impulsive Differential Equations in Banach Spaces,” International Journal of Nonlinear Science, Vol. 10, 2010, pp. 88-95.

[14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific, Singapore, 1989.

[15] M. Li, M. Wang and F. Zhang, “Controllability of Impulsive Functional Differential Systems in Banach Spaces,” Chaos, Solitons and Fractals, Vol. 29, No. 1, 2006, pp. 175-181. doi:10.1016/j.chaos.2005.08.041

[16] Z. H. Liu and X. W. Li, “On the Controllability of Impulsive Fractional Evolution Inclusions in Banach Spaces,” Journal of Optimizition Theory and Application, Vol. 156, No. 1, 2013, pp. 167–182. doi:10.1007/s10957-012-0236-x

[17] A. M. Samoilenko and N. A. Perestyuk, “Impulsive Differential Equations,” World Scientific, Singapore, 1995.

[18] V. Obukhovski and P. Zecca, “Controllability for Systems Governed by Semilinear Differential Inclusions in a Banach Space with a Noncompact Semigroup,” Nonlinear Analysis, Vol. 70, No. 9, 2009, pp. 3424-3436. doi:10.1016/j.na.2008.05.009

[19] C. Travis, G. Webb, “Existence and Stability for Partial Functional Differential Equations,” Transactions of American Mathematical Society, Vol. 200, 1974, pp. 395-418. doi:10.1090/S0002-9947-1974-0382808-3

[20] F. Wang, Z. H. Liu and J. Li, “Complete Controllability of Fractional Neutral Differential Systems in Abstract Space, Abstract and Applied Analysis,” Article ID 529025,2013, pp. 1-11.

[21] G. Webb, “An Abstract Semilinear Volterra Integrodifferential Equations,” Proceedings of American Math-ematic Society, Vol. 69, 1978, pp. 255-260. doi:10.1090/S0002-9939-1978-0467214-4

[22] R. Ye, “Existence of Solutions for Impulsive Partial Neutral Functional Differential Equation with Infinite Delay,” Nonlinear Analysis, Vol. 73, No. 1, 2010, pp. 155-162. doi:10.1016/j.na.2010.03.008