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Existence of solutions for impulsive partial neutral functional differential equation with infinite delay. (English) Zbl 1198.34177
The existence of mild solutions for first-order impulsive semilinear neutral functional differential equations with infinite delay in Banach spaces $$\cases \frac{d}{dt}(x(t)+g(t,x_t))=Ax(t)+f(t,x_t), \,\,\, t\in J=[0,b]\\ x_0=\phi\in {\Cal B},\\ \Delta x(t_i)=I_i(x_{t_i}), \,\, i=1,2,\dots,n, \endcases$$ is investigated. Here, $A$ is the infinitesimal generator of an analytic semigroup of linear operators on a Banach space $X,$ $x_t: (-\infty,0]\to X,$ $x_t(\theta)=x(t+\theta),$ belongs to some abstract space ${\Cal B}$ defined axiomatically, $g, f, I_i, i=1,2,\dots, n$ are appropriate functions, $0<t_1<\dots<t_n<b$ are fixed numbers and $\Delta \xi(t)$ represent the jump of the functions $\xi$ at $t,$ defined by $\Delta\xi(t)=\xi(t^+)-\xi(t^-).$ Existence results are obtained without the compactness assumption on the associated semigroup, via the Hausdorff measure of noncompactness. An example illustrating the abstract results is also presented.

34K30Functional-differential equations in abstract spaces
34K40Neutral functional-differential equations
34K45Functional-differential equations with impulses
47D03(Semi)groups of linear operators
Full Text: DOI
[1] Benchohra, M.; Henderson, J.; Ntouyas, S. K.: Existence results for impulsive multivalued semilinear neutral functional inclusions in Banach spaces, J. math. Anal. appl. 263, 763-780 (2001) · Zbl 0998.34064 · doi:10.1006/jmaa.2001.7663
[2] Chang, Y. -K.; Anguraj, A.; Arjunan, M. Mallika: Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear anal. Hybrid syst. 2, 209-218 (2008) · Zbl 1170.35467 · doi:10.1016/j.nahs.2007.10.001
[3] Hernández, E.: A second-order impulsive Cauchy problem, Int. J. Math. math. Sci. 31, No. 8, 451-461 (2002) · Zbl 1013.34061 · doi:10.1155/S0161171202012735
[4] M., E. Hernández; Henriquez, H. R.: Impulsive partial neutral differential equations, Appl. math. Lett. 19, 215-222 (2006) · Zbl 1103.34068 · doi:10.1016/j.aml.2005.04.005
[5] M., E. Hernández; Rabello, M.; Henriaquez, H.: Existence of solutions for impulsive partial neutral functional differential equations, J. math. Anal. appl. 331, 1135-1158 (2007) · Zbl 1123.34062 · doi:10.1016/j.jmaa.2006.09.043
[6] Liu, James H.: Nonlinear impulsive evolution equations, Dyn. contin. Discrete impuls. Syst. 6, No. 1, 77-85 (1999) · Zbl 0932.34067
[7] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, Series in modern appl. Math. 6 (1989) · Zbl 0719.34002
[8] Rogovchenko, Yuri V.: Impulsive evolution systems: Main results and new trends, Dyn. contin. Discrete impuls. Syst. 3, No. 1, 57-88 (1997) · Zbl 0879.34014
[9] Rogovchenko, Yuri V.: Nonlinear impulse evolution systems and applications to population models, J. math. Anal. appl. 207, No. 2, 300-315 (1997) · Zbl 0876.34011 · doi:10.1006/jmaa.1997.5245
[10] Travis, C. C.; Webb, G. F.: Existence and stability for partial functional differential equations, Trans. amer. Math. soc. 200, 395-418 (1974) · Zbl 0299.35085 · doi:10.2307/1997265
[11] Webb, G. F.: Autonomos nonlinear functional differential equations and nonlinear semigroups, J. math. Anal. appl. 46, 1-12 (1974) · Zbl 0277.34070 · doi:10.1016/0022-247X(74)90277-7
[12] Benchohra, M.; Djebali, S.; Moussaoui, T.: Boundary value problems for double perturbed first order ordinary differential systems, Electron. J. Qual. theory differ. Equ., No. 11, 1-10 (2006) · Zbl 1118.34011 · http://www.math.u-szeged.hu/ejqtde/2006/200611.html
[13] Baghlt, S.; Benchohra, M.: Perturbed functional and neutral functional evolution equations with infinite delay in Fréchet spaces, Electron. J. Differential equations 2008, No. 69, 1-19 (2008) · Zbl 1190.34098 · emis:journals/EJDE/Volumes/2008/69/abstr.html
[14] Dong, Q.: Double perburbed evolution equations with infinite delay in Banach spaces, J. yangzhou univ. (Natural science edition) 11, No. 4, 7-11 (2008)
[15] Dong, Q.; Li, Z. Fang: Existence of solutions to nonlocal neutral functional differential and integrodifferential equations, Int. J. Nonlinear sci. 5, No. 2, 140-151 (2008) · Zbl 1177.35240
[16] Hernández, E.: Existence results for a class of semi-linear evolution equations, Electron. J. Differential equations 2001, 1-14 (2001) · Zbl 0973.35009 · emis:journals/EJDE/Volumes/2001/24/abstr.html
[17] Hernández, E.; Henríquez, H. R.: Existence results for partial neutral functional differential equations with unbounded delay, J. math. Anal. appl. 221, 452-475 (1998) · Zbl 0915.35110 · doi:10.1006/jmaa.1997.5875
[18] Hernández, E.; Henríquez, H. R.: Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. math. Anal. appl. 221, 499-522 (1998) · Zbl 0926.35151 · doi:10.1006/jmaa.1997.5899
[19] Hino, Y.; Murakami, S.; Naito, T.: Functional-differential equations with infinite delay, Lecture notes in math. 1473 (1991) · Zbl 0732.34051
[20] Hale, J. K.; Kato, J.: Phase space for retarded equations with infinite delay, Funkcial. ekvac. 21, 11-41 (1978) · Zbl 0383.34055
[21] Banas, J.; Goebel, K.: Measure of noncompactness in Banach space, Lecture notes in pure and applied matyenath (1980) · Zbl 0438.47051
[22] Agarwal, R.; Meehan, M.; O’regan, D.: Fixed point theory and applications, Cambridge tracts in mathematics (2001)