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PERIODIC BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS

Year 2007, Volume: 2 Issue: 2, 253 - 272, 01.12.2007

Abstract

 We prove existence results for the solutions of the periodic boundary valueproblem concerning the n-th order functional differential equation with impulses effects and the periodic boundary conditions. Our method is based upon the coincidence degree theory of Mawhin and some technicalinequalities. Examples are presented to illustrate the main results.


References

  • BAINOV, D.D., et al.; 1989, Periodic boundary value problems foe systems of first order impulsive equations, Differential and Integral Equations, 2(1), 37-43.
  • BAINOV, D.D.; SIMEONOV, P.S.; 1993, Impulsive differential equations: periodic solutions and applications, Hacloi: Longman Scientific and Technical.
  • CABADA, A.; 1994, The monotone method for first order problems with linear and nonlinear boundary conditions, Applied Mathematics and Computation, 63, 163-186.
  • CABADA, A.; NIETO, J. J.; FRANCO, D.; TROFIMCHUK, S. I.; 2000, A generalization of the monotone method for second order periodic boundary value problems with impulses at fixed points, Dynamics of Continuous Discrete and Impulsive Systems, 7, 145-158.
  • CHEN, L.; SUN, J.; 2006, Boundary value problems of second order impulsive functional differential equations, Journal of Mathematical Analysis and Applications, 323, 708-720.
  • CHU, J.; ZHOU, Z.; 2006, Positive solutions for singular third order periodic boundary value problems, Nonlinear Analysis, 64, 1528-1542.
  • CONG, F.; 1998, Periodic solutions for 2kth order ordinary differential equations with nonresonance, Nonlinear Analysis, 32, 787-793.
  • CONG, F.; HUANG, Q.; SHI, S.; 2000, existence and Uniqueness of Periodic Solutions for (2n + 1)th-Order Differential Equations, Journal of Mathematical Analysis and Applications, 241, 1-9.
  • CONG, F.; 2004, Existence of periodic solutions of (2n + 1)th-order ordinary differential equations, Applied Mathematics Letters, 17, 727-732.
  • DING, W.; HAN, M.; 2004, Periodic boundary value problem for the second order impulsive functional differential equations, Applied Mathematics and Computation, 155, 709-726.
  • DING, W.; HAN, M.; YAN, J.; 2004, Periodic BVPs for the second order functional differential equations, Journal of Mathematical Analysis and Applications, 298, 341-351.
  • DING, W.; HAN, M.; MI, J.; 2005, Periodic boundary value problem for the second-order impulsive functional differential equations, Computers & Mathematics with Applications, 50, 491-507.
  • FRANCO, D.; NIETO, J. J.; 1998, Maximum principles for periodic impulsive first order problems, Journal of Computational and Applied Mathematics, 88, 144-159.
  • GAINES, R. E.; MAWHIN, J. L.; 1977, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math. 568, Springer, Berlin.
  • GUO, D.; 1997, Periodic boundary value problems for second order impulsive integro-differential equations in Banach spaces, Nonlinear Analysis, 28, 983-997.
  • HAKL, R.; LOMTATIDZE, A.; PUZA, B.; 2003, On a boundary value problem for first order scalar functional differential equations, Nonlinear Analysis, 53, 391-405.
  • HE, Z.; YU, J.; 2002, Periodic boundary value problems for first order impulsive ordinary differential equations, Journal of Mathematical Analysis and Applications, 272, 67-78.
