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Higher-order singular multi-point boundary-value problems on time scales. (English) Zbl 1223.34122

The authors study the higher-order singular eigenvalue equation on time scales

(φ(u Δ n-1 )) +λa(t)f(u)=0,t(0,T) 𝕋

which satisfies some two multi-point boundary conditions, for instance,

u Δ i (0)= j=1 m α j u Δ i (ξ j ),i=0,,n-2,φ(u Δ n-1 (T))= j=1 m β j φ(u Δ n-1 (ξ j )),

where φ is a homeomorphism. The existence and uniqueness of positive solutions are obtained. The tool is the mixed monotone operator theory.

MSC:
34N05Dynamic equations on time scales or measure chains
34B15Nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
34B09Boundary eigenvalue problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations