Eigenvalue problems for second-order nonlinear dynamic equations on time scales (Q2488777)

The authors are concerned with the second-order nonlinear dynamic equation on time scales \[ u^{\Delta\Delta}(t)+\lambda a(t)f(u(\sigma(t)))=0, t\in [0,1], \] satisfying either the conjugate boundary conditions \(u(0)=u(\sigma(1))=0\) or the right focal boundary conditions \(u(0)=u^\Delta(\sigma(1))=0\), where \(a\) and \(f\) are positive. The number of positive solutions of the above boundary value problem for \(\lambda\) belonging to the half-line \((0,\infty)\) and the dependence of positive solutions of the problem on the parameter \(\lambda\) are discussed. It is proved that there exists a \(\lambda^*>0\) such that the problem has at least two, one and no positive solution(s) for \(0<\lambda<\lambda^*, \lambda=\lambda^*\) and \(\lambda>\lambda^*\), respectively. The main tool is a fixed-point index theorem on cones due to Guo-Lakshmikantham. Furthermore, by using the semi-order method on cones of Banach space, an existence and uniqueness criterion for a positive solution of the problem is established. In particular, such a positive solution \(u_\lambda(t)\) of the problem depends continuously on the parameter \(\lambda\), i.e., \(u_\lambda(t)\) is nondecreasing in \(\lambda\), \(\lim_{\lambda\rightarrow 0^+}\| u_\lambda\| =0\) and \(\lim_{\lambda\rightarrow +\infty}\| u_\lambda\| =+\infty\).