  • HU, S.; LAKSHMIKANTHAM, V.; 1989, Periodic boundary value problems for second order impulsive differential systems, Nonlinear Analysis, 13, 75-85.
  • KIGURADZE, I.; STANK, S.; 2002, On periodic boundary value problem for the equation
  • with one-sided growth restrictions on f, Nonlinear Analysis, 48, ( , , ') 1065-1075.
  • KONG, L.; WANG, S.; WANG, J.; 2001, Positive solution of a singular nonlinear third-order periodic boundary value problem, Journal of Computational and Applied Mathematics, 132, 247-253.
  • JIANG, D.; NIETO, J. J.; ZUO, W.; 2004, On monotone method for first order and second order periodic boundary value problems and periodic solutions of functional differential equations, Journal of Mathematical Analysis and Applications, 289, 691-699.
  • JIANG, D.; CHU, J.; ZHANG, M.; 2005, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, Journal of Differential Equations, 211, 282-302.
  • LADDE, G. S.; LAKSHMIKANTHAM, V. ; VATSALA, A. S.; 1985, Monotone iterative techniques for nonlinear differential equations, Pitman Advanced Publishing Program.
  • LAKSHMIKANTHAM, V.; LEELA, S.; 1984, Remarks on first and second order periodic boundary valuue problems, Nonlinear Analysis, 8, 281-287.
  • LI, F.; LI, Y.; LIANG, Z.; 2007, Existence and multiplicity of solutions to 2mth-order ordinary differential equations, Journal of Mathematical Analysis and Applications, 331, 958-977.
  • LI, J.; SHEN, J.; 2006, Periodic boundary value problems for delay differential equations with impulses, Journal of Computational and Applied Mathematics, 193, 563-573.
  • LIANG, R.; SHEN, J.; 2007, Periodic boundary value problem for second-order impulsive functional differential equations, Applied Mathematics and Computation, doi:10.1016/j.amc.2007.03.072.
  • LIU, X.; 1990, Nonlinear boundary value problems for first order impulsive integral-differential equations, Applied Analysis, 36, 119-130.
  • LIU, Y.; GE, W.; 2005, Periodic boundary value problems for n-th order ordinary differential equations with p
  • Laplacian, Journal of Applied Mathematics, 1, 1-21. −
  • LIU, Y.; YANG, P.; GE, W.; 2005, Periodic solutions of higher-order delay differential equations, Nonlinear Analysis, 63, 136-152.
  • NIETO, J. J.; ALVAREZ-NORIEGA, N.; 1996, Periodic boundary value problems for nonlinear first order ordinary differential equations, Acta Mathematica Hungarica, 71, 49-58.
  • NIETO, J. J.; 1997, Basic theory for nonresonance impulsive periodic problems of first order, Journal of Mathematical Analysis and Applications, 205, 423-433.
  • NIETO, J. J.; 2002, Impulsive resonance periodic problems of first order, Applied Mathematics Letters, 15, 489-493.
  • NIETO, J. J.; 2002, Periodic boundary value problems for first order impulsive ordinary differential equations, Nonlinear Analysis, 51, 1223-1232.
  • PIERSON-GOREZ, C.; 1993, Impulsive differential equations of first order with periodic boundary conditions, Differential Equations and Dynamical Systems, 185-196.
  • VATSALA, A. S.; SUN, Y.; 1992, Periodic boundary value problems of impulsive differential equations, Nonlinear Analysis, 44, 145-158.
  • YANG, X.; SHEN, J.; 2007, Periodic boundary value problems for second-order impulsive integro differential equations, Journal of Computational and Applied Mathematics, 209, 176-186.
Year 2007, Volume: 2 Issue: 2, 253 - 272, 01.12.2007

