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Positive solutions to singular multi-point dynamic eigenvalue problems with mixed derivatives. (English) Zbl 1168.34014
Summary: This paper considers a singular m-point dynamic eigenvalue problem on time scales $\Bbb T$: $$-(p(t)u^\Delta(t))^\nabla= \lambda f(t,u(t)), \quad t\in(0,1]\cap\Bbb T,$$ $$u(0)= \sum_{i=1}^{m-2} a_iu(\xi_i), \quad \gamma u(1)+\delta p(1)u^\Delta(1)= \sum_{i=1}^{m-2} b_ip(\xi_i)u^\Delta(\xi_i).$$ 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.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
39A10Additive difference equations
Full Text: DOI
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