Solvability and dependence on a parameter of a fourth-order periodic boundary value problem

In this paper, we investigate the solvability and dependence on a parameter for a fourth-order periodic boundary value problem of the form $$\begin{aligned} \left\{ \begin{array}{ll} u^{(4)}(t)-\rho ^4u(t)+\lambda f(t,u(t))=0,\quad 0\le t\le 2\pi , \\ u^{(i)}(0)=u^{(i)}(2\pi ),\quad i=0,1,2,3, \end{array} \right. \end{aligned}$$ where \(\rho \in (0,\frac{1}{2})\) is a constant, \(\lambda \) is a positive parameter and \(f\in C([0,2\pi ]\times \mathbb {R})\). By using a fixed point theorem of cone expansion/compression type, the existence and multiplicity of positive solutions for the above problem are obtained. The uniqueness of positive solutions and the dependence of solutions on the parameter are also obtained. Meanwhile, an iteration scheme which converges uniformly to the unique positive solution and the rate of convergence are also obtained. An example is given to illustrate our results.