Abstract

References

  • BAINOV, D.D., et al.; 1989, Periodic boundary value problems foe systems of first order impulsive equations, Differential and Integral Equations, 2(1), 37-43.
  • BAINOV, D.D.; SIMEONOV, P.S.; 1993, Impulsive differential equations: periodic solutions and applications, Hacloi: Longman Scientific and Technical.
  • CABADA, A.; 1994, The monotone method for first order problems with linear and nonlinear boundary conditions, Applied Mathematics and Computation, 63, 163-186.
  • CABADA, A.; NIETO, J. J.; FRANCO, D.; TROFIMCHUK, S. I.; 2000, A generalization of the monotone method for second order periodic boundary value problems with impulses at fixed points, Dynamics of Continuous Discrete and Impulsive Systems, 7, 145-158.
  • CHEN, L.; SUN, J.; 2006, Boundary value problems of second order impulsive functional differential equations, Journal of Mathematical Analysis and Applications, 323, 708-720.
  • CHU, J.; ZHOU, Z.; 2006, Positive solutions for singular third order periodic boundary value problems, Nonlinear Analysis, 64, 1528-1542.
  • CONG, F.; 1998, Periodic solutions for 2kth order ordinary differential equations with nonresonance, Nonlinear Analysis, 32, 787-793.
  • CONG, F.; HUANG, Q.; SHI, S.; 2000, existence and Uniqueness of Periodic Solutions for (2n + 1)th-Order Differential Equations, Journal of Mathematical Analysis and Applications, 241, 1-9.
  • CONG, F.; 2004, Existence of periodic solutions of (2n + 1)th-order ordinary differential equations, Applied Mathematics Letters, 17, 727-732.
  • DING, W.; HAN, M.; 2004, Periodic boundary value problem for the second order impulsive functional differential equations, Applied Mathematics and Computation, 155, 709-726.
  • DING, W.; HAN, M.; YAN, J.; 2004, Periodic BVPs for the second order functional differential equations, Journal of Mathematical Analysis and Applications, 298, 341-351.
  • DING, W.; HAN, M.; MI, J.; 2005, Periodic boundary value problem for the second-order impulsive functional differential equations, Computers & Mathematics with Applications, 50, 491-507.
  • FRANCO, D.; NIETO, J. J.; 1998, Maximum principles for periodic impulsive first order problems, Journal of Computational and Applied Mathematics, 88, 144-159.
  • GAINES, R. E.; MAWHIN, J. L.; 1977, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math. 568, Springer, Berlin.
  • GUO, D.; 1997, Periodic boundary value problems for second order impulsive integro-differential equations in Banach spaces, Nonlinear Analysis, 28, 983-997.
  • HAKL, R.; LOMTATIDZE, A.; PUZA, B.; 2003, On a boundary value problem for first order scalar functional differential equations, Nonlinear Analysis, 53, 391-405.
  • HE, Z.; YU, J.; 2002, Periodic boundary value problems for first order impulsive ordinary differential equations, Journal of Mathematical Analysis and Applications, 272, 67-78.
  • HU, S.; LAKSHMIKANTHAM, V.; 1989, Periodic boundary value problems for second order impulsive differential systems, Nonlinear Analysis, 13, 75-85.
  • KIGURADZE, I.; STANK, S.; 2002, On periodic boundary value problem for the equation
  • with one-sided growth restrictions on f, Nonlinear Analysis, 48, ( , , ') 1065-1075.
  • KONG, L.; WANG, S.; WANG, J.; 2001, Positive solution of a singular nonlinear third-order periodic boundary value problem, Journal of Computational and Applied Mathematics, 132, 247-253.
  • JIANG, D.; NIETO, J. J.; ZUO, W.; 2004, On monotone method for first order and second order periodic boundary value problems and periodic solutions of functional differential equations, Journal of Mathematical Analysis and Applications, 289, 691-699.
  • JIANG, D.; CHU, J.; ZHANG, M.; 2005, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, Journal of Differential Equations, 211, 282-302.
  • LADDE, G. S.; LAKSHMIKANTHAM, V. ; VATSALA, A. S.; 1985, Monotone iterative techniques for nonlinear differential equations, Pitman Advanced Publishing Program.
  • LAKSHMIKANTHAM, V.; LEELA, S.; 1984, Remarks on first and second order periodic boundary valuue problems, Nonlinear Analysis, 8, 281-287.
  • LI, F.; LI, Y.; LIANG, Z.; 2007, Existence and multiplicity of solutions to 2mth-order ordinary differential equations, Journal of Mathematical Analysis and Applications, 331, 958-977.
  • LI, J.; SHEN, J.; 2006, Periodic boundary value problems for delay differential equations with impulses, Journal of Computational and Applied Mathematics, 193, 563-573.
  • LIANG, R.; SHEN, J.; 2007, Periodic boundary value problem for second-order impulsive functional differential equations, Applied Mathematics and Computation, doi:10.1016/j.amc.2007.03.072.
  • LIU, X.; 1990, Nonlinear boundary value problems for first order impulsive integral-differential equations, Applied Analysis, 36, 119-130.
  • LIU, Y.; GE, W.; 2005, Periodic boundary value problems for n-th order ordinary differential equations with p
  • Laplacian, Journal of Applied Mathematics, 1, 1-21. −
  • LIU, Y.; YANG, P.; GE, W.; 2005, Periodic solutions of higher-order delay differential equations, Nonlinear Analysis, 63, 136-152.
  • NIETO, J. J.; ALVAREZ-NORIEGA, N.; 1996, Periodic boundary value problems for nonlinear first order ordinary differential equations, Acta Mathematica Hungarica, 71, 49-58.
  • NIETO, J. J.; 1997, Basic theory for nonresonance impulsive periodic problems of first order, Journal of Mathematical Analysis and Applications, 205, 423-433.
  • NIETO, J. J.; 2002, Impulsive resonance periodic problems of first order, Applied Mathematics Letters, 15, 489-493.
  • NIETO, J. J.; 2002, Periodic boundary value problems for first order impulsive ordinary differential equations, Nonlinear Analysis, 51, 1223-1232.
  • PIERSON-GOREZ, C.; 1993, Impulsive differential equations of first order with periodic boundary conditions, Differential Equations and Dynamical Systems, 185-196.
  • VATSALA, A. S.; SUN, Y.; 1992, Periodic boundary value problems of impulsive differential equations, Nonlinear Analysis, 44, 145-158.
  • YANG, X.; SHEN, J.; 2007, Periodic boundary value problems for second-order impulsive integro differential equations, Journal of Computational and Applied Mathematics, 209, 176-186.
There are 39 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Makaleler
Authors

Yuji Lıu This is me

Publication Date December 1, 2007
Published in Issue Year 2007 Volume: 2 Issue: 2

Cite

IEEE Y. Lıu, “PERIODIC BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 2, no. 2, pp. 253–272, 2007, doi: 10.29233/sdufeffd.134631.