Advances in Difference Equations (Apr 2017)

Reversed S-shaped connected component for a fourth-order boundary value problem

  • Jinxiang Wang,
  • Ruyun Ma

DOI
https://doi.org/10.1186/s13662-017-1167-5
Journal volume & issue
Vol. 2017, no. 1
pp. 1 – 11

Abstract

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Abstract In this paper, we investigate the existence of a reversed S-shaped component in the positive solutions set of the fourth-order boundary value problem { u ′′′′ ( x ) = λ h ( x ) f ( u ( x ) ) , x ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , $$\textstyle\begin{cases} u''''(x)=\lambda h(x)f(u(x)),\quad x\in(0,1),\\ u(0)=u(1)=u''(0)=u''(1)=0, \end{cases} $$ where λ > 0 $\lambda>0$ is a parameter, h ∈ C [ 0 , 1 ] $h\in C[0,1]$ and f ∈ C [ 0 , ∞ ) $f\in C[0,\infty )$ , f ( 0 ) = 0 $f(0)=0$ , f ( s ) > 0 $f(s)>0$ for all s > 0 $s>0$ . By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to parameter λ, and especially, we obtain the existence of three distinct positive solutions for λ being in a certain interval.

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