Eigenvalue Problems for Nonlinear Differential Equations on a Measure Chain

Values of λ are determined for which there exist positive solutions of the second order differential equation on a measure chain, uΔΔ(t)+λa(t)f(u(σ(t)))=0,t∈[0,1], satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=uΔ(σ(1))=0, where a and f are positive valued, and both limx→0+f(x)x and limx→∞f(x)x exist.