On the existence of solutions for nonlinear impulsive periodic viable problems
Tóm tắt
In this paper we prove the existence of periodic solutions for nonlinear impulsive viable problems monitored by differential inclusions of the type x′(t)∈F(t,x(t))+G(t,x(t)). Our existence theorems extend, in a broad sense, some propositions proved in [10] and improve a result due to Hristova-Bainov in [13].
Tài liệu tham khảo
D.D. Bainov, V. Covachev: Impulsive differential equations with a small parameter, World Scientific, Series on Advances in Math. for Applied sciences, 1994.
D.D. Bainov, P.S. Simeonov: Systems with impulsive effect. Stability, theory and applications, Ellis Horwood Series in Maths and Appl., Ellis Horwood, Chicester, 1989.
D.D. Bainov, P.S. Simeonov: Impulsive differential equations. Asymptotic properties of the solutions, World Scientific, Series on Advances in Math. for Applied Sciences, 1995.
M. Benchohra, A. Boucherif: “Initial value problems for impulsive differential inclusions of first order”, Diff. Eqns. Dyn. Syst., Vol. 8, (2000), pp. 51–66.
M. Benchohra, A. Boucherif: “An existence result for first order initial value problems for impulsive differential inclusions in Banach spaces”, Arch. Math., Vol. 36, (2000), pp. 159–169.
M. Benchohra, A. Boucherif, J.J. Nieto: “On initial value problems for a class of first order impulsive differential inclusions”, Disc. Math. Diff. Incl. Control Optim., Vol. 21, (2001), pp. 159–171.
M. Benchohra, J. Henderson, S.K. Ntouyas: “On a periodic boundary value problem for first order impulsive differential inclusions”, Dyn. Syst. Appl., Vol. 10, (2001), pp. 477–488.
M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahabi: “Existence results for impulsive lower semicontinuous differential inclusions”, Int. J. Pure Appl. Math., Vol. 1, (2002), pp. 431–443.
M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahabi: “Existence results for impulsive functional and neutral functional differential inclusions with lower semicontinuous right hand side”, Electron. J. Math. Phys. Sci., Vol. 1, (2002), pp. 72–91.
T. Cardinali, R. Servadei: “Periodic solutions of nonlinear impulsive differential inclusions with constraints”, Proc. AMS, Vol. 132, (2004), pp. 2339–2349.
B.C. Dhage, A. Boucherif, S.K. Ntouyas: “On periodic boundary value problems of first-order perturbed impulsive differential inclusions”, Electron. J. Diff. Eqns., Vol. 84, (2004), pp. 1–9.
M. Frigon, D. O’Regan: “Existence results for first order impulsive differential equations”, J. Math. Anal. Appl., Vol. 193, (1995), pp. 96–113.
S.G. Hristova, D.D. Bainov: “Existence of periodic solutions of nonlinear systems of differential equations with impulse effect”, J. Math. Anal. Appl., Vol. 125, (1987), pp. 192–202.
S. Hu, N.S. Papageorgiou: Handbook of multivalued analysis, Kluwer, Dordrecht, The Netherlands, 1997.
V. Lakshmikantham, D.D. Bainov, P.S. Simeonov: Theory of impulsive differential equations, World Scientific, Singapore, 1989.
V.D. Mil’man, A.D. Myshkis: “On the stability of motion in the presence of impulses”, Siberian Math. J., Vol. 1, (1960), pp. 233–237.
A.M. Samoilenko, N.A. Perestyuk: “Differential equations with impulse effect”, Visca Skola, Kiev, 1987. [in Russian]
P.J. Watson: “Impulsive differential inclusions”, Nonlin. World, Vol. 4, (1997), pp. 395–402.