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2014 | 12 | 1 | 128-140

Tytuł artykułu

Second order BVPs with state dependent impulses via lower and upper functions

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Języki publikacji

EN

Abstrakty

EN
The paper deals with the following second order Dirichlet boundary value problem with p ∈ ℕ state-dependent impulses: z″(t) = f (t,z(t)) for a.e. t ∈ [0, T], z(0) = z(T) = 0, z′(τ i+) − z′(τ i−) = I i(τ i, z(τ i)), τ i = γ i(z(τ i)), i = 1,..., p. Solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses.

Wydawca

Czasopismo

Rocznik

Tom

12

Numer

1

Strony

128-140

Opis fizyczny

Daty

wydano
2014-01-01
online
2013-10-30

Twórcy

  • Palacký University
autor
  • Palacký University

Bibliografia

  • [1] Afonso S.M., Bonotto E.M., Federson M., Schwabik Š., Discontinuous local semiflows for Kurzweil equations leading to LaSalle’s invariance principle for differential systems with impulses at variable times, J. Differential Equations, 2011, 250(7), 2969–3001 http://dx.doi.org/10.1016/j.jde.2011.01.019
  • [2] Akhmet M.U., On the general problem of stability for impulsive differential equations, J. Math. Anal. Appl., 2003, 288(1), 182–196 http://dx.doi.org/10.1016/j.jmaa.2003.08.001
  • [3] Akhmetov M.U., Zafer A., Stability of the zero solution of impulsive differential equations by the Lyapunov second method, J. Math. Anal. Appl., 2000, 248(1), 69–82 http://dx.doi.org/10.1006/jmaa.2000.6864
  • [4] Bainov D., Simeonov P., Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monogr. Surveys Pure Appl. Math., 66, Longman Scientific & Technical, Harlow, 1993
  • [5] Bajo I., Liz E., Periodic boundary value problem for first order differential equations with impulses at variable times, J. Math. Anal. Appl., 1996, 204(1), 65–73 http://dx.doi.org/10.1006/jmaa.1996.0424
  • [6] Belley J.-M., Virgilio M., Periodic Duffing delay equations with state dependent impulses, J. Math. Anal. Appl., 2005, 306(2), 646–662 http://dx.doi.org/10.1016/j.jmaa.2004.10.023
  • [7] Belley J.-M., Virgilio M., Periodic Liénard-type delay equations with state-dependent impulses, Nonlinear Anal., 2006, 64(3), 568–589 http://dx.doi.org/10.1016/j.na.2005.06.025
  • [8] Benchohra M., Graef J.R., Ntouyas S.K., Ouahab A., Upper and lower solutions method for impulsive differential inclusions with nonlinear boundary conditions and variable times, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 2005, 12(3–4), 383–396
  • [9] Benchohra M., Henderson J., Ntouyas S.K., Ouahab A., Impulsive functional differential equations with variable times, Comput. Math. Appl., 2004, 47(10–11), 1659–1665 http://dx.doi.org/10.1016/j.camwa.2004.06.013
  • [10] Córdova-Lepe F., Pinto M., González-Olivares E., A new class of differential equations with impulses at instants dependent on preceding pulses. Applications to management of renewable resources, Nonlinear Anal. Real World Appl., 2012, 13(5), 2313–2322 http://dx.doi.org/10.1016/j.nonrwa.2012.01.026
  • [11] Devi J.V., Vatsala A.S., Generalized quasilinearization for an impulsive differential equation with variable moments of impulse, Dynam. Systems Appl., 2003, 12(3–4), 369–382
  • [12] Domoshnitsky A., Drakhlin M., Litsyn E., Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments, J. Differential Equations, 2006, 228(1), 39–48 http://dx.doi.org/10.1016/j.jde.2006.05.009
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  • [17] Kaul S., Lakshmikantham V., Leela S., Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times, Nonlinear Anal., 1994, 22(10), 1263–1270 http://dx.doi.org/10.1016/0362-546X(94)90109-0
  • [18] Lakshmikantham V., Bainov D.D., Simeonov P.S., Theory of Impulsive Differential Equations, Ser. Modern Appl. Math., 6, World Scientific, Teaneck, 1989
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  • [20] Li Y., Cong F., Lin Z., Boundary value problems for impulsive differential equations, Nonlinear Anal. TMA, 1997, 29(11), 1253–1264 http://dx.doi.org/10.1016/S0362-546X(96)00177-0
  • [21] Liu L., Sun J., Existence of periodic solution for a harvested system with impulses at variable times, Phys. Lett. A, 2006, 360(1), 105–108 http://dx.doi.org/10.1016/j.physleta.2006.07.080
  • [22] Rachůnková I., Tomeček J., A new approach to BVPs with state-dependent impulses, Bound. Value Probl., 2013, #22
  • [23] Samoilenko A.M., Perestyuk N.A., Impulsive Differential Equations, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 14, World Scientific River Edge, 1995
  • [24] Qi J., Fu X., Existence of limit cycles of impulsive differential equations with impulses at variable times, Nonlinear Anal., 2001, 44(3), 345–353 http://dx.doi.org/10.1016/S0362-546X(99)00268-0
  • [25] Vatsala A.S., Vasundhara Devi J., Generalized monotone technique for an impulsive differential equation with variable moments of impulse, Nonlinear Stud., 2002, 9(3), 319–330

Typ dokumentu

Bibliografia

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bwmeta1.element.doi-10_2478_s11533-013-0324-7
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