Abstract
The paper deals with the boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses of the form
Here, T > 0, ϕ : ℝ → ℝ is an increasing homeomorphism, ϕ(ℝ) = ℝ, ϕ(0) = 0, f : [0, T] × ℝ2 → ℝ satisfies Carathéodory conditions, M : ℝ → ℝ is continuous and γ : ℝ → (0, T) is continuous, Δ z′(t) = z′(t+) − z′(t−). Sufficient conditions for the existence of at least one solution to this problem having no pulsation behaviour are provided.
References
[1] Agarwal, R. P.—O’Regan, D.: Multiple nonnegative solutions for second order impulsive differential equations, Appl. Math. Comput. 114 (2000), 51–59.10.1016/S0096-3003(99)00074-0Search in Google Scholar
[2] Azbelev, N. V.—Maksimov, V. P.—Rakhmatullina, L. F.: Introduction to the Theory of Functional Differential Equations: Methods and Applications. Contemp. Math. Appl. 3, Hindawi Publ. Corp., New York, 2007.10.1155/9789775945495Search in Google Scholar
[3] Bai, L.—Dai, B.: An application of variational method to a class of Dirichlet boundary value problems with impulsive effects, J. Franklin Inst. 348 (2011), 2607–2624.10.1016/j.jfranklin.2011.08.003Search in Google Scholar
[4] Bai, L.—Dai, B.: Three solutions for a p-Laplacian boundary value problem with impulsive effects, Appl. Math. Comput. 217 (2011), 9895–9904.10.1016/j.amc.2011.03.097Search in Google Scholar
[5] Bainov, D. D.—Simeonov, P. S.: Impulsive Differential Equations: Periodic Solutions and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics 66, Longman Scientific and Technical, Essex, England, 1993.Search in Google Scholar
[6] Bajo, I.—Liz, E.: Periodic boundary value problem for first order differential equations with impulses at variable times, J. Math. Anal. Appl. 204 (1996), 65–73.10.1006/jmaa.1996.0424Search in Google Scholar
[7] Belley, J. M.—Virgilio, M: Periodic Duffing delay equations with state dependent impulses, J. Math. Anal. Appl. 306 (2005), 646–662.10.1016/j.jmaa.2004.10.023Search in Google Scholar
[8] Belley, J. M.—Virgilio, M: Periodic Liénard-type delay equations with state-dependent impulses, Nonlinear Anal. 64 (2006), 568–589.10.1016/j.na.2005.06.025Search in Google Scholar
[9] 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. 12 (2005), 383–396.Search in Google Scholar
[10] Cabada, A.—Liz, E.—Lois, S.: Green’s function and maximum principle for higher order ordinary differential equations with impulses, Rocky Mountain J. Math. 30 (2000), 435–444.10.1216/rmjm/1022009274Search in Google Scholar
[11] Cabada, A.—Thompson, B.: Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions, Nonlinear Anal. 74 (2011), 7198–7209.10.1016/j.na.2011.07.047Search in Google Scholar
[12] Cabada, A.—Tomeček, J.: Extremal solutions for nonlinear functional ϕ-Laplacian impulsive equations, Nonlinear Anal. 67 (2007), 827–841.10.1016/j.na.2006.06.043Search in Google Scholar
[13] Chu, J.—Nieto, J. J.: Impulsive periodic solutions of first-order singular differential equations, Bull. London Math. Soc. 40 (2008), 143–150.10.1112/blms/bdm110Search in Google Scholar
[14] Feng, M.—Du, B.—Ge, W: Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian, Nonlinear Anal. 70 (2009), 3119–3126.10.1016/j.na.2008.04.015Search in Google Scholar
[15] Ferrara, M.—Heidarkhani, S.: Multiple solutions for perturbed p-Laplacian boundary-value problems with impulsive effects, Electron. J. Differential Equations 2014 (2014), 1–14.10.1155/2014/485647Search in Google Scholar
[16] Frigon, M.—O’Regan, D.: First order impulsive initial and periodic problems with variable moments, J. Math. Anal. Appl. 233 (1999), 730–739.10.1006/jmaa.1999.6336Search in Google Scholar
[17] Frigon, M.—O’Regan, D.: Second order Sturm-Liouville BVP’s with impulses at variable moments, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 8 (2001), 149–159.Search in Google Scholar
[18] Galewski, M.: On variational impulsive boundary value problems, Cent. Eur. J. Math. 10 (2012), 1969–1980.10.2478/s11533-012-0084-9Search in Google Scholar
