# zbMATH — the first resource for mathematics

Existence results for impulsive differential inclusions with nonlocal conditions. (English) Zbl 1231.34107
Summary: We establish sufficient conditions for the existence of mild solutions for nonlocal impulsive differential inclusions. On the basis of the fixed point theorems for multivalued maps and the technique of approximate solutions, new results are obtained. Examples are also provided to illustrate our results.

##### MSC:
 34G25 Evolution inclusions 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B37 Boundary value problems with impulses for ordinary differential equations 34A60 Ordinary differential inclusions
Full Text:
##### References:
 [1] Benchohra, M.; Henderson, J.; Ntouyas, S.K., Impulsive differential equations and inclusions, vol. 2, (2006), Hindawi Publishing Corporation New York · Zbl 1130.34003 [2] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 [3] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003 [4] Ahmed, N.U., Measure solutions for impulsive evolution equations with measurable vector fields, J. math. anal. appl., 319, 1, 74-93, (2006) · Zbl 1101.34044 [5] Cardinali, T.; Rubbioni, P., Impulsive semilinear differential inclusion: topological structure of the solution set and solutions on non-compact domains, Nonlinear anal., 14, 73-84, (2008) · Zbl 1147.34045 [6] Chang, Y.K.; Anguraj, A.; Arjunan, M. Mallika, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear anal., 2, 209-218, (2008) · Zbl 1170.35467 [7] Cuevas, C.; Hernández, E.; Rabelo, M., The existence of solutions for impulsive neutral functional differential equations, Comput. math. appl., 58, 744-757, (2009) · Zbl 1189.34155 [8] Hernández, E.; Rabelo, M.; Henríquez, H.R., Existence of solutions for impulsive partial neutral functional differential equations, J. math. anal. appl., 331, 1135-1158, (2007) · Zbl 1123.34062 [9] Migorski, S.; Ochal, A., Nonlinear impulsive evolution inclusion of second order, Dynam. systems appl., 16, 155-174, (2007) · Zbl 1128.34038 [10] Ji, S.; Wen, S., Nonlocal Cauchy problem for impulsive differential equations in Banach spaces, Int. J. nonlinear sci., 10, 1, 88-95, (2010) · Zbl 1225.34082 [11] Ji, S.; Li, G.; Wang, M., Controllability of impulsive differential systems with nonlocal conditions, Appl. math. comput., 217, 6981-6989, (2011) · Zbl 1219.93013 [12] Byszewski, L.; Lakshmikantham, V., Theorem about the existence and uniqueness of solutions of a nonlocal Cauchy problem in a Banach space, Appl. anal., 40, 11-19, (1990) · Zbl 0694.34001 [13] 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 [14] Ntouyas, S.; Tsamotas, P., Global existence for semilinear evolution equations with nonlocal conditions, J. math. anal. appl., 210, 679-687, (1997) · Zbl 0884.34069 [15] Ntouyas, S.; Tsamotas, P., Global existence for semilinear integrodifferential equations with delay and nonlocal conditions, Appl. anal., 64, 99-105, (1997) · Zbl 0874.35126 [16] Liang, J.; Liu, J.; Xiao, T., Nonlocal Cauchy problems governed by compact operator families, Nonlinear anal., 57, 183-189, (2004) · Zbl 1083.34045 [17] Aizicovici, S.; McKibben, M., Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear anal., 39, 649-668, (2000) · Zbl 0954.34055 [18] Aizicovici, S.; Lee, H., Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. math. lett., 18, 401-407, (2005) · Zbl 1084.34002 [19] 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, 763-780, (2001) · Zbl 0998.34064 [20] 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, 1613-1622, (2007) · Zbl 1119.35105 [21] García-Falset, J., Existence results and asymptotic behavior for nonlocal abstract Cauchy problems, J. math. anal. appl., 338, 639-652, (2008) · Zbl 1140.34026 [22] Liu, J., A remark on the mild solutions of nonlocal evolution equations, Semigroup forum, 66, 63-67, (2003) · Zbl 1015.37045 [23] Xue, X., Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces, Nonlinear anal., 70, 2593-2601, (2009) · Zbl 1176.34071 [24] Zhu, L.; Li, G., On a nonlocal problem for semilinear differential equations with upper semicontinuous nonlinearities in general Banach spaces, J. math. anal. appl., 341, 1, 660-675, (2008) · Zbl 1145.34034 [25] Liang, J.; Liu, J.H.; Xiao, T.J., Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. comput. modelling, 49, 798-804, (2009) · Zbl 1173.34048 [26] Fan, Z.; Li, G., Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. funct. anal., 258, 1709-1727, (2010) · Zbl 1193.35099 [27] Thiems, H., Integrated semigroup and integral solutions to abstract Cauchy problem, J. math. anal. appl., 152, 416-447, (1990) [28] Agarwal, R.; Meehan, M.; O’Regan, D., () [29] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023 [30] Zhu, L.; Li, G., Nonlocal differential equations with multivalued perturbations in Banach spaces, Nonlinear anal., 69, 2843-2850, (2008) · Zbl 1163.34041 [31] Deng, K., Exponential decay of solutions of semilinear parabolic equations with non-local initial conditions, J. math. anal. appl., 179, 630-637, (1993) · Zbl 0798.35076 [32] Jackson, D., Existence and uniqueness of solutions of a semilinear nonlocal parabolic equations, J. math. anal. appl., 172, 256-265, (1993) · Zbl 0814.35060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.