Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems

In this work, we consider the nonlinear eigenvalue problems u″+ra(t)f(u)=0,00f(s)>0 for s∈(0,s1)∪(s1,+∞)s∈(0,s1)∪(s1,+∞), f(s)<0f(s)<0 for s∈(−∞,s2)∪(s2,0)s∈(−∞,s2)∪(s2,0), the limits f0=lim|s|→0f(s)s, f∞=lim|s|→∞f(s)s exist. Using global bifurcation techniques, we study the global behavior of the components of nodal solutions of the above asymptotically linear eigenvalue problems.