Positivity

pp 1–16 | Cite as

Positive solutions of nonlinear multi-point boundary value problems

Article
  • 16 Downloads

Abstract

This paper deals with the existence of positive solutions of nonlinear differential equation
$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$
subject to the boundary conditions
$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$
where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy   \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality
$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$

Keywords

Differential equation Nonlinear boundary value problems Positive solutions Fixed point theorem 

Mathematics Subject Classification

34B15 34B18 

Notes

Acknowledgements

The author would like to thank the anonymous referees and editor for their helpful comments and suggestions.

References

  1. 1.
    Agarwal, R.P., Bohner, M., Rehak, P.: Half-linear dynamic equations. In: Agarwal, R.P., O’Regan, D. (eds.) Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday, pp. 1–57. Kluwer Academic Publisher, Dordrecht (2003)Google Scholar
  2. 2.
    Aulbach, B., Neidhart, L.: Integration on measure chains. In: Proceedings of the Sixth International Conference on Difference Equations, pp. 239–252. CRC, Boca Raton, FL (2004)Google Scholar
  3. 3.
    Bai, D., Feng, H.: Three positive solutions for positive for \(m\)-point boundary value problems with one-dimensional \(p\)-Laplacian. Electron. J. Differ. Equ. 2011(75), 1–10 (2011)MathSciNetMATHGoogle Scholar
  4. 4.
    Cheung, W.S., Ren, J.: Twin positive solutions for quasi-linear multi-point boundary value problems. Nonlinear Anal. 62, 167–177 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dogan, A.: The existence of positive solutions for a semipositone second-order \(m\)-point boundary value problem. Dyn. Syst. Appl. 24, 419–428 (2015)MathSciNetMATHGoogle Scholar
  6. 6.
    Ehrke, J., Henderson, J., Kunkel, C., Sheng, Q.: Boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations. J. Math. Anal. Appl. 333, 191–203 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Feng, H., Ge, W.: Existence of three positive solutions for \(m\)-point boundary value problems with one dimensional \(p\)-Laplacian. Nonlinear Anal. 68, 2017–2026 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Feng, H., Ge, W.: Triple symmetric positive solutions for multipoint boundary-value problem with one dimensional \(p\)-Laplacian. Math. Comput. Model. 47, 186–195 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Feng, H., Ge, W., Jiang, M.: Multiple positive solutions for \(m\)-point boundary value problems with a one dimensional \(p\)-Laplacian. Nonlinear Anal. 68, 2269–2279 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Graef, J.R., Yang, B.: Positive solutions to a multi-point higher order boundary value problem. J. Math. Anal. Appl. 316, 409–421 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gupta, C.P.: A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. Math. Comput. 89, 133–146 (1998)MathSciNetMATHGoogle Scholar
  12. 12.
    Henderson, J., Luca, R.: Positive solutions for a system of second-order multi-point boundary value problems. Appl. Math. Comput. 218, 6083–6094 (2012)MathSciNetMATHGoogle Scholar
  13. 13.
    Ji, D., Tian, Y., Ge, W.: Positive solutions for one dimensional \(p\)-Laplacian boundary value problems with sign changing nonlinearity. Nonlinear Anal. 71, 5406–5416 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ji, D., Ge, W.: Multiple positive solutions for some \(p\)-Laplacian boundary value problems. Appl. Math. Comput. 187, 1315–1325 (2007)MathSciNetMATHGoogle Scholar
  15. 15.
    Ji, D., Feng, M., Ge, W.: Multiple positive solutions for multipoint boundary value problems with sign changing nonlinearity. Appl. Math. Comput. 196, 511–520 (2008)MathSciNetMATHGoogle Scholar
  16. 16.
