Global Bifurcation for Fourth-Order Differential Equations with Periodic Boundary-Value Conditions

We establish the global structure of positive solutions of fourth-order periodic boundary-value problems $u''''(t)+Mu(t)=\lambda f(t,u(t))$, $t\in[0,T]$, $u^{k}(0)=u^{(k)}(T)$, $k=0,1,2,3,$ with $M\in\big(0,4({2\pi M_4}/{T})^4\big)$ and $u^{(4)}(t)-Mu(t)+\lambda g(t,u(t))=0$, $t\in[0,T]$, $u^{k}(0)=u^{(k)}(T)$, $k=0,1,2,3,$ with $M\in \big(0,({2\pi M_4}/{T})^4\big)$; here $g, f\in C([0,T]\times[0,\infty),[0,\infty))$, $M$ is constant, and $\lambda>0$ is a real parameter. The main results are based on a global bifurcation theorem.