Abstract

By applying the Krasnoselskii fixed point theorem in cones and the fixed point index theory, we study the existence of positive solutions of the non linear third-order three point boundary value problem $u'''(t)+a(t)f(t,u(t))=0$, $t\in(0,1)$; $u'(0)=u'(1)=\alpha u(\eta)$, $u(0)=\beta u(\eta)$, where $\alpha$, $\beta$ and $\eta$ are constants with $\alpha\in[0,\frac{1}{\eta})$, and $0<\eta<1$. The results obtained here generalize the work of Torres [Positive solution for a third-order three point boundary value problem, Electronic J. Diff. Equ. 2013 (2013), 147, 1--11].

Keywords: Third-order differential equations; three point boundary value problem; Krasnoselski fixed point in a cone; fixed point index theory

MSC: 34B10

Pages:  12$-$25     

Volume  68 ,  Issue  1 ,  2016