We consider the classical nonlinear fourth-order two-point boundary value problem $$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=\lambda h\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right)),0<t<1,\hfill \\ u\left(0\right)=u^{\text{'}}\left(1\right)=u^{\text{'}\text{'}}\left(0\right)=u^{\text{'}\text{'}\text{'}}\left(1\right)=0.\hfill \end{array}\right.$$ In this problem, the nonlinear term $h\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right))$ contains the first and second derivatives of the unknown function, and the function $h\left(t\right)f(t,x,y,z)$ may be singular at $t=0$, $t=1$ and at $x=0$, $y=0$, $z=0$. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.

We consider the existence of positive solutions of the equation $1/\lambda \left(t\right){\left(\lambda \left(t\right){\phi }_p\left(x^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right))=0$, where ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s$, p > 1, subject to some singular Sturm-Liouville boundary conditions. Using the Krasnosel’skiĭ fixed point theorem for operators on cones, we prove the existence of positive solutions under some structure conditions.

Values of $\lambda$ are determined for which there exist positive solutions of the system of three-point boundary value problems, $u^{\text{'}\text{'}}+\lambda a\left(t\right)f\left(v\right)=0$, $v^{\text{'}\text{'}}+\lambda b\left(t\right)g\left(u\right)=0$, for $0<t<1$, and satisfying, $u\left(0\right)=\beta u\left(\eta \right)$, $u\left(1\right)=\alpha u\left(\eta \right)$, $v\left(0\right)=\beta v\left(\eta \right)$, $v\left(1\right)=\alpha v\left(\eta \right)$. A Guo-Krasnosel’skii fixed point theorem is applied.