Positivity

, Volume 22, Issue 5, pp 1387–1402

# Positive solutions of nonlinear multi-point boundary value problems

Article

## 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

34B15 34B18

## 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)
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)
4. 4.
Cheung, W.S., Ren, J.: Twin positive solutions for quasi-linear multi-point boundary value problems. Nonlinear Anal. 62, 167–177 (2005)
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)
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)
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)
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)
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)
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)
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)
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)
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)
14. 14.
Ji, D., Ge, W.: Multiple positive solutions for some $$p$$-Laplacian boundary value problems. Appl. Math. Comput. 187, 1315–1325 (2007)
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)
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)
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)
18. 18.
Luca, R.: Positive solutions for a higher-order $$m$$-point boundary value problem. Mediterr. J. Math. 9, 379–392 (2012)
19. 19.
Ma, R.: Positive solutions of a nonlinear $$m$$-point boundary value problem. Comput. Math. Appl. 42, 755–765 (2001)
20. 20.
Ma, R., Castaneda, N.: Existence of solutions of nonlinear $$m$$-point boundary value problems. J. Math. Anal. Appl. 256, 556–567 (2001)
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)
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)
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)
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)
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)
26. 26.
Wang, Y., Hou, C.: Existence of multiple positive solutions for one-dimensional $$p$$-Laplacian. J. Math. Anal. Appl. 315, 144–153 (2006)
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)
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)
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)
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)
31. 31.
Xu, F.: Multiple positive solutions for nonlinear singular $$m$$-point boundary value problem. Appl. Math. Comput. 204, 450–460 (2008)
32. 32.
Yang, J.: Constant sign solutions for second-order $$m$$-point boundary value problems. Electron. J. Differ. Equ. 2013(65), 1–7 (2013)
33. 33.
Yang, Y., Xiao, D.: On existence of multiple positive solutions for $$\phi$$-Laplacian multipoint boundary value. Nonlinear Anal. 71, 4158–4166 (2009)
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)
35. 35.
Yang, Z.: Existence and uniqueness of positive solutions for an integral boundary value problem. Nonlinear Anal. 69, 3910–3918 (2008)
36. 36.
Zhang, Y.: Existence and multiplicity results for a class of generalized one-dimensional $$p$$-Laplacian problem. Nonlinear Anal. 72, 748–756 (2010)
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)