Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains. (English) Zbl 1147.34045

The following impulsive Cauchy problem is considered:
\[ \begin{aligned} &x'(t) \in A(t)x(t) + F(t,x(t))\quad \text{a.e.} \,\, t\in [0,b],\,\,\, t \neq t_{k}, k=1,2,\dots ,m,\\ &x(t_{k}^{+}) = x(t_{k}) + I_{k}(x(t_{k})),\quad k=1,2,\dots ,m, \\ &x(0) = a_0 \in E, \end{aligned} \] where \(\{A(t)\}_{t\in [0,b]}\) is a family of linear operators (not necessarily bounded) in a Banach space \(E\) generating an evolution operator, \(F\) is a multifunction of Carathéodory type, \(0<t_0<t_1<\ldots<t_m<t_{m+1}=b,\) \(I_k:E\to E, k=1,\ldots,m,\) are impulse functions and \(x(t^+)=\lim_{s\to t^+}x(s).\) First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on noncompact domains. An example illustrating the abstract results is also presented.


34G25 Evolution inclusions
34A37 Ordinary differential equations with impulses
Full Text: DOI


[1] Bainov, D. D.; Covachev, V., (Impulsive Differential Equations with a Small Parameter. Impulsive Differential Equations with a Small Parameter, Series on Advances in Math. for Applied Sciences, vol. 24 (1994), World Scientific) · Zbl 0828.34001
[2] Bainov, D. D.; Lakshmikantham, V.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[3] Ballinger, G.; Liu, X., Boundness for impulsive delay differential equations and applications to population growth models, Nonlinear Anal., 53, 1041-1062 (2003) · Zbl 1037.34061
[4] Benchohra, M.; Henderson, J.; Ntouyas, S. K., Existence results for first order impulsive semilinear evolution inclusions, Electron. J. Qual. Theory Differ. Equ., 1, 12 (2001), (electronic only) · Zbl 0984.34048
[5] Cardinali, T.; Rubbioni, P., On the existence of mild solutions of semilinear evolution differential inclusions, J. Math. Anal. Appl., 308, 620-635 (2005) · Zbl 1083.34046
[6] Cardinali, T.; Rubbioni, P., Mild solutions for impulsive semilinear evolution differential inclusions, J. Appl. Funct. Anal., 1, 3, 303-325 (2006) · Zbl 1109.34043
[7] Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions, (Lect. Notes in Math., vol. 580 (1977), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York) · Zbl 0346.46038
[8] Chang, Y. K.; Li, W. T.; Nieto, J. J., Controllability of evolution differential inclusions in Banach spaces, Nonlinear Anal., 67, 2, 623-632 (2007) · Zbl 1128.93005
[9] Deimling, K., Multivalued Differential Equations (1992), De Gruyter: De Gruyter Berlin · Zbl 0760.34002
[10] D’Onofrio, A., On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Math. Lett., 18, 729-732 (2005) · Zbl 1064.92041
[11] Franco, D.; Nieto, J. J., First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions, Nonlinear Anal., 42, 163-173 (2000) · Zbl 0966.34025
[12] Gao, S.; Chen, L.; Nieto, J. J.; Torres, A., Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24, 6037-6045 (2006)
[13] Guo, M.; Xue, X.; Li, R., Controllability of impulsive evolution inclusions with nonlocal conditions, J. Optim. Theory Appl., 120, 355-374 (2004) · Zbl 1048.93008
[14] Hu, S.; Papageorgiou, N. S., Handbook of Multivalued Analysis, Vol. I: Theory (1997), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht, Boston, London · Zbl 0887.47001
[15] Kamenskii, M.; Obukhovskii, V.; Zecca, P., (Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl., vol. 7 (2001), Walter de Gruyter: Walter de Gruyter Berlin, New York) · Zbl 0988.34001
[16] Krein, S. G., Linear Differential Equations in Banach Spaces (1971), Amer. Math. Soc.: Amer. Math. Soc. Providence · Zbl 0636.34056
[17] Jiang, G.; Lu, Q., Impulsive state feedback control of a predator-prey model, J. Comput. Appl. Math., 200, 193-207 (2007) · Zbl 1134.49024
[18] Li, J.; Nieto, J. J.; Shen, J., Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl., 325, 226-236 (2007) · Zbl 1110.34019
[19] Li, W.; Huo, H., Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics, J. Comput. Appl. Math., 174, 227-238 (2005) · Zbl 1070.34089
[20] Liu, B., Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Anal., 60, 1533-1552 (2005) · Zbl 1079.93008
[21] Liu, J. H., Nonlinear impulsive evolution equations, Dyn. Contin. Discrete Impuls. Syst., 6, 1, 77-85 (1999) · Zbl 0932.34067
[22] Lou, X. Y.; Cui, B. T., Global asymptotic stability of delay BAM neural networks with impulses, Chaos Solitons Fractals, 29, 1023-1031 (2006) · Zbl 1142.34376
[23] Mil’man, V. D.; Myshkis, A. D., On the solvability of motion in the presence of impulses, Siberian Math. J., 1, 2, 233-237 (1960) · Zbl 1358.34022
[24] Nieto, J. J., Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Anal., 51, 1223-1232 (2002) · Zbl 1015.34010
[25] Nieto, J. J.; Rodriguez-Lopez, R., Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. Math. Anal. Appl., 318, 593-610 (2006) · Zbl 1101.34051
[26] Nieto, J. J.; Rodriguez-Lopez, R., New comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl., 328, 2, 1343-1368 (2007) · Zbl 1113.45007
[27] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0516.47023
[28] Qian, D.; Li, X., Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl., 303, 288-303 (2005) · Zbl 1071.34005
[29] Saker, S. H., Oscillation and global attractivity of impulsive periodic delay respiratory dynamics model, Chinese Ann. Math. Ser. B, 26, 511-522 (2005) · Zbl 1096.34053
[30] Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore, Visca Skola, Kiev, 1987 (in Russian) · Zbl 0837.34003
[31] Sesekin, A. N.; Zavalishchin, S. T., (Dynamic Impulse Systems. Theory and Applications. Dynamic Impulse Systems. Theory and Applications, Mathematics and its Applications, vol. 394 (1997), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht) · Zbl 0880.46031
[32] Tang, S.; Chen, L., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol., 44, 185-199 (2002) · Zbl 0990.92033
[33] Yan, J.; Zhao, A.; Nieto, J. J., Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems, Math. Comput. Modelling, 40, 509-518 (2004) · Zbl 1112.34052
[34] Zeidler, E., Nonlinear Functional Analysis and its Applications II/A (1990), Springer-Verlag: Springer-Verlag New York, Berlin, Heidelberg
[35] Zhang, W.; Fan, M., Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Math. Comput. Modelling, 39, 479-493 (2004) · Zbl 1065.92066
[36] Zhang, X.; Shuai, Z.; Wang, K., Optimal impulsive harvesting policy for single population, Nonlinear Anal. RWA, 4, 639-651 (2003) · Zbl 1011.92052
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