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The method of lower and upper solutions for third order singular four point boundary value problems

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Abstract

We mainly study the existence of positive solutions for the following third order singular four point boundary value problem

$$\begin{cases}x^{(3)}(t)+f(t,x,x',-x'')=0,\quad 0<t<1,\\x(0)-\alpha x(\xi)=0,\quad x'(1)-\beta x'(\eta)=0,\quad x''(0)=0.\end{cases}$$

where 0≤α<1, 0≤β<1, 0<ξ<1,0<η<1. And we obtain some necessary and sufficient conditions for the existence of C 2[0,1] positive solutions by means of the lower and upper solution method. Our nonlinearity f(t,x,y,z) may be singular at x,y,z,t=0 and/or t=1.

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Authors and Affiliations

  1. School of Mathematics and System Sciences, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

    Jian Zhang & Zhongli Wei

  2. Department of Mathematics, Shandong Jianzhu University, Jinan, Shandong, 250101, People’s Republic of China

    Zhongli Wei

  3. Hebei University of Engineering, Handan, Hebei, 056021, China

    Wei Dong

Authors
  1. Jian Zhang
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  2. Zhongli Wei
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  3. Wei Dong
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Correspondence to Zhongli Wei.

Additional information

Research supported by the NNSF-China (10971046) and the NSF of Shandong Province (ZR2009AM004).

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Zhang, J., Wei, Z. & Dong, W. The method of lower and upper solutions for third order singular four point boundary value problems. J. Appl. Math. Comput. 36, 275–289 (2011). https://doi.org/10.1007/s12190-010-0403-x

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  • DOI: https://doi.org/10.1007/s12190-010-0403-x

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