Unilateral Global Bifurcation for Fourth-Order Problems and Its Applications

We will establish unilateral global bifurcation result for a class of fourth-order problems. Under some natural hypotheses on perturbation function, we show that is a bifurcation point of the above problems and there are two distinct unbounded continua, and , consisting of the bifurcation branch from , where is the th eigenvalue of the linear problem corresponding to the above problems. As the applications of the above result, we study the existence of nodal solutions for the following problems: , , where is a parameter and are given constants; with on any subinterval of ; and is continuous with for We give the intervals for the parameter which ensure the existence of nodal solutions for the above fourth-order Dirichlet problems if or where and We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.