DOI QR코드

DOI QR Code

THE EXISTENCE OF TWO POSITIVE SOLUTIONS FOR $m$-POINT BOUNDARY VALUE PROBLEM WITH SIGN CHANGING NONLINEARITY

  • Liu, Jian (School of Mathematical Sciences, Qufu Normal University)
  • Received : 2011.08.09
  • Accepted : 2011.10.10
  • Published : 2012.05.30

Abstract

In this paper, the existence theorem of two positive solutions is established for nonlinear m-point boundary value problem by using an inequality for the following third-order differential equations $$({\phi}(u^{\prime\prime}))^{\prime}+a(t)f(t,u(t))=0,\;t{\in}(0,1)$$, $${\phi}(u^{\prime\prime}(0))=\sum^{m-2}_{i=1}a_i{\phi}(u^{\prime\prime}({\xi}_i)),\;u^{\prime}(1)=0,\;u(0)=\sum^{m-2}_{i=1}b_iu({\xi}_i)$$, where ${\phi}:R{\rightarrow}R$ is an increasing homeomorphism and homomorphism and $\phi(0)=0$. The nonlinear term f may change sign, as an application, an example to demonstrate our results is given.

Keywords

References

  1. D.R.Anderson, Green's function for a third-order generalized right focal problem,J.Math. Anal. Appl. 288 (2003) 1-14. https://doi.org/10.1016/S0022-247X(03)00132-X
  2. Y.P.Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306 (2005) 589-603. https://doi.org/10.1016/j.jmaa.2004.10.029
  3. C.L.Zhou, D.X.Ma, Existence and iteration of positive solutions for a generalized rightfocal boundary value problem with p-Laplacian operator, J. Math. Anal. Appl. 324 (2006) 409-424. https://doi.org/10.1016/j.jmaa.2005.10.086
  4. D.Ji, M.Feng, W.Ge, Multiple positive solutions for multipoint boundary value problems with sign changing nonlnearity, Appl. Math. Comput. 196 (2008) 515-520.
  5. C.Bai, J. Fang, Existence of multiple positive solutions for nonlinear m-point boundary value problems, J.Math.Appl.Appl. 81 (2003) 76-85.
  6. R.Ma, Positive solutions of nonlinear m-point boundary value problems, Comput. Math. Appl.42 (2001) 755-765. https://doi.org/10.1016/S0898-1221(01)00195-X
  7. Y.Sang, H.Su, Positive solutions of nonlinear third-order m-point BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity, J.Comput.Appl.Math. 225 (2009) 288-300. https://doi.org/10.1016/j.cam.2008.07.039
  8. D.Guo, V.Lakshmikanthan, Nonlinear problems in Abstract Cones, Academic Press, San Diego, 1988.