Controllability of Neutral Impulsive Differential Inclusions with Non-Local Conditions

Dimplekumar N. Chalishajar, Falguni S. Acharya

.

DOI: 10.4236/am.2011.212211                

Abstract

In this short article, we have studied the controllability result for neutral impulsive differential inclusions with nonlocal conditions by using the fixed point theorem for condensing multi-valued map due to Martelli [1]. The system considered here follows the P.D.E involving spatial partial derivatives with α-norms.

Keywords

Controllability, Neutral Impulsive Differential Inclusions, Spatial Partial Derivative, Martelli Fixed Point Theorem

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	M. Martelli, “A Rothe’s Type Theorem for Noncompact Acyclic-Valued Map,” Bollettino dell’Unione Mathematica Italiana, Vol. 2, 1975, pp. 70-76.
[2]	M. Benchohra and S. K. Ntouyas, “Existence Results for Nondensely Defined Impulsive Semilinear Functional Differential Inclusion with Infinite Delay,” Journal of Fixed Point Theory and Applications, Vol. 2, No. 1, 2007, pp. 11-51.
[3]	C. C. Travis and G. F. Webb, “Existence, Stability and Compactnes with α-Norm for Partial Functional Differential Equations,” Transaction of American Mathematical Society, Vol. 240, 1978, pp. 129-143.
[4]	L. Byszewski, “Theorems about the Existence and Uniqueness of a Semilinear Evolution Nonlocal Cauchy Problem,” Journal of Mathematical Analysis and Applications, Vol. 162, 1991, pp. 496-505.
[5]	K. Deng, “Exponential Decay of Solutions of Semilinear Paprabolic Equations with Nonlocal Initial Conditions,” Journal Mathematical Analysis and Applications, Vol. 179, 1993, pp. 630-637. doi:10.1006/jmaa.1993.1373
[6]	D. N. Chalishajar, “Controllability of Second Order Semilinear Neutral Functional Differential Inclusions with Non-Local Conditions in Banach Spaces,” Nonlinear Functional Analysis and Applications, Vol. 14, No. 1, 2009, pp.1-15.
[7]	D. N. Chalishajar and F. S. Acharya, “Controllability of Semilinear Impulsive Partial Neutral Functional Differential Equations with Infinite Delay,” Proceeding of Fourth International Conference on Neural, Parallel Scientific and Computations, Atlanta, 2010, pp. 83-89.
[8]	D. N. Chalishajar and F. S. Acharya, “Controllability of Second Order Semi-linear Neutral Impulsive Differential Inclusions on Unbounded Domain with Infinite Delay in Banach Spaces,” Bulletin of Korean Mathematical Society, Vol. 48, No. 4, 2011, pp. 813-838.
[9]	D. N. Chalishajar, R. K. George, A. K. Nandakumaran and F. S. Acharya, “Trajectory Controllability of Nonlinear Integro-Differential System,” Journal of Franklin Institute, Vol. 347, 2010, pp. 1065-1075.
[10]	M. Benchohra and S. K. Ntouyas, “Existence and Controllability Results for Nonlinear Differential Inclusions with Nonlocal Conditions in Banach Spaces,” Journal of Applied Analysis, Vol. 1, No. 8, 2002, pp. 45-52.
[11]	M. Benchohra and S. K. Ntouyas, “On Second Order Impulsive Functional Differential Equations in Banach Spaces,” Journal of Applied Mathematics and Stochastic Analysis, Vol. 15, 2002, pp. 45-52. doi:10.1155/S1048953302000059
[12]	E. Hernandez, M. Rabello and H. R. Henriquez; “Existence of Solutions for Impulsive Partial Neutral Functional Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 31, 2007, pp. 1135-1158. doi:10.1016/j.jmaa.2006.09.043
[13]	D. N. Chalishajar, “Controllability of Nonlinear Integro-Differential Third Order Dispersion System,” Journal Mathematical Analysis and Applications, Vol. 348, 2008, pp. 480-486. doi:10.1016/j.jmaa.2008.07.047
[14]	X. L. Fu and Y. J. Cao, “Existence for Neutral Impulsive Differential Inclusions with Nonlocal Conditions,” Non-linear Analysis—TMA, Vol. 68, 2008, pp. 3707-3718.
[15]	R. Triggani, “Addendum: ‘A Note on Lack of Exact Controllability for Mild Solution in Banach Spaces’,” SIAM, Journal of Control and Optimisation, Vol. 18, No. 1, 1980, pp. 98-99.
[16]	K. Yosida, “Functional Analysis,” 6th Edition, Springer-Verlag; Berlin, 1980.
[17]	K. Deimling, “Multivalued Differential Equations,” De Gruyter, Berlin, 1992. doi:10.1515/9783110874228
[18]	L. Byszewski, “Existence of Solutions of Semilinear Functional-Differential Evolution Nonlocal Problem,” Nonlinear Analysis, Vol. 34, 1998, pp. 65-72. doi:10.1016/S0362-546X(97)00693-7
[19]	A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, New York, 1983. doi:10.1007/978-1-4612-5561-1
[20]	H. Themies, “Integrated Semigroup and Integral Solutions to Abstract Cauchy Problems,” Journal Mathematical Analysis and Applications, Vol. 152, 1990, pp. 416-447. doi:10.1016/0022-247X(90)90074-P
[21]	C. C. Travis and G. F. Webb, “Partial Functional Differential Equations with Deviating Arguments in Time Variable,” Journal of Mathematical Analysis and Applications, Vol. 56, 1976, pp. 397-409. doi:10.1016/0022-247X(76)90052-4
[22]	Y. K. Chang and D. N. Chalishajar, “Controllability of Mixed Volterra-Fredholm-Type Integro-Differential Inclusions in Banach Spaces,” Journal of Franklin Institute, Vol. 345, 2008, pp. 449-507. doi:10.1016/j.jfranklin.2008.02.002