Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

Abstract

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem

$$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0<t<1,\\ x(0)-\sum_{i=1}^{m_1}\alpha_i x(\xi_i)=0,\qquad x'(0)-\sum_{i=1}^{m_2}\beta_i x'(\eta_i)=0,\qquad x'(1)=0. \end{array} \right . $$

where $0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1$, i=1,2,…,m ₁, $0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1$, $0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1$, j=1,2,…,m ₂, $0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1$. And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.