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Published in:
01-10-2014 | Original Research
Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems
Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2014
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Abstract
We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem 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.
$$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0<t<1,\\ x(0)-\sum_{i=1}^{m_1}\alpha_i x(\xi_i)=0,\qquad x'(0)-\sum_{i=1}^{m_2}\beta_i x'(\eta_i)=0,\qquad x'(1)=0. \end{array} \right . $$