Discontinuity, Nonlinearity, and Complexity
Existence of Semi Linear Impulsive Neutral Evolution Inclusions with Infinite Delay in Frechet Spaces
Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 1934  DOI:10.5890/DNC.2017.03.003
Dimplekumar N. Chalishajar; K. Karthikeyan; A. Anguraj
Department of Applied Mathematics, Virginia Military Institute (VMI), 431, Mallory Hall, Lexington, VA 24450, USA
Department of Mathematics, KSR College of Technology, Tiruchengode, Tamil Nadu 637215, India
Department of Mathematics, PSG College of Arts and Science, Coimbatore, Tamil Nadu 641 014, India
Download Full Text PDF
Abstract
In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semiinfinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractivemultivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We study a new phase space for impulsive system with infinite delay.
Acknowledgments
Authors wish to express their gratitude to the anonymous referees for their valuable suggestions and comments for improving this manuscript.
References

[1]  Bainov, D. D. and Simeonov, P. S. (1989), Systems with impulsive effect, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood, Chichester. 

[2]  Lakshimikantham, V. Bainov, D. D. and Simeonov, P. S. (1989), Theory of impulsive differential equations, World Scientific, Singapore, 1989. 

[3]  Samoilenko, A. M. and Perestyuk, N. A. (1995), Impulsive differential equations,World Scientific Series on Nonlinear ScienceSeries A, 14,World Scientific Publishing, Nejersey. 

[4]  Gurtin,M. E. and Pipkin, A. C. (1968), A general theory of heat conduction with finite wave speed, Archive for Rational Mechanics and Analysis, 31, 113126. 

[5]  Nunziato, J.W. (1971), On heat conduction in materials with memory, Quarterly of AppliedMathematics, 29, 187204. 

[6]  Wang, Z., Lam, J., and Burnham, K.J. (2002), Stability Analysis and Observer Design for Neutral Delay Systems, IEEE Transactions on Automatic Control, 47(3). 

[7]  Wu, L. and Wang, Z. (2009), Guaranteed cost control of switched systems with neutral delay via dynamic output feedback, International Journal of Systems Science, 40(7). 

[8]  Anguraj, A., Mallika Arjunan, M., and Eduardo Hernandez, M. (2007), Existence results for an impulsive neutral functional differential equation with statedependent delay, Applicable Analysis, 86, 861872. 

[9]  Chang, Y. K., Anguraj, A., and Karthikeyan, K. (2009), Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Nonlinear Anal. TMA, 71, 43774386. 

[10]  Hale, J. and Kato, J. (1978), Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21, 1141. 

[11]  Corduneanu, C. and Lakshmikantham, V. (1980), Equations with unbounded delay, Nonlinear Anal., 4, 831877. 

[12]  Graef, J. R. and Ouahab, A. (2006), Some existence and uniqueness results for first order Boundary value problems for impulsive functional differential equations with infinite delay in Frechet spaces, International Journal of Mathematics and Mathematical Sciences, 116. 

[13]  Baghli, S. and Benchohra, M. (2008), Uniqueness results for partial functional differential equations in Frechet spaces, Fixed Point Theory, 9(2), 395406. 

[14]  Baghli, S. and Benchohra, M. (2008),Multivalued evolution equations with infinite delay in Frechet spaces,Electronic journal of Qualitative theory of differential equations, (33), 124. 

[15]  Henderson, J. and Ouahab, A. (2005), Existence results for nondensely defined semilinear functional differential inclusions in Frechet spaces, Elect. J. Qual. Theor. Differ. Equat., 17, 117. 

[16]  Hernandez, E. M., Rabello, M., and Henriquez, H. (2007) Existence of solution of impulsive partial neutral functional differential equations,J. Math. Anal. Appl., 331, 11351158. 

[17]  Arthi, G. and Balachandran, K. (2011), Controllability of second order impulsive functional differential equations with state dependent delay,Bulletin of Korean Mathematical Society, 48(6), 12711290. 

[18]  Hino, Y., Murakami, S., and Naito, T. (1991), Functional differential equations with infinite delay, Lecture notes in Math., 1473, SpringerVerlag, Berlin. 

[19]  Chalishajar, D. N. (2012), Controllability of second order impulsive neutral functional differential inclusions with infinite delay,Journal of Optimization Theory and Applications, accepted for publication, 154(2). 

[20]  Chalishajar, D. N. and Acharya, F. S. (2011), Controllability of second order semi linear neutral functional impulsive differential inclusion with infinite delay in Banach spaces, Bulletin of the KoreanMathematical Society, 48(4), 813838. 

[21]  Triggiani, R. (1977) (1980), Addendum: A note on lack of exact controllability for mild solution in Banach spaces;[SIAM, Journal of Control and Optimization, 15, 407411, MR 55 8942]; SIAM, Journal of Control and Optimization, 18(1), 9899. 

[22]  Ntouyas, S. and O’Regan, D. (2009), Some Remarks on Controllability of Evolution Equations in Banach Spaces; Electonic Journal of Differential Equations, 2009(79), 16. 

[23]  Yosida, K. (1980), Functional Analysis, 6th edn. SpringerVerlag, Berlin. 

[24]  Kisielewicz, M. (1991), Differential inclusions and optimal control, Kluwer, Dordrecht, The Netherlands. 

[25]  Ahmed, N. U. (1991), Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series, 246, Longman Scientific and Technical, Harlow John Wiley and Sons,Inc., NY. 

[26]  Engel, K. J. and Nagel, R. (2000), Oneparameter semigroups for linear evolution equations, SpringerVerlag, NY. 

[27]  Pazy, A. (1988), Semigroups of linear operators and applications to partial differential equations, SpringerVerlag, NY. 

[28]  Aubin, J. P. and Cellina, A. (1984), Differential inclusions, BerlinHeidelbergNew YorkTokyo. 

[29]  Deimling, K. (1992), Multivalued differential equations,Walter de Gruyter, BerlinNew York. 

[30]  Gorniewicz, L. (1999), Topological fixed point theory of multivalued mappings, Mathematics and its applications, 495, Kluwer Academic Publishers, Dordrecht. 

[31]  Hu, Sh., and Papageorgiou, N. (1997), Handbook of multivalued analysis, volume I: Theory, Kluwer, Dordrecht, Boston, London. 

[32]  Tolstonogov, A. A. (2000), Differential inclusions in a Banach space, Kluwer Academic Publishers, Dordrecht. 

[33]  Frigon, M. (2002), Fixed point results for multivalued contractions on gauge spaces, Set valued mappings with applications in nonlinear analysis, 175181, Ser. Math. Anal. Appl., 4, Taylor & Francis, London. 

[34]  Frigon, M. (2007), Fixed point and continuation results for contractions in metric and gauge spaces, Fixed point theory and its applications, Banach Center Publ., Polish Acad. Sci., Warsaw, 77, 89114. 

[35]  Chalishajar, D. and Chalishajar, H. (2014), Trajectory Controllability of second order Nonlinear Integrodifferential System An Analytical and a Numerical estimations, J. of Differential Equations and Dynamical Systems, Springer. 

[36]  Castaing, C. and Valadier, M. (1977), Convex Analysis and Measurable multifunctions, Lecture notes in Mathematics, 580, SpringerVerlag, New York. 

[37]  Freidman, A. (1969), Partial differential equations, Holt, Rinehat and Winston, NY. 