摘要：In this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions $$\begin{aligned}& \left \{ \textstyle\begin{array}{l}u^{(4)}(t)=h(t)f(t,u(t),u'(t),u''(t)),\quad t\in(0,1),\\ u(0)=u(1)=\beta_)[u],\qquad u''(0)+\beta_,[u]=0,\qquad u''(1)+\beta_"[u]=0, \end{array}\displaystyle \right . \end{aligned}$$ where $f: [0,1]\times\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{-}\to \mathbb{R}_{+}$ is continuous, $h\in L^)(0,1)$ and $\beta_{i}[u]$ is Stieltjes integral ($i=1,2,3$). This equation describes the deflection of an elastic beam. Some inequality conditions on nonlinearity f are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in $C^,[0,1]$. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.
关键词：Positive solution ; Fixed point index ; Cone ;