[19] Li, J.—Nieto, J. J.—Shen, J.: Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl. 325 (2007), 226–236.10.1016/j.jmaa.2005.04.005Search in Google Scholar
[20] Li, P.—Wu, Y.: Triple positive solutions for nth-order impulsive differential equations with integral boundary conditions and p-Laplacian. Results Math. 61 (2012), 401–419.10.1007/s00025-011-0125-xSearch in Google Scholar
[21] Liang, S.—Zhang, J: The existence of countably many positive solutions for some nonlinear singular three-point impulsive boundary value problems, Nonlinear Anal. 71 (2009), 4588–4597.10.1016/j.na.2009.03.016Search in Google Scholar
[22] Nieto, J. J.: Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl. 205 (1997), 423–433.10.1006/jmaa.1997.5207Search in Google Scholar
[23] Nieto, J. J.—O’Regan, D.: Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl. 10 (2009), 680–690.10.1016/j.nonrwa.2007.10.022Search in Google Scholar
[24] Polášek, V.: Periodic BVP with ϕ-Laplacian and impulses, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 44 (2005), 131–150.Search in Google Scholar
[25] Rachůnková, I.—Rachůnek, L.: First-order nonlinear differential equations with state-dependent impulses, Bound. Value Probl. 2013 (2013), 1–18. (Article ID 195)10.1186/1687-2770-2013-1Search in Google Scholar
[26] Rachůnková, I.—Tomeček, J.: Existence principle for BVPs with state-dependent impulses, Topol. Methods Nonlinear Anal. 44 (2014), 349–368.10.12775/TMNA.2014.050Search in Google Scholar
[27] Rachůnková, I.—Tomeček, J.: A new approach to BVPs with state-dependent impulses, Bound. Value Probl. 2013 (2013), 1–13. (Article ID 22)10.1186/1687-2770-2013-22Search in Google Scholar
[28] Rachůnková, I.—Tomeček, J.: Existence principle for higher order nonlinear differential equations with state-dependent impulses via fixed point theorem, Bound. Value Probl. 2014 (2014), 1–15. (Article ID 118)10.1186/1687-2770-2014-118Search in Google Scholar
[29] Rachůnková, I.—Tomeček, J.: Fixed point problem associated with state-dependent impulsive boundary value problems, Bound. Value Probl. 2014 (2014) 1–17. (Article ID 172)10.1063/1.4912444Search in Google Scholar
[30] Rachůnková, I.—Tomeček, J.: Second order BVPs with state dependent impulses via lower and upper functions, Cent. Eur. J. Math. 12 (2014), 128–140.10.2478/s11533-013-0324-7Search in Google Scholar
[31] Rachůnková, I.—Tvrdý, M.: Second-order periodic problem with ϕ-Laplacian and impulses, Nonlinear Anal. 63 (2005), e257–e266.10.1016/j.na.2004.09.017Search in Google Scholar
[32] Samoilenko, A. M.—Perestyuk, N. A.: Impulsive Differential Equations, World Scientific, Singapore, 1995.10.1142/2892Search in Google Scholar
[33] Teng, K.—Zhang, C.: Existence of solution to boundary value problem for impulsive differential equations, Nonlinear Anal. Real World Appl. 11 (2010), 4431–4441.10.1016/j.nonrwa.2010.05.026Search in Google Scholar
[34] Tian, Y.—Ge, W.: Multiple positive solutions of second-order Sturm-Liouville boundary value problems for impulsive differential equations, Rocky Mountain J. Math. 40 (2010), 643–672.10.1216/RMJ-2010-40-2-643Search in Google Scholar
[35] Wang, L.—Ge, W.: Infinitely many solutions of a second-order p-Laplacian problem with impulsive conditions, Appl. Math. 55 (2010), 405–418.10.1007/s10492-010-0015-7Search in Google Scholar
[36] Wang, L.—Pei, M.: Infinitely many solutions of a Sturm-Liouville system with impulses, J. Appl. Math. Comput. 35 (2011), 577–593.10.1007/s12190-010-0379-6Search in Google Scholar
[37] Xu, J.—Kang, P.—Wei, Z.: Singular multipoint impulsive boundary value problem with p-Laplacian operator, J. Appl. Math. Comput. 30 (2009), 105–120.10.1007/s12190-008-0160-2Search in Google Scholar
[38] Zhang, D.—Dai, B.: Existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions, Math. Comput. Modelling 53 (2011), 1154–1161.10.1016/j.mcm.2010.11.082Search in Google Scholar
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