    Ji, D., Bai, Z., Ge, W.: The existence of countably many positive solutions for singular multipoint boundary value problems. Nonlinear Anal. 72, 955–964 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kim, C.G.: Existence and iteration of positive solutions for multi-point boundary value problems on a half-line. Comput. Math. Appl. 61, 1898–1905 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Luca, R.: Positive solutions for a higher-order \(m\)-point boundary value problem. Mediterr. J. Math. 9, 379–392 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ma, R.: Positive solutions of a nonlinear \(m\)-point boundary value problem. Comput. Math. Appl. 42, 755–765 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ma, R., Castaneda, N.: Existence of solutions of nonlinear \(m\)-point boundary value problems. J. Math. Anal. Appl. 256, 556–567 (2001)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ma, D.X., Du, Z.J., Ge, W.G.: Existence and iteration of monotone positive solutions for multipoint boundary value problem with \(p\)-Laplacian operator. Comput. Math. Appl. 50, 729–739 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sang, Y., Su, H.: Positive solutions of nonlinear third-order \(m\)-point BVP for an incresing homeomorphism and homomorphism with sign-changing nonlinearity. J. Comput. Appl. Math. 225, 288–300 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sun, B., Ge, W.G., Zhao, D.: Three positive solutions for multipoint one-dimensional \(p\)-Laplacian boundary value problems with dependence on the first order derivative. Math. Comput. Model. 45, 1170–1178 (2007)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sun, B., Zhao, J., Yang, P., Ge, W.G.: Successive iteration and positive solutions for a third-order multipoint generalized right-focal boundary value problem with \(p\)-Laplacian. Nonlinear Anal. 70, 220–230 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wang, Y., Zhao, M., Hu, Y.: Triple positive solutions for a multi-point boundary value problem with a one-dimensional \(p\)-Laplacian. Comput. Math. Appl. 60, 1792–1802 (2010)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Wang, Y., Hou, C.: Existence of multiple positive solutions for one-dimensional \(p\)-Laplacian. J. Math. Anal. Appl. 315, 144–153 (2006)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wang, Y., Ge, W.: Existence of triple positive solutions for multipoint boundary value problems with one-dimensional \(p\)-Laplacian. Comput. Math. Appl. 54, 793–807 (2007)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wang, Y., Ge, W.: Multiple positive solutions for multipoint boundary value problems with one-dimensional \(p\)-Laplacian. J. Math. Anal. Appl. 327, 1381–1395 (2007)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Wang, Y., Ge, W.: Existence of multiple positive solutions for multipoint boundary value problems with a one-dimensional \(p\)-Laplacian. Nonlinear Anal. 67, 476–485 (2007)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wang, Y., Ge, W.: Positive solutions for multipoint boundary value problems with a one-dimensional \(p\)-Laplacian. Nonlinear Anal. 66, 1246–1256 (2007)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Xu, F.: Multiple positive solutions for nonlinear singular \(m\)-point boundary value problem. Appl. Math. Comput. 204, 450–460 (2008)MathSciNetMATHGoogle Scholar
  32. 32.
    Yang, J.: Constant sign solutions for second-order \(m\)-point boundary value problems. Electron. J. Differ. Equ. 2013(65), 1–7 (2013)MathSciNetGoogle Scholar
  33. 33.
    Yang, Y., Xiao, D.: On existence of multiple positive solutions for \( \phi \)-Laplacian multipoint boundary value. Nonlinear Anal. 71, 4158–4166 (2009)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Yang, L., Shen, C.: On the existence of positive solution for a kind of multi-point boundary value problem at resonance. Nonlinear Anal. 72, 4211–4220 (2010)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Yang, Z.: Existence and uniqueness of positive solutions for an integral boundary value problem. Nonlinear Anal. 69, 3910–3918 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Zhang, Y.: Existence and multiplicity results for a class of generalized one-dimensional \(p\)-Laplacian problem. Nonlinear Anal. 72, 748–756 (2010)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Zhou, Y., Cao, Y.: Triple positive solutions of the multi-point boundary value problem for second-order differential equations. J. Math. Res. Expos. 30, 475–486 (2010)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Computer SciencesAbdullah Gul UniversityKayseriTurkey

Personalised recommendations