Existence of Solutions for a Fourth-Order Periodic Boundary Value Problem near Resonance

Abstract

We show the existence and multiplicity of solutions for the fourth-order periodic boundary value problem

$$\begin{cases} u''''(t)-\lambda u(t)=f(t,u(t))-h(t), \qquad t\in [0,1],\\ u(0)=u(1),\;u'(0)=u'(1),\; u''(0)=u''(1),\;u'''(0)=u'''(1), \end{cases}$$

where \(\lambda\in\mathbb{R}\) is a parameter, \(h\in L^1(0,1)\), and \(f:[0,1]\times \mathbb{R}\rightarrow\mathbb{R}\) is an \(L^1\)-Carathéodory function. Moreover, \(f\) is sublinear at \(+\infty\) and nondecreasing with respect to the second variable. We obtain that if \(\lambda\) is sufficiently close to \(0\) from the left or right, then the problem has at least one or two solutions, respectively. The proof of main results is based on bifurcation theory and the method of lower and upper solutions.