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Existence of positive solution to singular systems of second-order four-point BVPs

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Abstract

In this paper, we consider the following system of nonlinear second-order four-point boundary value problems

$$\left\{\begin{array}{l}-u^{\prime \prime }(t)=f(t,v(t)),\quad t\in (0,1),\\-v^{\prime \prime }(t)=g(t,u(t)),\quad t\in (0,1),\\u(0)=\alpha u(\xi ),\qquad u(1)=\beta u(\eta ),\\v(0)=\alpha v(\xi ),\qquad v(1)=\beta v(\eta ),\end{array}\right.$$

where 0<ξ<η<1, \(0\leq \alpha <\frac{1}{1-\xi },\) \(0\leq \beta <\frac{1}{\eta}\) , α ξ(1−β)+(1−α)(1−β η)>0; f(t,v) and g(t,u) may be singular at t=0 and/or t=1. Under suitable conditions, we show the existence of at least one positive solution by applying the functional expansion-compression fixed point theorem, which is due to Zhang and Sun.

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Authors and Affiliations

  1. Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People’s Republic of China

    Hai-E Zhang & Jian-Ping Sun

Authors
  1. Hai-E Zhang
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  2. Jian-Ping Sun
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Correspondence to Jian-Ping Sun.

Additional information

Supported by the NSF of Gansu Province of China (2007GS05333).

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Zhang, HE., Sun, JP. Existence of positive solution to singular systems of second-order four-point BVPs. J. Appl. Math. Comput. 29, 325–339 (2009). https://doi.org/10.1007/s12190-008-0133-5

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  • DOI: https://doi.org/10.1007/s12190-008-0133-5

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