Positive Solutions for Fourth-Order Equations with a Sign-Changing Weight and Clamped Beam Boundary Conditions

In this paper, we are concerned with the existence of positive solutions for nonlinear fourth-order equation with clamped beam boundary conditions $$\begin{aligned} \left\{ \begin{array}{ll} &{}u''''(x)=\lambda a(x)f(u(x))\ \ \ \text {for}\ \ x\in (0,1),\\ &{}u(0)=u(1)=u'(0)=u'(1)=0, \end{array} \right. \end{aligned}$$ where $$\lambda >0$$ is a parameter, $$a\in L^{1}(0,1)$$ may change sign and $$f\in C([0,\infty ),[0,\infty ))$$ . We establish some new results of existence of positive solutions to this problem if the nonlinearity f is monotone on $$[0,\infty )$$ . The proofs of our main results are based upon a monotone iteration technique and the Schauder’s fixed point theorem. Finally, an example is presented to illustrate the application of our main results.