Impulsive partial neutral differential equations. (English) Zbl 1103.34068
The authors establish conditions for the existence of mild and strong solutions of a partial neutral functional-differential equation with unbounded delay of the form
\[
(d/dt)(x(t)+F(t, x_t)) = Ax(t)+G(t, x_t)
\]
subject to pre-assigned moments of impulse effects. Here, \(A\) is the infinitesimal generator of a strongly continuous semigroup of linear operators on a Banach space, \(x_t\) is a Hale-type operator and \(F,G\) are given functions defined on a phase space.
They also revisit an example presented in an earlier paper of the authors [J. Math Anal. Appl. 221, No. 2, 452–475 (1998; Zbl 0915.35110)] now considering that impulse conditions are imposed on the system.
They also revisit an example presented in an earlier paper of the authors [J. Math Anal. Appl. 221, No. 2, 452–475 (1998; Zbl 0915.35110)] now considering that impulse conditions are imposed on the system.
Reviewer: Marcia Federson (São Paulo)
MSC:
| 34K45 | Functional-differential equations with impulses |
| 35K30 | Initial value problems for higher-order parabolic equations |
Keywords:
impulsive differential equations; neutral differential equations; semigroups of bounded linear operatorsCitations:
Zbl 0915.35110References:
| [1] | Ntouyas, S. K.; Sficas, Y. G.; Tsamatos, P. Ch., Existence results for initial value problems for neutral functional-differential equations, J. Differential Equations, 114, 2, 527-537 (1994) · Zbl 0810.34061 |
| [2] | Hale, J. K.; Lunel, S. M.V., Introduction to Functional-Differential Equations, Applied Mathematical Sciences, vol. 99 (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002 |
| [3] | Adimy, M.; Ezzinbi, K., Strict solutions of nonlinear hyperbolic neutral differential equations, Appl. Math. Lett., 12, 1, 107-112 (1999) · Zbl 0941.34075 |
| [4] | Datko, R., Linear autonomous neutral differential equations in a Banach space, J. Differential Equations, 25, 2, 258-274 (1977) · Zbl 0402.34066 |
| [5] | Hernández, E.; Henríquez, H. R., Existence results for partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl., 221, 2, 452-475 (1998) · Zbl 0915.35110 |
| [6] | Hernández, E.; Henríquez, H. R., Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl., 221, 2, 499-522 (1998) · Zbl 0926.35151 |
| [7] | Hernández, E., Existence results for partial neutral integrodifferential equations with unbounded delay, J. Math. Anal. Appl., 292, 1, 194-210 (2004) · Zbl 1056.45012 |
| [8] | Rogovchenko, Y. V., Impulsive evolution systems: main results and new trends, Dyn. Contin. Discrete Impuls. Syst., 3, 1, 57-88 (1997) · Zbl 0879.34014 |
| [9] | Liu, J. H., Nonlinear impulsive evolution equations, Dyn. Contin. Discrete Impuls. Syst., 6, 1, 77-85 (1999) · Zbl 0932.34067 |
| [10] | Benchohra, M.; Henderson, J.; Ntouyas, S. K., Impulsive neutral functional differential inclusions in Banach spaces, Appl. Math. Lett., 15, 8, 917-924 (2002) · Zbl 1219.34097 |
| [11] | Benchohra, M.; Ouahab, A., Impulsive neutral functional differential equations with variable times, Nonlinear Anal., 55, 6, 679-693 (2003) · Zbl 1043.34085 |
| [12] | Benchohra, M.; Ouahab, A., Impulsive neutral functional differential inclusions with variable times, Electron. J. Differential Equations, 67, 12 (2003), (electronic) · Zbl 1044.34041 |
| [13] | Benchohra, M.; Henderson, J.; Ntouyas, S. K., Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl., 263, 2, 763-780 (2001) · Zbl 0998.34064 |
| [14] | Benchohra, M.; Henderson, J.; Ntouyas, S. K., Semilinear impulsive neutral functional differential inclusions in Banach spaces, Appl. Anal., 81, 4, 951-963 (2002) · Zbl 1037.34076 |
| [15] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44 (1983), Springer-Verlag: Springer-Verlag New York, Berlin · Zbl 0516.47023 |
| [16] | Hino, Y.; Murakami, S.; Naito, T., Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473 (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0732.34051 |
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