Existence and uniqueness of positive solutions for Kirchhoff type beam equations

This paper is concerned with the existence and uniqueness of positive solution for the fourth order Kirchhoff type problem $$\{\begin{array}{l}{u}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(x)-(a+b{\int }_0^1(u^{\mathrm{\prime }}(x){)}^{2}dx)u^{\mathrm{\prime }\mathrm{\prime }}(x)=\lambda f(u(x)),x\in (0,1),\\ u(0)=u(1)=u^{\mathrm{\prime }\mathrm{\prime }}(0)=u^{\mathrm{\prime }\mathrm{\prime }}(1)=0,\end{array}$$ where $a>0,b\ge 0$ are constants, $\lambda \in \mathbb{R}$ is a parameter. For the case $f(u)\equiv u$, we use an argument based on the linear eigenvalue problems of fourth order equations and their properties to show that there exists a unique positive solution for all $\lambda >{\lambda }_{1,a}$, here ${\lambda }_{1,a}$ is the first eigenvalue of the above problem with $b=0$; For the case $f$ is sublinear, we prove that there exists a unique positive solution for all $\lambda >0$ and no positive solution for $\lambda <0$ by using bifurcation method.