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Three positive doubly periodic solutions of a nonlinear telegraph system

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Abstract

This paper studies existence of at least three positive doubly periodic solutions of a coupled nonlinear telegraph system with doubly periodic boundary conditions. First, by using the Green function and maximum principle, existence of solutions of a nonlinear telegraph system is equivalent to existence of fixed points of an operator. By imposing growth conditions on the nonlinearities, existence of at least three fixed points in cone is obtained by using the Leggett-Williams fixed point theorem to cones in ordered Banach spaces. In other words, there exist at least three positive doubly periodic solutions of nonlinear telegraph system.

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References

  1. Fucik, S. and Mawhin, J. Generated periodic solution of nonlinear telegraph equation. Nonlinear Analysis 2(5), 609–617 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  2. Kim, W. S. Doubly-periodic boundary value problem for nonlinear dissipative hyperbolic equations. J. Math. Anal. Appl. 145(1), 1–6 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kim, W. S. Multiple doubly periodic solutions of semilinear dissipative hyperbolic equations. J. Math. Anal. Appl. 197(2), 735–748 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Mawhin, J. Periodic solution of nonlinear telegraph equations. Dynamical Systems (eds. Bedlarek, A. R. and Cesari, L.), Academic Press, New York (1977)

    Google Scholar 

  5. Ortega, R. and Robles-Perez, A. M. A maximum principle for periodic solutions of the telegraph equations. J. Math. Anal. Appl. 221(2), 625–651 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berkovits, J. and Mustonuen, V. On nonresonance for system of semilinear wave equations. Nonlinear Analysis 29(6), 627–638 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. An, Yukun. Periodic solutions of telegraph-wave coupled system at nonresonance. Nonlinear Analysis 46(4), 525–533 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Li, Y. Positive doubly periodic solutions of nonlinear telegraph equations. Nonlinear Analysis 55(3), 245–254 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Wang, Fanglei and An, Yukun. Nonnegative doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl. 338(1), 91–100 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Davis, J. M., Eloe, P. W., and Henderson, J. Triple positive solutions and dependence on high order derivatives. J. Math. Anal. Appl. 237(2), 710–720 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Sun, Jianping and Li, Wantong. Multiple positive solutions of a discrete difference system. Appl. Math. Comput. 143(2), 213–221 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Leggett, R.W. and Williams, L. R. Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana University Mathematics Journal 28(4), 673–688 (1979)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P. R. China

    Fang-lei Wang  (王方磊) & Yu-kun An  (安玉坤)

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  1. Fang-lei Wang  (王方磊)
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  2. Yu-kun An  (安玉坤)
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Correspondence to Fang-lei Wang  (王方磊).

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Wang, Fl., An, Yk. Three positive doubly periodic solutions of a nonlinear telegraph system. Appl. Math. Mech.-Engl. Ed. 30, 81–88 (2009). https://doi.org/10.1007/s10483-009-0109-z

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  • DOI: https://doi.org/10.1007/s10483-009-0109-z

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