×

On impulsive hyperbolic differential inclusions with nonlocal initial conditions. (English) Zbl 1159.49042

Summary: This paper is focused mainly upon existence of solutions for a second-order impulsive hyperbolic differential inclusions with nonlocal initial conditions. By using some well-known fixed-point theorems, existence theorems are established when the multivalued map has convex or nonconvex values. As applications of these main theorems, some consequences are given for the sublinear growth cases.

MSC:

49N25 Impulsive optimal control problems
49J20 Existence theories for optimal control problems involving partial differential equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Benchora, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions. Hindawi, New York (2006)
[2] Haddad, W.M., Chellabonia, V., Nersesov, S.G., Sergey, G.: Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton University Press, Princeton (2006)
[3] Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) · Zbl 0719.34002
[4] Belarbi, A., Benchora, M.: Existence theory for perturbed impulsive hyperbolic differential inclusions with variable times. J. Math. Anal. Appl. 327, 1116–1129 (2007) · Zbl 1122.35148
[5] Benchora, M., Gōrniewicz, L., Ntouyas, S.K., Ouahab, A.: Existence results for impulsive hyperbolic differential inclusions. Appl. Anal. 82, 1085–1097 (2003) · Zbl 1047.34071
[6] Chang, Y.K., Li, W.T.: Existence results for second order impulsive functional differential inclusions. J. Math. Anal. Appl. 301, 477–490 (2005) · Zbl 1067.34083
[7] Chang, Y.K., Qi, L.M.: Existence results for second order impulsive functional differential inclusions. J. Appl. Math. Stoch. Anal. 2006, 1–12 (2006) · Zbl 1118.34078
[8] Chang, Y.K., Li, W.T.: Existence results for impulsive dynamic equations on time scales with nonlocal initial conditions. Math. Comput. Model. 43, 377–384 (2006) · Zbl 1134.39014
[9] Migorski, S., Ochal, A.: Nonlinear impulsive evolution inclusions of second order. Dyn. Syst. Appl. 16, 155–173 (2007) · Zbl 1128.34038
[10] Nieto, J.J.: Impulsive resonance periodic problems of first order. Appl. Math. Lett. 15, 489–493 (2002) · Zbl 1022.34025
[11] 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
[12] Nieto, J.J., Rodriguez-Lopez, R.: New comparison results for impulsive integro-differential equations and applications. J. Math. Anal. Appl. 328, 1343–1368 (2007) · Zbl 1113.45007
[13] Nieto, J.J., O’regan, D.: Variational approach to impulsive differential equations, Nonlinear Anal. doi: 10.1016/j.nonrwa.2007.10.022
[14] Li, S.: Estimation of coefficients in a hyperbolic equation with impulsive inputs. J. Inverse Ill-Posed Probl. 14, 891–904 (2006) · Zbl 1111.35123
[15] Sun, J., Zhang, Y.: Impulsive control of a nuclear spin generator. J. Comput. Appl. Math. 157, 235–242 (2003) · Zbl 1051.93086
[16] Liu, X., Willms, A.R.: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng. 2, 277–299 (1996) · Zbl 0876.93014
[17] Miele, A., Weeks, M.W., Ciarcia, M.: Optimal trajectories for spacecraft rendezvous. J. Optim. Theory Appl. 132, 353–376 (2007) · Zbl 1176.70036
[18] Miele, A., Ciarcia, M., Weeks, M.W.: Guidance trajectories for spacecraft rendezvous. J. Optim. Theory Appl. 132, 377–400 (2007) · Zbl 1176.70035
[19] Cui, B., Han, M.: Oscillation theorems for nonlinear hyperbolic systems with impulses. Nonlinear Anal. 98, 94–102 (2008) · Zbl 1135.35091
[20] Erbe, L.H., Freedman, H.I., Liu, X., Wu, J.H.: Comparison principles for impulsive parabolic equations with applications to models of single species growth. Aust. Math. Soc. J. Ser. B. 32, 382–400 (1991) · Zbl 0881.35006
[21] 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. Model. 40, 509–518 (2004) · Zbl 1112.34052
[22] Zhang, H., Chen, L., Nieto, J.J.: A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Anal. doi: 10.1016/j.nonrwa.2007.05.004 · Zbl 1154.34394
[23] 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)
[24] Sun, J., Qiao, F., Wu, Q.: Impulsive control of a financial model. Phys. Lett. A 335, 282–288 (2005) · Zbl 1123.91325
[25] Company, R., Jodar, L., Rubio, G., Villanueva, R.J.: Explicit solution of Black-Scholes option pricing mathematical models with an impulsive payoff function. Math. Comput. Model. 45, 80–92 (2007) · Zbl 1135.91018
[26] Belarbi, A., Benchohra, M.: Existence theory for perturbed hyperbolic differential inclusions. Electron. J. Differ. Equ. 2006, 1–11 (2006) · Zbl 1122.35078
[27] Benchohra, M., Ntouyas, S.K.: The method of lower and upper solutions to the Darboux problem for partial differential inclusions. Miskolc Math. Notes 4, 81–88 (2003) · Zbl 1049.35133
[28] Henderson, J., Ouahab, A.: Impulsive hyperbolic differential inclusions with infinite delay and variable moments. Commun. Appl. Nonlinear Anal. 13, 61–78 (2006) · Zbl 1125.35108
[29] Bohnenblust, H.F., Karlin, S.: On a theorem of Ville. In: Contributions to the Theory of Games, vol. 1, pp. 155–160. Princeton University Press, Princeton (1950) · Zbl 0041.25701
[30] Dugundij, J., Granas, A.: Fixed Point Theory. Monografie Mat. PWN, Warsaw (1982)
[31] Bressan, A., Colombo, G.: Existence and selections of maps with decomposable values. Studia Math. 90, 69–86 (1988) · Zbl 0677.54013
[32] Deimling, K.: Multivalued Differential Equations. De Gruyter, Berlin (1992) · Zbl 0760.34002
[33] Hu, S., Papageorgiou, N.: Handbook of Multivalued Analysis. Kluwer, Dordrecht/Boston (1997) · Zbl 0887.47001
[34] Lasota, A., Opial, Z.: An application of the Kakutani–Ky Fan theorem in the theory of ordinary differential equations. Acad. Polonaise Sci. Sér. Sci. Math. Astron. Phys. 13, 781–786 (1965) · Zbl 0151.10703
[35] Nieto, J.J.: Basic theory for nonresonance impulsive periodic problems of first order. J. Math. Anal. Appl. 205, 423–433 (1997) · Zbl 0870.34009
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.