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The Method of Upper and Lower Solutions to Impulsive Differential Equation with Sturm–Liouville Integral Boundary Conditions

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Abstract

In this paper, we consider a second-order impulsive differential equation with Sturm–Liouville integral boundary conditions. We provide the sufficient conditions that guarantee the existence and multiplicity to our problem. Our technique is based on the method of upper and lower solutions and Leray–Schäuder degree theory. In the end, an example is worked out to illustrate our main results.

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References

  1. Aronson, D., Crandall, M.G., Peletier, L.A.: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal. 6(10), 1001–1022 (1982)

    Article  MathSciNet  Google Scholar 

  2. Choi, Y.S., Ludford, G.S.S.: An unexpected stability result of the near-extincion diffusion flame for non-unity lewis numbers. Q. J. Mech. Appl. Math. 42(1), 143–158 (1989)

    Article  Google Scholar 

  3. Cohen, D.S.: Multiple Stable Solutions of Nonlinear Boundary Value Problems Arising in Chemical Reactor Theory. SIAM J. Appl. Math. 20(1), 1–13 (1971)

    Article  MathSciNet  Google Scholar 

  4. Dancer, E.N.: On the structure of solutions of an equation in catalysis theory when a parameter is large. J. Differ. Equ. 37(3), 404–437 (1983)

    Article  MathSciNet  Google Scholar 

  5. Csavinszky, P.: Universal approximate analytical solution of the Thomas–Fermi equation for ions. Phys. Rev. A 8(4), 1688–1701 (1973)

    Article  Google Scholar 

  6. Lee, M.G., Lin, S.S.: On the positive solutions for semilinear elliptic equations on annular domain with non-homogeneous dirichlet boundary condition. J. Math. Anal. Appl. 181(2), 348–361 (1994)

    Article  MathSciNet  Google Scholar 

  7. Parter, S.V.: Solutions of a differential equation arising in chemical reactor processes. SIAM J. Appl. Math. 26(4), 687–716 (1973)

    Article  MathSciNet  Google Scholar 

  8. Hong, A.H.L., Yeh, C.C.: The existence of positive solutions of Sturm–Liouville boundary value problems. Comput. Math. Appl. 35(9), 89–96 (1998)

    Article  MathSciNet  Google Scholar 

  9. Ge, W., Ren, J.: New existence theorems of positive solutions for Sturm–Liouville boundary value problems. Appl. Math. Comput. 148(3), 631–644 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Sun, J., Zhang, G.: Nontrivial solutions of singular sublinear Sturm–Liouville problems. J. Math. Anal. Appl. 326(1), 242–251 (2007)

    Article  MathSciNet  Google Scholar 

  11. Hu, L., Liu, L., Wu, Y.: Positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive differential equations. Appl. Math. Comput. 196(2), 550–562 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Benmezaï, A.: On the number of solutions of two classes of Sturm–Liouville boundary value problems. Nonlinear Anal. 70(4), 1504–519 (2009)

    Article  MathSciNet  Google Scholar 

  13. Cheng, X., Dai, G.: Positive solutions of sub-superlinear Sturm–Liouville problems. Appl. Math. Comput. 261, 351–359 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Yang, G.C., Feng, H.B.: New results of positive solutions for the Sturm–Liouville problem. Bound. Value Probl. 2016(1), 64 (2016)

    Article  MathSciNet  Google Scholar 

  15. Agarwal, R.P., Wong, P.J.Y.: Positive solutions of higher-order Sturm–Liouville boundary value problems with derivative-dependent nonlinear terms. Bound. Value Probl. 2016, 112 (2016)

    Article  MathSciNet  Google Scholar 

  16. Ding, W., Wang, Y.: New result for a class of impulsive differential equation with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 18(5), 1095–1105 (2013)

    Article  MathSciNet  Google Scholar 

  17. Infante, G., Pietramala, P., Zima, M.: Positive solutions for a class of nonlocal impulsive BVPs via fixed point index. Topol. Methods Nonlinear Anal. 36(2), 263–284 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Zhang, X., Feng, M.: Transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations. J. Comput. Appl. Math. 271(271), 117–129 (2014)

    Article  MathSciNet  Google Scholar 

  19. Chen, X., Du, Z.: Existence of positive periodic solutions for a neutral delay Predator–Prey Model with Hassell–Varley type functional response and impulse. Qual. Theory Dyn. Syst. 17(3), 1–14 (2017)

    MathSciNet  Google Scholar 

  20. Liang, R., Liu, Z.: Nagumo type existence results of Sturm–Liouville BVP for impulsive differential equations. Nonlinear Anal. 74(17), 6676–6685 (2011)

    Article  MathSciNet  Google Scholar 

  21. Zhao, J., Sun, B., Wang, Y.: Existence and iterative solutions of a new kind of Sturm–Liouville-type boundary value problem with one-dimensional p-Laplacian. Bound. Value Probl. 2016(1), 1–11 (2016)

    Article  MathSciNet  Google Scholar 

  22. Cannon, J.R.: The solution of the heat equation subject to specification of energy. Q. Appl. Math. 21, 155–160 (1963)

    Article  MathSciNet  Google Scholar 

  23. Ionkin, N.I.: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differ. Equ. 13, 294–304 (1997)

    Google Scholar 

  24. Chegis, R.Y.: Numerical solution of a heat conduction problem with an integral condition. Litov. Mat. Sb. 24(4), 209–15 (1984)

    MathSciNet  MATH  Google Scholar 

  25. Zhang, Y.: A multiplicity result for a singular generalized Sturm–Liouville boundary value problem. Math. Comput. Model. 50, 132–140 (2009)

    Article  MathSciNet  Google Scholar 

  26. Zhang, L., Xuan, Z.: Multiple positive solutions for a second-order boundary value problem with integral boundary conditions. Bound. Value Probl. 60(1), 2718–2727 (2016)

    MathSciNet  Google Scholar 

  27. Agarwal, R.P., ORegan, D.: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 114(1), 51–59 (2000)

    MathSciNet  Google Scholar 

  28. Lee, E.K., Lee, Y.H.: Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput. 158(3), 745–759 (2004)

    MathSciNet  MATH  Google Scholar 

  29. Wang, M.X., Cabada, A., Nieto, J.J.: Monotone method for nonlinear second order periodic boundary value problems with Caratheodory functions. Annales Polonici Mathematici 58, 221–235 (1993)

    Article  MathSciNet  Google Scholar 

  30. Cui, M., Zhu, Y., Pang, H.: Existence and uniqueness results for a coupled fractional order systems with the multi-strip and multi-point mixed boundary conditions. Adv. Differ. Equ. 1, 224–246 (2017)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Science, China Agricultural University, Beijing, 100083, People’s Republic of China

    Huihui Pang, Yuke Zhu & Mengyan Cui

Authors
  1. Huihui Pang
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  2. Yuke Zhu
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  3. Mengyan Cui
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Correspondence to Huihui Pang.

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The authors declare that there is no conflict of interests regarding the publication of this paper.

Additional information

This work was supported by Chinese Universities Scientific Fund (Project no. 2017LX003) and National Training Program of Innovation (Project no. 201710019252)

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Pang, H., Zhu, Y. & Cui, M. The Method of Upper and Lower Solutions to Impulsive Differential Equation with Sturm–Liouville Integral Boundary Conditions. Differ Equ Dyn Syst 30, 335–351 (2022). https://doi.org/10.1007/s12591-018-0428-4

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