Existence of Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Space

Radhakrishnan, B.

Research Article

Existence of Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Space

B. Radhakrishnan

Abstract

In this paper, we devoted to study the existence of mild solutions for quasilinear impulsive integrodi_erential equation in Banach spaces. The results are established by using Hausdor_'s measure of noncompactness and the _xed point theorems. Application is provided to illustrate the theory.

How to Cite

B. Radhakrishnan, Existence of Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Space, Int. J. Anal. Appl. 7 (2015), 22–37.

BibTeX

@article{Radhakrishnan2015,
  author  = {Radhakrishnan, B.},
  title   = {Existence of Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Space},
  journal = {Int. J. Anal. Appl.},
  volume  = {7},
  year    = {2015},
  pages   = {22--37},
  doi     = {}
}

References

[1] K. Balachandran and E. R. Anandhi, Neutral functional integrodifferential control systems in Banach spaces, Kybernetika, 39 (2003), 359-367.
[2] K. Balachandran and E. R. Anandhi, Controllability of neutral functional integrodifferential infinite delay systems in Banach spaces, Taiwanese Journal of Mathematics, 8 (2004), 689- 702.
[3] E. Hernandez and H. R. Henriquez, Impulsive partial neutral differential equations, Applied Mathematics Letters, 19 (2006), 215-222.
[4] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer-Verlag, New York, 1993.
[5] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1992), 494-505.
[6] L. Byszewski, H. Akca, Existence of solutions of a semilinear functional-differential evolution nonlocal problem, Nonlinear Anal. 34 (1998), 65-72.
[7] L. Byszewski, V. Lakshmikanthan, Theorems about the existence and uniqueness of solutions of a nonlocal Cauchy problem in Banach spaces, Applicable Anal. 40 (1990), 11-19.
[8] S. K. Ntouyas, P. Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl. 210 (1997), 679-687.
[9] Q. Dong, G. Li, J. Zhang, Quasilinear nonlocal integrodifferential equations in Banach spaces, Electronic J. Diff. Equa. 19 (2008), 1-8.
[10] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
[11] D. Guo, X. Liu, External solutions of nonlinear impulsive integrodifferential equations in Banach spaces, J. Math. Anal. Appl. 177 (1993), 538-552.
[12] J. Liang, J. H. Liu, T. J. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Computer Modelling. 49 (2009), 798-804.
[13] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
[14] Z. Luo, J. J. Nieto, New results for the periodic boundary value problem for impulsive integrodifferential equations, Nonlinear Anal. 70 (2009), 2248-2260.
[15] J. Banas, K. Goebel, Measure of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcle Dekker, New York, 1980.
[16] F.S. De Blasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie, 21 (1977), 259-262.
[17] J. M. Ayerbe, T. D. Benavides, G. L. Acedo, Measure of noncmpactness in in metric fixed point theorem, Birkhauser, Basel, 1997.
[18] J. Banas, W.G. El-Sayed, Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation, J. Math. Anal. Appl. 167 (1992), 133- 151.
[19] G. Emmanuele, Measures of weak noncompactness and fixed point theorems, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 25 (1981), 353-358.
[20] Q. Dong, G. Li, The Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces, Electronic J. Qualitative Theory of Diff. Equa. 47 (2009), 1-13.
[21] Z. Fan, Q. Dong, G. Li, Semilinear differential equations with nonlocal conditions in Banach spaces, International J. Nonlinear Sci. 2 (2006), 131-139.
[22] H.P. Heinz, On the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7 (1983), 1351-1371.
[23] M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter Series. Nonlinear Anal. Appl. Vol.7, de Gruyter, Berlin, 2001.
[24] D. Both, Multivalued perturbation of m-accretive differential inclusions, Israel J. Math. 108 (1998), 109- 138.
[25] K. Balachandran, J. Y. Park, Existence of solutions and controllability of nonlinear integrodifferential systems in Banach spaces, Mathematical Problems in Engineering, 2 (2003), 65-79.
[26] B. Radhakrishnan, K. Balachandran, Controllability results for semilinear impulsive integrodifferential evolution systems with nonlocal conditions, J. Control Theory and Appl. 10 (2012), 28-34.
[27] L. Wang and Z. Wang, Controllability of abstract neutral functional differential systems with infinite delay, Dynamics of Continuous, Discrete and Impulsive Systems Ser.B: Applications and Algorithms, 9 (2002), 59-70.