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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 \align &(\phi(u^{\Delta^{n-1}}))^{\nabla}+\lambda a(t)f(u)=0,\quad t\in (0,T)_{\mathbb{T}}\endalign which satisfies some two multi-point boundary conditions, for instance, \align &u^{{\Delta}^i} (0)=\sum_{j=1}^{m}\alpha_j u^{{\Delta}^i}(\xi_j),\quad i=0,\dots,n-2,\\ &\phi(u^{\Delta^{n-1}}(T))=\sum_{j=1}^{m}\beta_j \phi(u^{\Delta^{n-1}}(\xi_j)),\endalign where $\phi$ is a homeomorphism. The existence and uniqueness of positive solutions are obtained. The tool is the mixed monotone operator theory.

##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34B15 Nonlinear boundary value problems for ODE 34B16 Singular nonlinear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE 34B09 Boundary eigenvalue problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
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