EXISTENCE RESULTS OF SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY AND FRACTIONAL DAMPING

Shengli Xie

doi:10.11948/2017020

2017 Volume 7 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(1): 291-308

Next Article Previous Article

Shengli Xie. EXISTENCE RESULTS OF SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY AND FRACTIONAL DAMPING[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 291-308. doi: 10.11948/2017020

Citation:

Shengli Xie. EXISTENCE RESULTS OF SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY AND FRACTIONAL DAMPING[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 291-308. doi: 10.11948/2017020

EXISTENCE RESULTS OF SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY AND FRACTIONAL DAMPING

Shengli Xie

School of Mathematics & Physics, Anhui University of Architecture, Ziyun Road, 230601 Hefei, China

Fund Project:

Abstract

Abstract

In this paper we prove the existence, uniqueness, regularity and continuous dependence of mild solutions for second order impulsive functional differential equations with infinite delay and fractional damping in Banach spaces. We generalize the existence theorem of integer order differential equations to the fractional order case. The results obtained here improve and generalize some known results.
- Mild solutions /
- classical solutions /
- strong solution /
- continuous dependence /
- fixed point
MSC: 34G25;34K37;35L30;35L15;35L70

FullText(HTML)

References

[1]	K. Aissani and M. Benchohra, Semlinear fractional order integro-differential equations with infinite delay in Banach spaces, Arch. Math. (Brno), 49(2013)(2), 105-117. Google Scholar
[2]	M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for impulsive semilinear damped differential inclusions, Electron. J. Qual. Theory Differ. Equ., 11(2003), 1-19. Google Scholar
[3]	J. Dabas, A. Chauhan and M. Kumar, Existence of the mild solutions for impulsive fractional equations with infinite delay, Int. J. Differ. Equ., 2011(2011). Google Scholar
[4]	J. Dabas and A. Chauhan, Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Math. Comput. Model., 57(2013)(3-4), 754-763. Google Scholar
[5]	M. Fečkan, Y. Zhou and J. R. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17(2012)(7), 3050-3060. Google Scholar
[6]	H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Math. Stud., vol. 108, North-Holland, Amsterdam, 1985. Google Scholar
[7]	Z. Y. Guo and M. Liu, An integrod-differential equation with fractional derivatives in the nonlinearities, Acta Math. Univ., Comenianae, LXXXⅡ, 1(2013), 105-111. Google Scholar
[8]	T. L. Guo and W. Jiang, Impulsive fractional functional differential equations, Comput. Math. Appl., 64(2012)(10), 3414-3424. Google Scholar
[9]	D. J. Guo, V. Lakshmikantham and X. Z. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, 1996. Google Scholar
[10]	E. Hernández, K. Balachandran and N. Annapoorani, Existence results for a damped second order abstract functional differential equation with impulses, Math. Comput. Model., 50(2009)(11-12), 1583-1594. Google Scholar
[11]	H. R. Henriquez and C. H. Vásquez, Differentiability of solutions of the second order abstract Cauchy problem, Semigroup Forum, 64(2002)(3), 472-488. Google Scholar
[12]	H. R. Henriquez and C. H. Vásquez, Differentiability of solutions of secondorder functional differential equations with unbounded delay, J. Math. Anal. Appl., 280(2003)(2), 284-312. Google Scholar
[13]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial Ekvac, 21(1978), 11-41. Google Scholar
[14]	Y. Hino, S. Murakami and T. Naito, Functional-Differential Equations with Infinite Delay, in:Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. Google Scholar
[15]	Y. Jalilian and R. Jalilian, Existence of solution for delay fractional differential equations, Mediterr. J. Math., 10(2013)(4), 1731-1747. Google Scholar
[16]	H. P. Jiang, Existence results for fractional order functional differential equations with impulse, Comput. Math. Appl., 64(2012)(10), 3477-3483. Google Scholar
[17]	V. Kavitha, P. Z. Wang and R. Murugesu, Existence results for neutral functional fractional differential equations with state dependent-delay, Malaya Journal of Matematik, 1(2012)(1), 50-61. Google Scholar
[18]	J. Kisy'nski, On cosine operator functions and one parameter group of operators, Studia Math., 49(1972), 93-105. Google Scholar
[19]	H. Mönch, Boundary value problems for nonlinear ordinary equations of second order in Banach spaces, Nonlinear Analysis, 4(1980)(5), 985-999. Google Scholar
[20]	C. Ravichandran and M. Mallika Arjunan, Existence results for impulsive fractional semilinear functional integro-differential equations in Banach spaces,J. Fract. Calc. Appl., 3(2012)(8), 1-11. Google Scholar
[21]	X. B. Shu, Y. Z. Lai and Y. M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Analysis, 74(2011)(5), 2003-2011. Google Scholar
[22]	X. B. Shu and Q. Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1< α < 2, Comput. Math. Appl., 64(2012)(6), 2100-2110. Google Scholar
[23]	X. B. Shu and F. Xu, The existence of solutions for impulsive fractional partial neutral differential equations, Journal of Mathematics, 2013(2013), Article ID 147193. Google Scholar
[24]	N. E. Tatar, Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the nonlocal conditions, Adv. Difference Equ., 2011(2011), 1-18. Google Scholar
[25]	C. C. Travis and G. F. Webb, An abstract second order semilinear Volterra integrodifferential equation, SIAM J. Math. Anal., 10(1979)(2), 412-424. Google Scholar
[26]	C. C. Travis and G. F. Webb, Compactness, regularity and uniform continuity properties of strongly continuous cosine families, Houston J. Math., 3(1977)(4), 555-567. Google Scholar
[27]	J. R. Wang, M. Fečkan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynamics of PDE, 8(2011)(4), 345-361. Google Scholar
[28]	S. L. Xie, Solvability of impulsive partial neutral second-order functional integro-differential equations with infinite delay, Bound. Value Probl., 203(2013), 1-19. Google Scholar
[29]	S. L. Xie, Existence results of mild solutions for impulsive fractional integrodifferential evolution equations with infinite delay, Fract. Calc. Appl. Anal., 17(2014)(4), 1158-1174. Google Scholar
[30]	X. M. Zhang, X. Y. Huang and Z. H. Liu, The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, Nonlinear Analysis:Hybrid Systems, 4(2010), 775-781. Google Scholar