A class of differential equations with impulses at variable times on Banach spaces. (English) Zbl 1238.34113
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, e-Suppl., e1312-e1319 (2009).
Summary: Two typical classes of semilinear evolution equation with impulses at variable times on infinite dimensional spaces are considered. Introducing the reasonable PC-mild solutions, we discuss the local existence, maximal existence interval, global existence of PC-mild solution for two models. In addition to the pulse phenomena, an example is given for demonstration.
MSC:
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34A37 | Ordinary differential equations with impulses |
References:
| [1] | Benchohra, M.; Henderson, J.; Ntouyas, S., Impulsive Differential Equations and Inclusions (2006), Hindawi Publishing Corporation: 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: World Scientific Singapore · Zbl 0719.34002 |
| [3] | Benchohra, Mouffak; Ouahab, Abdelghani, Impulsive neutral functional differential equations with variable times, Nonlinear Analysis, 55, 679-693 (2003) · Zbl 1043.34085 |
| [4] | Frigon, M.; O’Regan, D., Impulsive differential equations with variable times, Nonlinear Analysis, 26, 1913-1922 (1996) · Zbl 0863.34015 |
| [5] | Kaul, S. K.; Lakshmikantham, V.; Leea, S., Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times, Nonlinear Analysis, 22, 1263-1270 (1994) · Zbl 0807.34064 |
| [6] | Kaul, S. K.; Liu, X. Z., Impulsive integro-differential equations with variable times, Nonlinear Studies, 8, 21-32 (2001) · Zbl 0993.45009 |
| [7] | Ahmed, N. U.; Teo, K. L.; Hou, S. H., Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis, 54, 907-925 (2003) · Zbl 1030.34056 |
| [8] | Ahmed, N. U., Measure solutions for impulsive evolution equations with measurable vector fields, Journal of Mathematical Analysis and Applications, 319, 1, 74-93 (2006) · Zbl 1101.34044 |
| [9] | Migorski, S.; Ochal, A., Nonlinear impulsive evolution inclusion of second order, Dynamic Systems and Applications, 16, 155-174 (2007) · Zbl 1128.34038 |
| [10] | Peng, Y.; Xiang, X.; Wei, W., Optimal feedback control for a class of strongly nonlinear impulsive evolution equations, Computers and Mathematics with Applications, 52, 759-768 (2006) · Zbl 1122.49030 |
| [11] | Peng, Y.; Xiang, X.; Wei, W., Nonlinear impulsive integro-differential equation of mixed type with time-varying generating operators and optimal controls, Dynamical Systems and Applications, 16, 481-496 (2007) · Zbl 1153.34046 |
| [12] | Y. Peng, X. Xiang, W. Wei, Necessary conditions of optimality for second order nonlinear impulsive differential equations, Advances in Differential Equations 2007, Article ID 40160, p. 17; Y. Peng, X. Xiang, W. Wei, Necessary conditions of optimality for second order nonlinear impulsive differential equations, Advances in Differential Equations 2007, Article ID 40160, p. 17 · Zbl 1140.49018 |
| [13] | Peng, Y.; Xiang, X., Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls, Journal of Industrial and Managment Optimization, 4, 1, 17-32 (2008) · Zbl 1175.34102 |
| [14] | Peng, Y.; Xiang, X., Second order nonlinear impulsive evolution equations with time-varying generating operators and optimal controls, Optimization, 57, 827-840 (2008) · Zbl 1166.34035 |
| [15] | Peng, Y.; Xiang, X.; Wei, W., Second-order nonlinear impulsive integro-differential equations of mixed type with time-varying generating operators and optimal controls, Computers and Mathematics with Applications, 57, 42-53 (2009) · Zbl 1165.45308 |
| [16] | Wei, W.; Xiang, X.; Peng, Y., Nonlinear impulsive integro-differential equation of mixed type and optimal controls, Optimization, 55, 141-156 (2006) · Zbl 1101.45002 |
| [17] | Xiang, X.; Wei, W.; Jiang, Y., Strongly nonlinear impulsive system and necessary conditions of optimality, Dynamics of Continuous, Discrete and Impulsive Systems-A, 12, 6, 811-824 (2005) · Zbl 1081.49027 |
| [18] | Ahemd, N. U., Semigroup Theory with Application to Systems and Control (1991), Longman Science and Technical: Longman Science and Technical New York · Zbl 0727.47026 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
