Variational approach to a class of impulsive differential equations. (English) Zbl 1319.34047
A periodic boundary value problem for a second-order nonlinear impulsive differential equation is studied. Two types of sufficient conditions for the existence at least one classical solution are obtained by application of a variational method.
Reviewer: Snezhana Hristova (Plovdiv)
MSC:
| 34B37 | Boundary value problems with impulses for ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 58E50 | Applications of variational problems in infinite-dimensional spaces to the sciences |
References:
| [1] | Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989. · Zbl 0719.34002 · doi:10.1142/0906 |
| [2] | Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995. · Zbl 0837.34003 |
| [3] | Benchohra M, Henderson J, Ntouyas SK: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York; 2006. · Zbl 1130.34003 · doi:10.1155/9789775945501 |
| [4] | Agarwal RP, O’Regan D: Multiple nonnegative solutions for second order impulsive differential equations.Appl. Math. Comput. 2000, 114:51-59. · Zbl 1047.34008 · doi:10.1016/S0096-3003(99)00074-0 |
| [5] | Yan B: Boundary value problems on the half line with impulses and infinite delay.J. Math. Anal. Appl. 2001, 259:94-114. · Zbl 1009.34059 · doi:10.1006/jmaa.2000.7392 |
| [6] | Agarwal RP, O’Regan D: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem.Appl. Math. Comput. 2005, 161:433-439. · Zbl 1070.34042 · doi:10.1016/j.amc.2003.12.096 |
| [7] | Kaufmann ER, Kosmatov N, Raffoul YN: A second-order boundary value problem with impulsive effects on an unbounded domain.Nonlinear Anal. 2008, 69:2924-2929. · Zbl 1159.34023 · doi:10.1016/j.na.2007.08.061 |
| [8] | Guo D: Positive solutions of an infinite boundary value problem fornth-order nonlinear impulsive singular integro-differential equations in Banach spaces.Nonlinear Anal. 2009, 70:2078-2090. · Zbl 1159.45002 · doi:10.1016/j.na.2008.02.110 |
| [9] | Guo D: Multiple positive solutions for first order impulsive superlinear integro-differential equations on the half line.Acta Math. Sci. Ser. B 2011,31(3):1167-1178. · Zbl 1240.45017 · doi:10.1016/S0252-9602(11)60307-X |
| [10] | Guo D, Liu X: Multiple positive solutions of boundary value problems for impulsive differential equations.Nonlinear Anal. 1995, 25:327-337. · Zbl 0840.34015 · doi:10.1016/0362-546X(94)00175-H |
| [11] | Guo D: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space.Appl. Math. Comput. 2003, 143:233-249. · Zbl 1030.45009 · doi:10.1016/S0096-3003(02)00356-9 |
| [12] | Guo D: Multiple positive solutions of a boundary value problem fornth-order impulsive integro-differential equations in Banach spaces.Nonlinear Anal. 2005, 63:618-641. · Zbl 1096.34042 · doi:10.1016/j.na.2005.05.023 |
| [13] | Xu X, Wang B, O’Regan D: Multiple solutions for sub-linear impulsive three-point boundary value problems.Appl. Anal. 2008, 87:1053-1066. · Zbl 1168.34021 · doi:10.1080/00036810802429001 |
| [14] | Jankowski J: Existence of positive solutions to second order four-point impulsive differential problems with deviating arguments.Comput. Math. Appl. 2009, 58:805-817. · Zbl 1189.34128 · doi:10.1016/j.camwa.2009.06.001 |
| [15] | Liu Y, O’Regan D: Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations.Commun. Nonlinear Sci. Numer. Simul. 2011, 16:1769-1775. · Zbl 1221.34072 · doi:10.1016/j.cnsns.2010.09.001 |
| [16] | Tian Y, Ge W: Applications of variational methods to boundary value problem for impulsive differential equations.Proc. Edinb. Math. Soc. 2008, 51:509-527. · Zbl 1163.34015 · doi:10.1017/S0013091506001532 |
| [17] | Nieto JJ, O’Regan D: Variational approach to impulsive differential equations.Nonlinear Anal., Real World Appl. 2009, 10:680-690. · Zbl 1167.34318 · doi:10.1016/j.nonrwa.2007.10.022 |
| [18] | Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses.Nonlinear Anal., Real World Appl. 2010, 11:155-162. · Zbl 1191.34039 · doi:10.1016/j.nonrwa.2008.10.044 |
| [19] | Chen H, Sun J: An application of variational method to second-order impulsive differential equations on the half line.Appl. Math. Comput. 2010, 217:1863-1869. · Zbl 1213.34020 · doi:10.1016/j.amc.2010.06.040 |
| [20] | Bai L, Dai B: Existence and multiplicity of solutions for an impulsive boundary value problem with a parameter via critical point theory.Math. Comput. Model. 2011, 53:1844-1855. · Zbl 1219.34039 · doi:10.1016/j.mcm.2011.01.006 |
| [21] | Guo D: A class of second-order impulsive integro-differential equations on unbounded domain in a Banach space.Appl. Math. Comput. 2002, 125:59-77. · Zbl 1030.45011 · doi:10.1016/S0096-3003(00)00115-6 |
| [22] | Krasnoselskii MA: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, Oxford; 1964. · Zbl 0111.30303 |
| [23] | Guo D: The number of nontrivial solutions to Hammerstein nonlinear integral equations.Chin. Ann. Math., Ser. B 1986,7(2):191-204. · Zbl 0596.45010 |
| [24] | Zaanen AC: Linear Analysis. Interscience, New York; 1958. |
| [25] | Ambrosetti A, Rabinowitz PH: Dual variational method in critical point theory and applications.J. Funct. Anal. 1973, 14:349-381. · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7 |
| [26] | Rabinowitz, PH, Variational methods for nonlinear eigenvalue problems (1974) |
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