Abstract
We consider, for q > 1 q>1 , the one-dimensional Kirchhoff-type problem − A ( ∫ 0 1 ( u ′ ( s ) ) q d s ) u ( t ) a m p ; = λ f ( u ( t ) ) , t ∈ ( 0 , 1 ) u ( 0 ) a m p ; = 0 u ′ ( 1 ) a m p ; = 0 \begin{equation} \begin {split} -A\left (\int _0^1\big (u’(s)\big )^q\ ds\right )u(t)&=\lambda f\big (u(t)\big )\text {, }t\in (0,1)\\ u(0)&=0\\ u’(1)&=0\notag \end{split} \end{equation} and demonstrate the existence of at least one positive solution to this problem. The main contribution is to show that by using a nonstandard order cone together with topological fixed point theory much weaker conditions than usual can be imposed on the coefficient function A A .
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