Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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EXISTENCE RESULTS FOR NONLINEAR FIRST-ORDER PERIODIC BOUNDARY VALUE PROBLEM OF IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES

  • Guan, Wen (Department of Mathematics, Lanzhou University of Technology) ;
  • Wang, Da-Bin (Department of Mathematics, Lanzhou University of Technology)
  • Received : 2009.09.24
  • Accepted : 2009.12.12
  • Published : 2010.05.30
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Abstract

In this paper, existence criteria of one solution to a nonlinear first-order periodic boundary value problem of impulsive dynamic equation on time scales are obtained by using the well-known Schaefer fixed-point theorem.

Keywords

References

  1. R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its, applications, Results Math. 35(1999), 3-22. https://doi.org/10.1007/BF03322019
  2. R. P. Agarwal and D. O'Regan, Multiple nonnegative solutions for second order impulsive differential equations, Appl. Math, Comput. 114(2000), 51-59. https://doi.org/10.1016/S0096-3003(99)00074-0
  3. D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability Theory and Applications, Horwood, Chichester, 1989.
  4. D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, Harlow, 1993.
  5. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001.
  6. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
  7. M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, On first order impulsive dynamic equations on time scales, J. Difference Equ. Appl. 6(2004), 541-548.
  8. M. Benchohra, S. K. Ntouyas and A, Ouahab, Existence results for second order bounary value problem of impulsive dynamic equations on time scales, J. Math. Anal, Appl. 296(2004), 65-73. https://doi.org/10.1016/j.jmaa.2004.02.057
  9. A. Cabada, Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales, J. Math. Anal. Appl. 290(2004, )35-54. https://doi.org/10.1016/j.jmaa.2003.08.018
  10. J. H. Chen, C. C. Tisdell and R. Yuan, On the solvability of periodic boundary .value problems with impulse, J. Math. Anal. Appl. 331(2007), 902-912. https://doi.org/10.1016/j.jmaa.2006.09.021
  11. F. J. Geng, D. M. Zhu and Q. Y. Lu, A new existence result for impulsive dynamic equations on timescales, Appl. Math. Lett. 20(2007), 206-212. https://doi.org/10.1016/j.aml.2006.03.013
  12. F. Geng, Y. Xu and D. Zhu, Periodic boundary value problems for first-order impulsive dynamic equations on time scales, Nonlinear Anal. 69(2008), 4074-4087. https://doi.org/10.1016/j.na.2007.10.038
  13. J. R. Graef and A. Ouahab, Extremal solutions for nonresonance impulsive functional dynamic equations on time scales, Appl. Math. Comput. 196(2008), 333-339. https://doi.org/10.1016/j.amc.2007.05.056
  14. J. Henderson, Double solutions of impulsive dynamic boundary value problems on a time scale, J. Difference Equ. Appl. 8(2002), 345-356. https://doi.org/10.1080/1026190290017405
  15. S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18(1990), 18-56. https://doi.org/10.1007/BF03323153
  16. Z. M. He and J. S. Yu, Periodic boundary value problem for first-order impulsive junctional differential equations, J. Comput. Appl. Math. 138(2002), 205-217. https://doi.org/10.1016/S0377-0427(01)00381-8
  17. Z. M. He and X. M. Zhang, Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions, Appl. Math. Comput. 156(2004), 605-620. https://doi.org/10.1016/j.amc.2003.08.013
  18. B. Kaymakcalan, V. Lakshmikantham and S. Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996.
  19. N. G. Lloyd, Degree Theory, Cambridge University Press, Cambridge, 1978.
  20. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  21. J. L. Li, J. J. Nieto and J. H. Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl. 325(2007), 226-236. https://doi.org/10.1016/j.jmaa.2005.04.005
  22. J. L. Li and J. H. Shen, Positive solutions for first-order difference equation with impulses, Int. J. Differ. Equ. 2(2006), 225-239.
  23. J. L. Li and J. H. Shen, Existence results for second-order impulsive boundary value problems on time scales, Nonlinear Anal. 70(2009), 1648-1655. https://doi.org/10.1016/j.na.2008.02.047
  24. X. Liu, Nonlinear boundary value problems for first order impulsive integro-differential equations, Appl. Anal. 36(1990), 119-130. https://doi.org/10.1080/00036819008839925
  25. J. J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl. .205(1997), 423-433. https://doi.org/10.1006/jmaa.1997.5207
  26. J. J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Lett. 15(2002), 489-493. https://doi.org/10.1016/S0893-9659(01)00163-X
  27. J. J. Nieto, Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Anal. 51(2002), 1223-1232. https://doi.org/10.1016/S0362-546X(01)00889-6
  28. J. P. Sun and W. T. Li, Existence of solutions to nonlinear first-order PBVPs on time scales, Nonlinear Anal. 67(2007), 883-888. https://doi.org/10.1016/j.na.2006.06.046
  29. D. B. Wang, Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales, Comput. Math. Appl. 56(2008), 1496-1504. https://doi.org/10.1016/j.camwa.2008.02.038