Global structure of nodal solutions for a fourth-order two-point boundary value problem

Abstract

We consider the fourth-order two-point boundary value problem x⁗+kx″+lx=a(t)f(x),0<t<1, x(0)=x(1)=x′(0)=x′(1)=0, where a∈C([0,1],[0,∞)) with a(t)≢0 on any subinterval of [0,1],f∈C1((R⧹{0}),R)∩C(R,R) satisfies f(x)x>0 for all x≠0. We give conditions on the constants k, l and the function f(x) that guarantee the existence and multiplicity results of nodal solutions. The proofs of our main results are based upon disconjugate operator theory and the global bifurcation techniques.