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Zbl 1223.34122
Dogan, Abdulkadir; Graef, John R.; Kong, Lingju
Higher-order singular multi-point boundary-value problems on time scales. (English)
[J] Proc. Edinb. Math. Soc., II. Ser. 54, No. 2, 345-361 (2011). ISSN 0013-0915; ISSN 1464-3839/e

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.
[Minghe Pei (Jilin)]

MSC 2000:: *34N05
34B15 Nonlinear boundary value problems of ODE
34B16 Singular nonlinear boundary value problems
34B10 Multipoint boundary value problems
34B09 Boundary value problems with an indefinite weight
34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations

Keywords: positive solutions; singular boundary-value problems; existence; uniqueness; dependence; mixed monotone operator; time scales

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