Positive solutions to singular multi-point dynamic eigenvalue problems with mixed derivatives

Abstract

This paper considers a singular m -point dynamic eigenvalue problem on time scales T : − ( p ( t ) u Δ ( t ) ) ∇ = λ f ( t , u ( t ) ) , t ∈ ( 0 , 1 ] ∩ T , u ( 0 ) = ∑ i = 1 m − 2 a i u ( ξ i ) , γ u ( 1 ) + δ p ( 1 ) u Δ ( 1 ) = ∑ i = 1 m − 2 b i p ( ξ i ) u Δ ( ξ i ) . We allow f ( t , w ) to be singular at w = 0 and t = 0 . By constructing the Green’s function and studying its positivity, eigenvalue intervals in which there exist positive solutions of the above problem are obtained by making use of the fixed point index theory.