Abstract:

This paper discusses the existence of the positive solution of the fourth-order boundary value problem$\left\{\begin{array}{ll} u^{(4)}(x)=f(x,u(x),u''(x)),\quad x\in [0,\,1],\\ u'(0)=u'''(0)=u(1)=u''(1)=0,\end{array}\right.$which models the deformations of a statically elastic beam, where $ \,f:[0,\,1]\times\mathbb{R}^{+}\times\mathbb{R}^{-}\to\mathbb{R}^{+} $ is continuous. Under that the nonlinearity $ f(x,\,u,\,v) $ satisfies some inequality conditions, the existence results of positive solutions of this problem are obtained by applying the fixed point index theory in cones.

Key words: Fourth-order boundary value problem, Positive solution, Cone, Fixed point index