Existence of solutions for nonlinear BVPs of second-order differential equations on measure chains. (Q1600510)

The authors consider the following boundary value problem on a measure chain \(T\) \[ u^{\Delta\Delta}+\mu f(t,u^\sigma (t))=0, \quad t\in [a,b]\cap T, \] \[ \alpha u(a)-\beta u^\Delta (a)=0,\qquad \gamma u(\sigma(b))+\delta u^\Delta (\sigma(b))=0. \] Sufficient conditions on \(f\) are offered to ensure that the boundary value problem has at least a solution. Criteria for the existence of multiple solutions are then extended from the main result. The treatment unifies the continuous and discrete cases which are available in the literature.