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Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. (English) Zbl 1173.34048

Summary: We study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive differential equation with nonlocal conditions

$\left\{\begin{array}{c}{u}^{\text{'}}\left(t\right)=Au\left(t\right)+f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}0\le t\le K,\phantom{\rule{4pt}{0ex}}t\ne {t}_{i},\hfill \\ u\left(0\right)+g\left(u\right)={u}_{0},\hfill \\ {\Delta }u\left({t}_{i}\right)={I}_{i}\left(u\left({t}_{i}\right)\right),\phantom{\rule{1.em}{0ex}}i=1,2,\cdots ,p,\phantom{\rule{4pt}{0ex}}0<{t}_{1}<{t}_{2}<\cdots <{t}_{p}

by combining and extending some earlier work on equations with nonlocal conditions and equations with impulsive conditions. Here, $A$ is the generator of a strongly continuous semigroup in a Banach space, $g$ constitutes a nonlocal condition, and ${\Delta }u\left({t}_{i}^{+}\right)-u\left({t}_{i}^{-}\right)$ constitutes an impulsive condition. New results are obtained.

##### MSC:
 34K30 Functional-differential equations in abstract spaces 34K45 Functional-differential equations with impulses
##### Keywords:
nonlocal and impulsive conditions
##### References:
 [1] Ahmed, N. U.: Optimal feedback control for impulsive systems on the space of finitely additive measures, Publ. math. Debrecen 70, 371-393 (2007) · Zbl 1164.34026 [2] Aizicovici, S.; Lee, Haewon: Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. math. Lett. 18, 401-407 (2005) · Zbl 1084.34002 · doi:10.1016/j.aml.2004.01.010 [3] Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. math. Anal. appl. 162, 494-505 (1991) · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U [4] Byszewski, L.: Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems, Dynam. systems appl. 5, 595-605 (1996) · Zbl 0869.47034 [5] Benchohra, M.; Henderson, J.; Ntouyas, S.: Impulsive differential equations and inclusions, Contemporary mathematics and its applications 2 (2006) · Zbl 1130.34003 [6] Byszewski, L.; Lakshmikantham, V.: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Applicable anal. 40, 11-19 (1990) · Zbl 0694.34001 · doi:10.1080/00036819008839989 [7] Boucherif, A.; Precup, R.: On the nonlocal initial value problem for first order differential equations, Fixed point theory 4, No. 2, 205-212 (2003) · Zbl 1050.34001 [8] Ezzinbi, K.; Fu, X.; Hilal, K.: Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear anal. 67, No. 5, 1613-1622 (2007) · Zbl 1119.35105 · doi:10.1016/j.na.2006.08.003 [9] Guo, D.; Liu, X.: Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces, J. math. Anal. appl. 177, 538-552 (1993) · Zbl 0787.45008 · doi:10.1006/jmaa.1993.1276 [10] Henriquez, H. R.; Hernandez, E.; Akca, H.: Global solutions for an abstract Cauchy problem with nonlocal conditions, Internat. J. Math. manuscripts 1 (2007) [11] Jackson, D.: Existence and uniqueness of solutions of a semilinear nonlocal parabolic equations, J. math. Anal. appl. 172, 256-265 (1993) · Zbl 0814.35060 · doi:10.1006/jmaa.1993.1022 [12] Liu, J. H.: Nonlinear impulsive evolution equations, Dynam. contin. Discrete impuls. Sys. 6, 77-85 (1999) · Zbl 0932.34067 [13] Liang, J.; Van Casteren, J.; Xiao, T. J.: Nonlocal Cauchy problems for semilinear evolution equations, Nonlinear anal. 50, 173-189 (2002) · Zbl 1009.34052 · doi:10.1016/S0362-546X(01)00743-X [14] Lin, Y.; Liu, J. H.: Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear anal. 26, 1023-1033 (1996) · Zbl 0916.45014 · doi:10.1016/0362-546X(94)00141-0 [15] Liang, J.; Liu, J. H.; Xiao, T. J.: Nonlocal Cauchy problems governed by compact operator families, Nonlinear anal. 57, 183-189 (2004) · Zbl 1083.34045 · doi:10.1016/j.na.2004.02.007 [16] Liang, J.; Xiao, T. J.: Semilinear integrodifferential equations with nonlocal initial conditions, Comput. math. Appl. 47, 863-875 (2004) · Zbl 1068.45014 · doi:10.1016/S0898-1221(04)90071-5 [17] Liu, X.; Willms, A.: Stability analysis and applications to large scale impulsive systems: A new approach, Canad. appl. Math. quart. 3, 419-444 (1995) · Zbl 0849.34044 [18] Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems, (1995) [19] N’guérékata, Gaston M.: Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions, , 843-849 (2006) · Zbl 1147.35329 [20] Rogovchenko, Y.: Impulsive evolution systems: Main results and new trends, Dynam. contin. Discrete impuls. Sys. 3, 57-88 (1997) · Zbl 0879.34014 [21] Zavalishchin, A.: Impulse dynamic systems and applications to mathematical economics, Dynam. systems appl. 3, 443-449 (1994) · Zbl 0805.34009